Ricci-flat deformations of orbifolds and asymptotically locally Euclidean manifolds Christian Overgaard Lund St John’s College Department of Pure Mathematics and Mathematical Statistics University of Cambridge This dissertation is submitted for the degree of Doctor of Philosophy September 2018 Declaration This thesis is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the acknowledgements and specified in the text is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the acknowledgements and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University of similar institution except as declared in the acknowledgements and specified in the text. Christian Overgaard Lund September 2018 Abstract In this thesis we study Ricci-flat deformations of Ricci-flat Kähler metrics on compact orbifolds and asymptotically locally Euclidean(ALE) manifolds. In both cases we also study the moduli space of Ricci-flat structures. For this purpose, it is convenient to assume that the initial Ricci-flat metrics are Kähler. Our work extends results by Koiso about Einstein-deformations of Kähler-Einstein metrics on compact manifolds. Orbifolds differ from manifolds by being locally modelled on a quotient of Euclidean space by the action of a finite group Γ. We adapt a slice construction by Ebin and the Cal- abi conjecture to orbifolds and show that for compact complex orbifolds with vanishing orbifold first Chern class and all infinitesimal complex deformations integrable, Ricci-flat deformations of Ricci-flat Kähler metric are Kähler, possibly with respect to a perturbed complex structure. We also show that the moduli space of Ricci-flat structures is, up to the action of a finite group, a finite dimensional manifold and we express its dimension in terms of the dimension of certain Dolbeault and sheaf cohomology groups. The strategy is to lift the problem locally to a Γ-invariant problem on a manifold. ALE manifolds are non-compact manifolds with one end, for which the metric at in- finity approximates a flat metric. We study ALE Ricci-flat Kähler manifolds that arise as the complement of a divisor D in a compact Kähler manifold X¯ and use the deforma- tion theory by Kawamata for the pair (X¯,D). By working with suitably chosen weighted Sobolev and Hölder spaces we recover the relevant elliptic theory for the linearisation of the Ricci operator and the linearisation of the complex Monge-Ampère equation. We prove that integrability of infinitesimal deformations of the pair (X¯,D) implies that ALE Ricci-flat deformations of ALE Ricci-flat Kähler metrics are Kähler, possibly with respect to a perturbed complex structure. We also show that the moduli space of ALE Ricci-flat structures is, up to the action of a finite group, a finite dimensional manifold and we ex- press its dimension in terms of the dimension of certain Dolbeault and sheaf cohomology groups. Acknowledgements I would like to thank my supervisor Dr Alexei Kovalev for suggesting the two research questions dealing with orbifolds and ALE manifolds that I have answered in this thesis and for our many meetings and discussions during the course of my PhD. I would like to thank the EPSRC, Department of Pure Mathematics and Mathematical Statistics and the Cambridge Trust for funding me. I would also like to thank St John’s College for economic support at the end of my degree. I would like to thank my teacher Benny Børgesen for opening my eyes to the fascinat- ing world of mathematics. I would also like to thank Professor Ian Kiming and Professor Henrik Schlichtkrull for their brilliant supervision for my Bachelor’s thesis and Master’s thesis at the University of Copenhagen. I would like to thank Nils Prigge and Dr Alan Thompson for many interesting and useful conversations about mathematics and for reading a late draft of my thesis. I would like to thank Anna Saroldi for her love and for always being there for me. I would also like to thank Dr Claudius Zibrowius, Dr Nina Friedrich, Nils Prigge, Eric Ernst Faber, Julius Bier Kirkegaard, Lera Schumaylova, Katarzyna Wyczesany, Brunella Tor- ricelli, Marius Leonhardt, Georgios Charalambous and Tom Brown for many interesting lunch time conversations and for providing a good social life around the department. Last but not least I would like to thank my family for all the support they have given me over the years. I would especially like to thank my father, who unfortunately never got to see me finish, and my mother for her eternal patience and support. Contents 1 Introduction 1 2 Preliminaries 5 2.1 Tools from analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Space of metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Laplace operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Einstein metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Kähler metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Deformation theory of complex structures . . . . . . . . . . . . . . . . . 22 2.7 Koiso’s deformation theory on compact manifolds . . . . . . . . . . . . . 23 3 Orbifold Ricci-flat deformations 29 3.1 Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Tools from analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Complex orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Calabi conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.6 Slice construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.7 Ricci-flat deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.8 Moduli space of Ricci-flat structures . . . . . . . . . . . . . . . . . . . . 67 4 Examples: Orbifold K3 surfaces 70 I Contents 5 ALE Ricci-flat deformations 75 5.1 Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Tools from analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3 ALE differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.4 Slice construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.5 ALE Ricci-flat deformations . . . . . . . . . . . . . . . . . . . . . . . . 93 5.6 Deformations of a pair (X¯,D) . . . . . . . . . . . . . . . . . . . . . . . 100 5.7 Stability results for deformations of a pair (X¯,D) . . . . . . . . . . . . . 103 5.8 Proof of Theorem 5.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.9 Moduli space of ALE Ricci-flat structures . . . . . . . . . . . . . . . . . 108 5.10 ALE Ricci-flat deformations revisited . . . . . . . . . . . . . . . . . . . 110 5.11 Asymptotically conical manifolds . . . . . . . . . . . . . . . . . . . . . 111 6 Examples: ALE manifolds 115 Concluding remarks 119 Bibliography 121 Notation Index 126 II Chapter 1 Introduction The starting point for this thesis is the basic observation that the set of Riemannian metrics M on a smooth manifold M is vast and that there a priori is no preferred choice of metric. The set of Riemannian structures on M is the quotient M˜ = M/D, where D denotes the group of diffeomorphism of M . In [BES87, Chapter 0] Besse asked, for a compact smooth manifold M , Are there any best (or nicest, or distinguished) Riemannian structures on M? For manifolds of dimension two the uniformization theorem provides an answer. Every connected two-dimensional manifold admits a complete metric with constant Gaussian curvature. In higher dimensions the concept of curvature is more complicated, and so is any potential answer involving a curvature condition. A single ’distinguished’ Riemannian structure seems out of reach on higher dimensional manifolds in general, but one could hope that a suitable curvature condition could provide a collection of ’best’ Riemannian structures which is significantly smaller than M˜. The three main types of curvature are sectional curvature, Ricci curvature and scalar curvature. All manifolds admit constant scalar curvature metrics, but they come in such abundances that the space of constant scalar curvature metrics is infinite dimensional in general([BES87, section 4.F]). A con- stant scalar curvature condition therefore seems too weak. On the other had, assuming constant sectional curvature is too restrictive, as many manifolds do not admit any con- stant sectional curvature metrics. This leaves us with a condition on the Ricci-curvature. The Einstein condition is a condition on the Ricci curvature. If we normalise volume, Einstein metrics are critical points of the total scalar curvature functional, which at least 1 Chapter 1. Introduction partially merits a status as a distinguished metric. Also, according to [BES87, remark 0.21] no compact manifolds of dimension ≥ 5 are known not to admit an Einstein metric. We take a closer look at the set of Einstein structures. Denote by M1 the metrics with volume 1. For a connected compact Riemannian manifold (M, g) Ebin has shown that there almost exists a slice S ⊂ M1 for the action of D, i.e. the space M˜1 is locally homeomorphic to S/Iso(M, g), where Iso(M, g) denotes the group of isometries. If we denote by R the subspace of Einstein metrics inM1, then the quotient space R˜ = R/D is called the moduli space of Einstein structures. Koiso carried out a detailed study of this moduli space in [KOI83]. He showed that for a slice through an Einstein metric g there is a finite dimensional manifold inside the slice containing the nearby Einstein metrics of the slice as a real analytic subset. If we furthermore assume that the compact Einstein manifold (M, g) is Kähler, has non-positive first Chern class and that all infinitesimal complex deformations are integrable, then Koiso showed that all infinitesimal Einstein deformations will in fact integrate into curves of Einstein metrics. This implies that all Einstein deformations of a Kähler-Einstein metric are Kähler, possibly with respect to a perturbed complex structure. When the first Chern class is non-positive then the isometry group Iso(M, g) acts as a finite group on the space of Einstein metrics in the slice S. It follows that the local model for the moduli space of Einstein structures is, up to an action of a finite group, a finite dimensional manifold. Koiso also found an expression for its dimension in terms of cohomology groups. Einstein structures therefore seem like a decent candidate for a ’distinguished’ Riemannian structure on a compact manifold. Despite the innocently looking condition Ric(g) = λg, Einstein metrics are notori- ously hard to construct, [BES87, 0.23]. On Kähler manifolds we have a powerful tool though. The Calabi conjecture [CAL54] constructs Ricci-flat Kähler metrics on compact Kähler manifolds with vanishing first Chern class. The combination of the Calabi conjec- ture and the results by Koiso give us a decent understanding of Einstein structures which is part of the reason for promoting them as a candidate for a distinguished class of struc- tures. The focus of this thesis is to explore if this understanding holds on more general objects than compact manifolds. A lot of work has already been done on proving the Cal- abi conjecture on more general spaces, so our main objective is to generalize the results by Koiso. The sign of the first Chern class plays a role in the understanding of Einstein metrics in general and in particular also for Koiso’s results. When the sign is negative or zero the problem can be solved with the above assumptions. When the sign is positive one 2 needs the extra assumption that no holomorphic vector fields exist. According to [BES87, 12.101] no such manifolds are known. In this thesis we explore two ways of relaxing the hypothesis of Koiso’s results. In Chapter 3 we show that similar results can be obtained when the compact manifold is replaced by a compact orbifold. In Chapter 5 we show that a version of Koiso’s results can be proved for a class of non-compact manifolds known as asymptotically locally Euclidean manifolds. An orbifold generalizes the concept of a manifold by admitting a more complicated local model. An orbifold is locally homeomorphic to a quotient of Euclidean space by a finite group. The action might have isotropy, which lead to the presence of quotient singularities. At such singular points the structure of the orbifold differs from that of a manifold. Orbifolds can be considered natural in the sense that they arise as the result of some natural operations in differential geometry. For instance, Koiso’s results show that the moduli space of Einstein structures on a compact manifold is an instance of an orbifold. It seems natural to ask if the understanding of Einstein metrics we have on manifolds also holds for orbifolds. In this thesis we provide an affirmative answer to this question. We show that for a compact complex orbifold with vanishing first Chern class, integrability of infinitesimal complex deformations implies that Ricci-flat deformations of Ricci-flat Kähler metrics are Kähler, possibly with respect to a perturbed complex structure. We also show that the moduli space of Ricci-flat structures in a neighbourhood of a Ricci-flat Kähler structure is, up to the action of a finite group, a finite dimensional manifold and we find an expression for the dimension of it. For completeness we also provide a proof of the Calabi conjecture on orbifolds. Locally an orbifold is a quotient of a manifold by a finite group Γ. Our strategy is locally to lift the problem on an orbifold to a Γ-invariant problem on a manifold. An asymptotically locally Euclidean (ALE) manifold is a non-compact manifold to- gether with a metric which at infinity approximates a flat metric. Proving a version of Koiso’s results for non-compact manifolds provide a number of challenges. For instance, it is used that elliptic operators are Fredholm, which need not be true on non-compact manifolds. For non-compact manifolds there is also no Kodaira-Spencer stability result saying that a smooth family of complex deformations through a complex structure with a Kähler metrics admits a smooth family of compatible Kähler metrics. To remedy the prob- lems related to the elliptic operators we work with appropriately chosen weighted Hölder 3 Chapter 1. Introduction and Sobolev spaces of sections. To tackle the deformation theory of complex structures we consider ALE manifolds that arise as the complement of a smooth divisor in a com- pact Kähler manifold and use the deformation theory by Kawamata for such pairs. We consider an ALE Ricci-flat Kähler manifold X = X¯\D where X¯ is a compact Kähler manifold and D is a smooth divisor satisfying KX¯ = −βLD for some β ≥ 1. We show that if all infinitesimal deformations of the pair (X¯,D) are integrable, then ALE Ricci- flat deformations of g are Kähler, possibly with respect to a perturbed complex structure. We also show that the moduli space of ALE Ricci-flat structures in a neighbourhood of a Ricci-flat Kähler structure is, up to the action of a finite group, a finite dimensional man- ifold. Joyce proved in [JOY00, Chapter 8] that the Calabi conjecture also holds for ALE metrics. The basic understanding of Einstein metrics regarding the existence of Einstein metrics on Kähler manifolds, their deformation theory and the finiteness of the dimension of their moduli space we get from the combination of the Calabi conjecture and the results by Koiso therefore carry over to orbifolds and ALE manifolds. It should be mentioned that our inspiration for studying Ricci-flat Kähler metrics on compact orbifolds and ALE manifolds is not the only one. Another reason for studying Ricci-flat Kähler metrics on compact orbifolds and ALE manifolds is the role they play in the construction of compact manifolds with holonomy G2 and Spin(7). This will not be treated in this thesis, but it is a vast field of active research. See for instance [JOY00, Chapter 11-15] and [KOV-NOR10]. This thesis is divided into five chapters. In Chapter 2 we present some background material. In Chapter 3 we generalize Koiso’s results to compact orbifolds. In Chapter 4 we apply the orbifold version of Koiso’s results to compute the dimension of the moduli space of Ricci-flat structures for some orbifold K3 surfaces. In Chapter 5 we prove a version of Koiso’s results for ALE manifolds and in Chapter 6 we discuss examples of ALE manifolds. 4 Chapter 2 Preliminaries In this chapter we give some background material and fix basic notation. The material is standard and is mostly taken from [LAN62], [BES87], [GRI-HAR94], [JOY00] and [HUY05]. All metrics in this thesis will be Riemannian and all finite dimensional manifolds will be of strictly positive dimension. We denote by (M, g) a smooth manifold with a metric g. If the manifold (M, g) is also complex, we denote it by (M,J) or (M,J, g). If it is furthermore Kähler, we use the symbol ω for the Kähler form of g with respect to J and we sometimes denote the manifold by (M,J, g, ω). We will usually use n for the real dimension of M and m for the complex dimension. Sometimes we write Mn to indicate that M is a smooth manifold of real dimension n. For a Riemannian manifold (M, g) we denote by dVg = dvolg the volume form of g and by vol(g) the total volume∫ M dVg. We denote the tensor bundle on M by T (r,s)M , where r is the covariant index and s the contravariant index, i.e. T (0,1)M = TM and T (1,0)M = T ∗M . Elements of T (r,s)M are called (r, s)-tensors or tensors of type (r, s). Covariant indices are raised and contravariant indices are lowered. In local coordinates {x1, . . . , xn} we write elements of TM as X = ∑ X i ∂ ∂xi and elements of T ∗M as α = ∑ αidx i. Locally a metric g is given by gijdxi⊗dxj . The component of the inverse matrix (gij)−1 we denote gij . We make use of the Einstein summation convention αiX i = ∑n i=1 αiX i. Smooth sections of a tensor bundle T (r,s)M is denoted by C∞(T (r,s)M). The musical isomorphisms, [ : TM → T ∗M and ] : T ∗M → TM are given by X[(Y ) = g(X, Y ) which in local coordinates reads X[ = gijX idxj . The contravariant tensor α] is given by α](ω) = ω(α]), which in local coordinates reads α] = gijαj ∂∂xi . 5 Chapter 2. Preliminaries The trace of a (1, 1)-tensors ω is tr( ∑n i,j=1 ω i jdx i ⊗ ∂ ∂xj ) = ∑n i,j=1 ω i j . For a symmetric (2, 0)-tensor h the trace with respect to the metric g is trgh = trh] = gijhij . For complex tensors we denote by trC the complex linear trace. Denote by Ωk(M) the space of differential k-forms on M and by d : Ωk(M) → Ωk+1(M) the exterior derivative. The differential, push-forward or linearization of a smooth function f : M → N between smooth manifolds is the map dfp : TpM → Tf(p)N , where the map dfp in local coordinates is the Jacobi map dfp = ( ∂fi∂xj )i,j=1,...,n. For α ∈ Ω1(N) the pull-back f ∗α ∈ Ω1(M) is f ∗αp(X) = αf(p)(dfp(X)). If f is a diffeomor- phism and α ∈ Ω1(M) the push-forward f∗α ∈ Ω1(N) is f∗αp(X) = αf−1(p)((df−1p (X)). Using the metric g we can construct a pointwise inner product on T (r,s)M which we denote (·, ·)p. If M is compact there is an L2-inner product on sections of T (r,s)M given by (η, ρ)L2 = ∫ M (ηp, ρp)p dVg. If there are more than one metric on M , then we denote the L2-inner product with respect to g by (·, ·)g. For general vector bundles pi : E → M we denote by C∞(E) the space of smooth sections. For a connection on E we denote by ∇ the corresponding covariant derivative ∇ : C∞(E)→ C∞(E⊗T ∗M). We say that a tensor T is parallel if∇T = 0. For any met- ric g there exists a unique linear covariant derivative ∇ on TM satisfying ∇Zg(X, Y ) = g(∇ZX, Y ) + g(X,∇ZY ) and which is torsion free, i.e. ∇XY −∇YX − [X, Y ] = 0. It is the Levi-Civita covariant derivative. For a complex manifold (M,J) we denote by J the multiplicative extension of J to ∧∗TCM .We use Greek indices to express the following modified tensors. For the con- travariant index a of the tensor T a...... we define the tensors T α... ... = 1 2 (T a...... − iJaj T j...... ) and T α¯...... = 1 2 (T a...... + iJ a j T j... ... ). For the covariant index b of tensors T ... b... we define the tensors T ...β... = 1 2 (T ...b... − iJ jbT ...j...) and T ...β¯... = 12(T ...b... + iJ jbT ...j...). These operations are projections and they satisfy T a...... = T α... ... + T α¯... ... and T ... b... = T ... β... + T ... β¯... . When we give definitions in the text we emphasize them. Throughout Chapter 2 we use the symbol M to denote a connected and oriented smooth manifold. It will often be compact, but not always. In Chapter 3 and 4 we use V for a smooth connected and oriented orbifold. It will often be compact, but not always. In Chapter 5 and 6 we use X for a smooth, connected and oriented non-compact manifold. It will often be ALE, but not always. 6 2.1. Tools from analysis 2.1 Tools from analysis In this section we introduce various function spaces and fundamental results about them. The material is mostly borrowed from [JOY00, Chapter 1]. A Banach space is a vector space X together with a norm || · ||X for which the metric d(x, y) = ||x − y||X is complete. If the Banach norm is induced by an inner produce (·, ·)X , i.e. ||x|| = √ (x, x)X , then (X, (·, ·)X) is a Hilbert space. A Hausdorff topological vector space is a Fréchet space if its topology can be induces from a family of semi-norms || · ||k, k = 1, 2, . . . and its metric given by these semi-norms is complete. We say that a topological space X is a Hilbert-, Banach- or Fréchet manifold if it admits an atlas of neighbourhoods homeomorphic to open sets in a Hilbert-, Banach- or Fréchet space respectively. For a Riemannian manifold (M, g) denote by L1(M) the set of measurable function f : M → R for which ∫ M f dVg < ∞. We say that a function f : M → R is locally integrable if for all compact subsets K ⊂ M , f ∈ L1(K). We denote the space of locally integrable functions on M by L1loc(M). Definition 2.1.1. For p ≥ 1 define the Lebesgue space Lp(M) to be the subset of L1loc(M) for which the Lp-norm ||f ||Lp = (∫ M |f |pdVg ) 1 p is finite. Smooth functions on M with compact support are called test functions. The set of test functions is denoted by C∞c (M). A function f ∈ L1loc(M) is said to be k-times weakly differentiable if for some f˜ ∈ L1loc(M), ∫ M f∇αφ dVg = (−1)|α| ∫ M f˜φ dVg for all test functions φ ∈ C∞c (M) and all multiindex |α| ≤ k. Definition 2.1.2. Define the Sobolev space Lpk(M) to be the set of those f ∈ Lp(M) which are k-times weakly differentiable and for which |∇if |g ∈ Lp(M) for all i = 1, . . . , k. Equip Lpk(M) with the norm ||f ||Lpk = ( k∑ i=0 ∫ M |∇if |p dVg ) 1 p . 7 Chapter 2. Preliminaries Denote by Ck(M) the set of bounded continuous functions f : M → R which have k times continuous bounded derivatives. Equip Ck(M) with the norm ||f ||Ck = k∑ i=0 sup M |∇if(x)|. We say that a function f : M → R is Hölder continuous with exponent α ∈ (0, 1) if [f ]α = sup x 6=y∈M |f(x)− f(y)| d(x, y)α <∞ where d(x, y) is the distance from x to y on M using g. We can extend this notion of continuity to tensors over M in the following way. Let δ(g) be the injectivity radius of g. For T ∈ T (r,s)M and x, y ∈M with d(x, y) < δ(g) we can make sense of |T (x)− T (y)|g by choosing the unique geodesic joining x and y. Parallel translating along this geodesic we can identify (T (r,s)M)x and (T (r,s)M)y as vector spaces, so the subtraction and the norm makes sense. Definition 2.1.3. Define the Hölder space Ck,α(M) as the set of those f ∈ Ck(M) for which supM [∇kf ]α <∞. Equip Ck,α(M) with the norm ||f ||Ck,α = ||f ||Ck + [∇kf ]α. where for k > 1 the supremum in [∇kf ]α is taken inside the injectivity radius of the metric g. Lemma 2.1.4. Let (M, g) be a Riemannian manifold. Let k ≥ 0 and p > 1 be inte- gers and α ∈ (0, 1). Equipping Lp(M), Lpk(M), Ck(M) and Ck,α(M) with the norms introduced above turns them into Banach spaces. The spaces L2(M) and L2k(M) are also Hilbert spaces with respect to the inner products (f1, f2)L2 = ∫ M f1f2 dVg and (f1, f2)L2k = ∑k j=0 ∫ M (∇jf1)(∇jf2) dVg respectively. Lemma 2.1.4 also holds for the spaces of Ck, Ck,α, Lp and Lpk sections of vector bundles over M . A normed vector space X is continuously embedded into a normed vector space Y if the inclusion map ι : X → Y is continuous. A linear operatorA : (X, ||·||X)→ (Y, ||·||Y ) is compact if for any sequence {xn} in X the sequence {Axn} has a Cauchy subsequence. 8 2.1. Tools from analysis We say that X is compactly embedded into Y if the inclusion map is compact. The next result is the Sobolev Embedding theorem (continuity) and Kondrachov’s theorem (com- pactness) combined. Theorem 2.1.5. Let (Mn, g) be a compact Riemannian manifold. Let k, l be integers with 0 ≤ l ≤ k and p, q real numbers with p, q ≥ 1 and α ∈ (0, 1). If 1 q ≤ 1 p + k − l n Then Lqk(M) is continuously embedded into L p l (M). If the inequality is strict then the embedding is also compact. If 1 q ≤ k − l − α n then Lqk(M) embeds continuously into C l,α(M). If the inequality is strict, then the em- bedding is also compact. Also, for any k ≥ 0 the embedding Ck,α(M) → Ck(M) is compact. Theorem 2.1.5 holds also for the space of sections of vector bundles over a compact manifold. For vector bundles V and W over M a linear differential operator P of order k is a map that takes sections of V to sections of W and is of the form Pu = Ai1...ik∇i1...iku+Bi1...ik−1∇i1...ik−1u+ · · ·+Ki1∇i1u+ Lu where ∇i1...ik = ∇i1 . . .∇ik and A, . . . ,K are symmetric tensors and L is a smooth function. The principal symbol of P is σ(P ) : T ∗M × V → W with ση(P, x) = Ai1...ik(x)ηi1 . . . ηik : Vx → Wx. It is a homogeneous polynomial of degree k in η. We say that P is elliptic if for all x ∈ M and all η ∈ T ∗xM with η 6= 0 we have ση(P, x) 6= 0. A non-linear differential operator is of the form Pu = Q(x, u(x),∇u(x), . . . ,∇ku(x)) for some continuous function Q. Let P be a differential operator of order k and u a section of V with k derivatives. The linearization of P at u is LuPv = d dt |t=0P (u+ tv) = lim t→0 P (u+ tv)− P (u) t . 9 Chapter 2. Preliminaries The linearized operator LuP is again of order k. If P is a linear differential operator, then LuP = P . We say that a non-linear differential operator is elliptic if its linearization is elliptic. Theorem 2.1.6 (Elliptic regularity). Let (Mn, g) be a compact Riemannian manifold. Let V,W be vector bundles over M and let P be a smooth linear elliptic differential operator of order k. Assume that we have found v ∈ L1(V ) and w ∈ L1(W ) such that P (v) = w. Let p and l be integers with p > 1 and l ≥ 0, and let α ∈ (0, 1). If w ∈ C∞(W ), then v ∈ C∞(V ). If w ∈ Lpl (W ), then v ∈ Lpl+k(V ) and ||v||Lpl+k ≤ C(||w||Lpl + ||v||L1). If w ∈ C l,α(W ), then v ∈ C l+k,α(V ) and ||v||Cl+k,α ≤ C(|w||Cl,α + ||v||C0). In both cases C > 0 is some constant independent of v and w. Theorem 2.1.7. Let (M, g) be a compact manifold. Let V and W be vector bundles over M of the same dimension and let P be a smooth linear elliptic operator of order k from V to W . Let l ≥ 0 and p > 1 be integers and α ∈ (0, 1). The operator P acts by P : C∞(V ) → C∞(W ), P : Lpk+l(V ) → Lpl (W ) and P : Ck+1,α(V ) → C l(W ). The kernel ker(P ) is the same for each of the actions and it is a finite dimensional vector subspace of C∞(V ). Let V and W be vector bundles over a compact manifold M and let them be equipped with metrics. For a linear differential operator k from V to W there exists a unique linear operator P ∗ such that (Pv, w)W = (v, P ∗w)V . This operator is called the formal adjoint operator of P . Theorem 2.1.8 (Fredholm Alternative). Let (M, g) be a compact Riemannian manifold and let V,W be vector bundles over M equipped with metrics. Let P be a smooth linear elliptic operator from V to W of order k. Let l ≥ 0 and p > 1 be integers and α ∈ (0, 1). Then the images of the maps P : Lpk+l(V )→ Lpl (W ) and P : Ck+l,α(V )→ C l,α(W ) are closed vector subspaces of Lpl (W ) and C l,α(W ) respectively. For w ∈ Lpl (W ) the equation Pv = w admits a solution v ∈ Lpk+l(V ) if and only if 10 2.2. Space of metrics w ⊥ ker(P ∗). If v ⊥ ker(P ) then v is unique. For w ∈ C l,α(W ) the equation Pv = w admits a solution v ∈ C l+k,α(V ) if and only if w ⊥ ker(P ∗). If v ⊥ ker(P ) then v is unique. Theorem 2.1.9 (Inverse function theorem for Banach spaces). LetX, Y be Banach spaces and U ⊂ X an open neighbourhood of x ∈ X . Suppose that the function F : U → Y is Ck for some k ≥ 1, with F (x) = y, and that the first derivative dFx : X → Y of F at x is an isomorphism between X and Y both as vector spaces and as topological spaces. Then there exist open neighbourhoods U ′ ⊂ U of x ∈ X and V ′ of y ∈ Y , such that F : U ′ → V ′ is a Ck-isomorphism. Theorem 2.1.10 (Implicit function theorem for Banach spaces). Let X, Y, Z be Banach spaces and let U ⊂ X and V ⊂ Y be open neighbourhoods of 0 in X and Y respectively. Assume that we have a Ck-function F : U × V → Z with F (0, 0) = 0. If dF(0,0)|Y : Y → Z is a linear homeomorphism, then there exists an open subset U ′ ⊂ U and a unique Ck-map G : U ′ → Y such that G(0) = 0 and F (x,G(x)) = 0 for all x ∈ U ′. Let U be a open subset of Rn and let V be an open subset of Rm. A function f : U → V : x 7→ (f1(x), . . . , fm(x)) is real analytic if around every point p ∈ U there is a neighbourhood U ′ such that each component fj|U ′ of f |U ′ admits a sequence of homogeneous polynomials Pi of degree i in n variables such that fj|U ′ can be written as fj|U ′(x) = ∑ i Pi(x− p). A real analytic manifold M is a topological manifold for which all transition functions are real analytic. A real analytic set is the kernel of a real analytic map. The next result is a real analytic implicit function theorem. It is taken from [KOI83, Lemma 13.6]. Theorem 2.1.11. Let V andW be Hilbert spaces and f a real analytic mapping from V to W defined on an open neighbourhood of the origin 0 ∈ V . Assume that f(0) = 0 and that the image of the differential df0 at 0 is closed in W . Then there is an open neighbourhood U of 0 ∈ V such that the set f−1(0)∩U is a real analytic set in a real analytic submanifold Z of U whose tangent space T0Z coincides with ker(df0). 2.2 Space of metrics In this section we introduce the space of Riemannian metrics. 11 Chapter 2. Preliminaries For a Riemannian manifold (M, g) denote by Sym2(T ∗M) the tensor bundle of sym- metric (2, 0)-tensors. Denote by M the set of smooth Riemannian metrics on M . In Section 2.1 the space Ck(Sym2(T ∗M)) was equipped with a Banach space topology. The space C∞(Sym2(T ∗M)) = ∩∞k=1Ck(Sym2(T ∗M)) is a Fréchet space with respect to the family of Ck-norms. Assume that M is compact. All Riemannian metrics are bounded andM is a subset of C∞(Sym2(T ∗M)). Equip M with the subspace topology. The space M is a convex cone in C∞(Sym2(T ∗M)) consisting of sections which are positive definite. Positive def- initeness is an open condition and M is an open subset of C∞(Sym2(T ∗M)). Denote by CkM the set of positive definite sections of Ck(Sym2(T ∗V )) and equip it with the subspace topology. Define Mk = L2k(Sym2(T ∗M)) ∩ C0M, and equip it with the subspace topology of L2k(Sym 2(T ∗M)). Lemma 2.2.1. Let (Mn, g) be a compact smooth manifold and let k > n/2 and r ≥ 0. ThenMk is a Hilbert manifold, CrM is a Banach manifold andM is a Fréchet manifold. Proof. For k > n/2 it follows from Theorem 2.1.5 that L2k(Sym 2(T ∗V )) embeds contin- uously into C0(Sym2(T ∗V )). the spaceMk is the preimage of the open subset C0M in C0(Sym2(T ∗V )) under the continuous embedding, so the spaceMk is an open subset of L2k(Sym 2(T ∗M)). Hence it is a Hilbert manifold. The space CkM is an open subset of the Banach space Ck(Sym2(T ∗M)), so it is a Banach manifold. The family of semi-norms on the space C∞(Sym2(T ∗M)) = ∩k≥0Ck(Sym2(T ∗M)), which we get from the Ck-norms on Ck(Sym2(T ∗M)), turns it into a Fréchet space and asM is an open subset of it,M is a Fréchet manifold. 2.3 Laplace operators In this section we give a brief introduction to three Laplace operators. We also introduce some notation from differential geometry related to curvature and some relevant differen- tial operators. The material is borrowed from [BES87, Chapter 1]. For a manifold (M, g) and a covariant derivative∇ : C∞(TM)→ C∞(TM ⊗T ∗M). 12 2.3. Laplace operators The Riemann curvature tensor R = R(∇) = F∇ ∈ Ω2M(End(TM)) is the (3, 1)-tensor field R(X, Y )Z = ∇X∇YZ − ∇Y∇XZ − ∇[X,Y ]Z. It can locally be written as R = R lijk dx i ⊗ dxj ⊗ dxk ⊗ ∂ ∂xl . The first two indices is the form part of R and the latter two the endomorphism part. For a diffeomorphism φ, R satisfies Rφ∗g(φ ∗X,φ∗Y )φ∗Z = φ∗(Rg(X, Y )Z). (2.1) Locally ∇ can be written as d∇s = ds + As, where A is the underlying connection of ∇ and ds is the exterior derivative applied componentwise to s. The curvature of ∇ can be written as R(∇) = dA+A∧A and it measures the non-commutativity of∇. A covariant derivative satisfying R(∇) = 0 is said to be flat. Metrics with a flat covariant derivative are locally isometric to Euclidean space. The Ricci curvature tensor is a symmetric (2, 0)- tensor Ric = Ric(g) defined as Ric(X, Y ) = tr(V 7→ R(X, V )Y ). Locally it is given by Ric = Rijdxi ⊗ dxj where Rij = R kkij . The scalar curvature is the trace of the Ricci tensor. It is the function sg ∈ C∞(M) given by sg = trg(Ric) = ∑n i,j=1 g ijRij . The total scalar curvature is Tg = ∫ M sg dVg. Assume that (M, g) is compact. The Levi-Civita Covariant derivative extends to a covariant derivative ∇ : C∞(T (r,s)M) → C∞(T (r+1,s)M). It admits a formal adjoint ∇∗ : C∞(T (r+1,s)M) → C∞(T (r,s)M) for the L2-inner product. On (k, 0)-tensors the operator∇∗ is locally given by (∇∗η)(X1, . . . , Xk) = − n∑ i=1 (∇eiη)(ei, X1, . . . , Xk), where {ei}ni=1 is an orthonormal basis of TM and X1, . . . , Xk are vector fields on M . The exterior derivative d : Ωk(M) → Ωk+1(M) similarly admits a formal adjoint d∗ : Ωk+1(M) → Ωk(M). It is given by d∗ω = (−1)n(k+1)+1 ∗g d ∗g ω, where the Hodge star operator ∗g : ∧k(T ∗xM) → ∧n−k(T ∗xM) satisfies α ∧ ∗β = (α, β)g dVg for all α, β ∈ ∧k(T ∗xM). The operator d∗ is the skew-symmetric part of ∇∗, i.e. d∗η is the restriction of ∇∗η to ∧k+1(M). Define the following operator on symmetric tensors: δ∗ = Sym ◦ ∇|Symk(T ∗M) : C∞(Symk(T ∗M))→ C∞(Symk+1(T ∗M)). 13 Chapter 2. Preliminaries The formal adjoint of δ∗ with respect to the L2-inner product is δ = ∇∗|Symk+1(T ∗M) : C∞(Symk+1(T ∗M))→ C∞(Symk(T ∗M)). For α ∈ Ω1(M) the covariant derivative decomposes as ∇α = δ∗α + 1 2 dα. The Hessian of a smooth function f ∈ C∞(M) is the (2, 0)-tensor field Hess(f) = ∇(df) and it is often denoted∇2(f). The divergence of a vector field X is the smooth function div(X) = δ(X[) = −∑ni=1 g(∇eiX, ei) where {ei}ni=1 is an orthonormal basis. Let X ∈ C∞(TM) and let θt be the flow of X . The Lie derivative of a tensor field A ∈ C∞(T (r,s)M) in the direction of X is a map LX : C∞(T (r,s)M) → C∞(T (r,s)M) given by (LXA)p = ddt |t=0 d(θ−t)θt(p)(Aθt(p)). It satisfies (LXA)p = ddt |t=0((θt)∗A)p, which means that Lie derivative of a tensor fieldA in the direction ofX is the linearization of the pull-back action of the flow of X on A. We say that a vector field X is Killing if LXg = 0. This happens exactly when the local flow of X are all isometries. We introduce three second order linear elliptic differential operators. The Hodge Laplacian ∆ : Ωk(M)→ Ωk(M) is defined as ∆ = dd∗ + d∗d. On functions, i.e. 0-forms, it takes the form ∆f = −trg(Hess(f)) = −trg(∇(df)). The rough Laplacian is the operator ∇∗∇ : C∞(T (r,s)M)→ C∞(T (r,s)M). It satisfies ∇∗∇ = −tr(∇2). The curvature tensor R acts on symmetric (2, 0)-tensors as a linear map. We denote this action by ◦ R. It is defined as ( ◦ R h)(X, Y ) = n∑ i=1 h(R(X, ei)Y, ei), (2.2) where {ei}ni=1 is an orthonormal basis of TM . The Lichnerowicz Laplacian is the dif- ferential operator ∆L : C∞(T (k,0)M) → C∞(T (k,0)M) given by ∆LT = ∇∗∇T + ΓT , where ΓT is a 0’th order correction term. For the precise definition see [BES87, 1.143]. 14 2.4. Einstein metrics On Symmetric (2, 0)-tensors the Lichnerowicz Laplacian is ∆Lh = ∇∗∇h+ 2 s n h− 2 ◦R h, (2.3) and for α ∈ Ωk(M), ∆Lα = ∆α. Also note that the Lichnerowicz Laplacian satisfies ∆L(trh) = tr(∆Lh). We shall see later that the Lichnerowicz Laplacian defines the first order deformations of Einstein metrics. For more about the Lichnerowicz Laplacian see [BER-EBI69]. The Hodge Laplacian and the rough Laplacian are related via the following Weitzenböck formula. Lemma 2.3.1. Let (M, g) be a compact connected Riemannian manifold. Then for any α ∈ Ωk(M), ∆α = ∇∗∇α + R˜α where R˜α is a smooth alternating (k, 0)-tensor with coefficients (R˜α)r1...rk = k∑ j=1 gabRicrjbαr1...a...rk − 2 ∑ i 2 Einstein metrics therefore have constant scalar curvature and constant Einstein coefficient λ. Einstein metrics behave well under pull-backs by diffeomorphisms and rescaling by constants. From (2.1) it follows that φ∗Ric(g) = Ric(φ∗g), and from [BES87, Theorem 1.159] we know that Ric(efg) = Ric(g) + (2− n)(∇df − df ⊗ df) + (∆f − (n− 2)||df ||2)g, so rescaling g by a non-zero constant c satisfies Ric(cg) = Ric(g). Denote by M1 the subspace of all Riemannian metrics with vol(g) = 1. One rea- son why Einstein metrics could be considered ’distinguished’ is [BES87, Theorem 4.21], which says that Theorem 2.4.1. Let M be a compact manifold. The critical points of the total scalar curvature functional Tg = ∫ M sg dVg restricted toM1 are exactly the Einstein metrics. In physics, the total scalar curvature functional introduced in Theorem 2.4.1 is often named the Einstein-Hilbert action. It is usually denoted by Sg, but as we wish to keep that notation free for later use we have chosen the slightly unusual Tg for the total scalar curvature. We make a few remarks about the existence of Einstein metrics on manifolds. It is convenient to consider 2, 3, 4 and ≥ 5 dimensional manifolds separately. For 2-dimensional manifolds the Uniformization Theorem asserts that any connected oriented surface admits a complete metric with constant sectional curvature. On surfaces 16 2.4. Einstein metrics a metric is Einstein exactly if it has constant sectional curvature. So any connected and oriented 2-dimensional manifold therefore admits an Einstein metric. For 3-dimensional manifolds a metric is Einstein exactly if it has constant sectional curvature. Contrary to the 2-dimensional case, not all 3-dimensional manifolds admit a constant sectional curvature metric. We give an example of one such manifold (from [BES87, 6.16]). Example 2.4.2. The manifold S1 × S2 does not admit any Einstein metrics. We briefly sketch the argument for Example 2.4.2. Let (M, g) be a connected complete 3-manifold and denote by (M˜, pi∗g) the universal cover of M . If (M, g) satisfies secg > 0, then (M˜, pi∗g) satisfies secpi∗g > 0. If M˜ is compact, then M˜ ' S3 by a theorem by Hamilton [HAM82]. If M˜ is complete and non-compact, then M˜ ' R3 by the Cheeger- Gromoll-Meyer theorem. If (M, g) satisfies secg ≤ 0, then by the Cartan-Hadamard the- orem the exponential map expg is a covering map and M˜ ' R3. The manifold S1 × S2 has universal cover R × S2, which is neither diffeomorphic to R3 or S3 and it therefore does not admit a constant sectional curvature metric. In particular, it does not carry any Einstein metrics. For 4-dimensional manifolds examples of manifolds that do not admit Einstein metrics exist. One such example is S1 × S3, see [BES87, 6.32]. For n-dimensional manifolds with n ≥ 5 no manifolds are known not to admit any Einstein metrics, [BES87, 0.21]. The Einstein operator is the map E :M→ C∞(Sym2(T ∗M)) given by E(g) = Ric(g)− sg n g. (2.6) A metric g is Einstein exactly if E(g) = 0. Einstein metrics on manifolds with dim(M) ≥ 3 have constant scalar curvature, so Tg = sg ∫ M dVg = sg vol(g). We can therefore express λ in terms of the total scalar curvature, λ = sg n = Tg n·vol(g) . For compact M with dim(M) ≥ 3 we can equivalently express the Einstein operator as E(g) = Ric(g) − Tg n·vol(g)g. The Einstein operator is a non-linear second order differential operator on the space of metrics. 17 Chapter 2. Preliminaries From [BES87, Theorem 1.174] we know that the linearization of g 7→ Ric(g) is Ric′g(h) = 1 2 ∆Lh− δ∗gδgh− 1 2 ∇gd(trgh) = ∇∗∇h− δ∗gδgh− 1 2 ∇gd(trgh)− 2 ◦ R h. Let g be an Einstein metric and let gt be a smooth curve of metrics with g0 = g. The linearized Einstein operator at g is E ′g(h) = d dt E(gt)|t=0 = d dt (Ric(gt)− Sgt n gt)|t=0 = Ric′g(h)− ( (Tg) ′ n g′0g + Tg n g′0 ) = Ric′g(h)− Tg n h where the last inequality follows because g is a critical point of the total scalar curvature functional. It leaves us with the following expression for the linearization of the Einstein operator, E ′g(h) = 1 2 ∆Lh− δ∗g(δg)h− 1 2 ∇gd(trgh)− Tg n h. 2.5 Kähler metrics In this section we fix some notation from complex geometry. We introduce Kähler metrics and the first Chern class of a complex vector bundle and explain the relation between Ricci-forms and the first Chern class on Kähler manifolds. We also present Yau’s solution to the Calabi conjecture. The material is borrowed from [JOY00] and [HUY05]. Another relevant reference is [YAU78]. A complex manifold of complex dimension m is a smooth manifold of real dimension 2m with a holomorphic atlas, i.e. an atlas such that each chart is homeomorphic to a polydisc in Cn and all transition functions are holomorphic. An almost complex structure J on a smooth manifold M is an endomorphism Jp : TpM → TpM satisfying J2p = −Id for each p ∈ M . An almost complex structure J is integrable if its Nijenhuis tensor NJ vanishes. An integrable almost complex structure is called a complex structure. A consequence of the Newlander-Nierenberg theorem is that we could equivalently have defined a complex manifold as a smooth manifold with an integrable almost complex 18 2.5. Kähler metrics structure. We write (M,J) for a complex manifold with a complex structure J . We will usually denote its complex dimension by m. On a complex manifold (M,J) the complex linear extension of J to the complex tangent bundle TM ⊗R C = T 1,0M ⊕ T 0,1M where T 1,0M is the i′-th eingenspace of J and T 0,1M is the −i′-eigenspace. The complex (∧k(TCM)∗, dC) splits naturally as ∧p,qM = ∧p(T 1,0M)∗ ⊗C ∧q(T 0,1M)∗ and we denote by Ap,q the sheaf of sections of ∧p,qM . The global sections Ap,q(M) are the forms of bidegree (p, q). Extending the exterior differential complex linearly to TM⊗C provides a differential dC. The projections of dC(Ap,q(M)) onto Ap+1,0(M) is denoted ∂ and the projection onto A0,q+1(M) is ∂¯. The holomorphic tangent bundle is TM = T 1,0M and the dual bundle ΩM . The sheaf of holomorphic k-forms is ΩkM for 0 ≤ k ≤ m. The canonical bundle is the determinant bundle KM = det(ΩM) = ΩmM . The (p, q)-Dolbeault cohomology is denoted H p,q(M) = Hp,q ∂¯ (M). This construction extends to holomorphic vector bundles E and is denoted by Hp,q(M,E). A real (1, 1)-form β can be converted into a symmetric bilinear (2, 0)-form b via b(X, Y ) = β(JX, Y ). We call β the fundamental form of b and we say that β is posi- tive, vanishing or negative if the corresponding (2, 0)-form b is positive definite, vanishing or negative definite respectively. On a complex manifold (M,J) a Riemannian metric g is said to be Hermitian if it is compatible with J , i.e. g(JX, JY ) = g(X, Y ). Definition 2.5.1. Let (M,J) be a complex manifold with a Hermitian metric g. We say that g is Kähler if its fundamental form ω is closed, i.e. dω = 0. Locally the Kähler condition translates to ωij = Jki gkj . We say that a complex mani- fold (X, J) is Kählerienne, if it admits Kähler metrics, even if we have not specified one. A Kähler form is non-degenerate, so ωn is proportional to the volume form of g. The relation is ωn = n! dVg. Remark 2.5.2. On a complex Hermitian manifold (M,J, g) with the Levi-Civita covariant derivative ∇ and where ω is the fundamental form of g, the Kähler condition can be expressed in various ways. We have g Kähler ⇔ dω = 0 ⇔ ∇ω = 0 ⇔ ∇J = 0. The condition ∇J = 0 can be rewritten as ∇XJY = J∇XY for all X, Y ∈ TpM and all 19 Chapter 2. Preliminaries p ∈M . Let (M,J) be a complex manifold and let pi : E → M be a complex vector bundle with a covariant derivative∇. Taking the trace on the endomorphism part of the Riemann curvature tensor produces a differential 2-form. It turns out that this form is closed, so it defines a class in cohomology. The cohomology class [ i 2pi trC(F∇)] ∈ H2(M,R) is there- fore well-defined. For any other covariant derivative ∇′ the forms trC(F∇) and trC(F∇′) differ only by an exact form, so [ i 2pi trC(F∇)] is independent of the choice of covariant derivative. This is the Chern-Weil construction of Chern classes. Definition 2.5.3. Let (M,J) be a complex manifold and let pi : E → M be a complex vector bundle with a covariant derivative ∇. We define the first Chern class of ∇ on E to be c1(E,∇) = i 2pi [trC(F∇)] ∈ H2(M,R). where the trace is taken in the endomorphism part of the curvature tensor. Definition 2.5.4. We define the first Chern class of a complex manifold (M,J) to be c1(M) = c1(TM) ∈ H2(M,R) where TM is the holomorphic tangent bundle of M . The first Chern class depends on the choice of complex structure on X and we will therefore often denote it c1(J). For any complex vector bundle we have c1(E) = c1(det(E)). The canonical bundle KM is the dual line bundle to the determinant bundle of TM , i.e. KM = − det(TM). We can therefore express the first Chern class of M via the canonical line bundle as c1(M) = −c1(KM). Similarly to the relation between the metric g and the Kähler form ω, where ω(u, v) = g(Ju, v), then the fundamental form of the Ricci curvature tensor Ric(u, v) is the real (1, 1)-form ρ = ρ(g) ∈ A1,1(M) defined by ρ(u, v) = Ric(Ju, v), which is called the Ricci-form of g. Locally this relation is ρij = Jki rkj . Proposition 2.5.5. Let (M,J, g) be a Kähler manifold and let ρ be the Ricci-form of g. Then ρ = i trC(F∇), where the trace is taken in the endomorphism part of the tensor. 20 2.5. Kähler metrics On a Kähler manifold we therefore have c1(J) = i 2pi [trC(F∇)] = 1 2pi [ρ] ∈ H2(M,R). A Riemannian metric g on a complex manifold (M,J) is Kähler-Einstein if it is both Kähler and Einstein. For such metrics ρ = λω, where ω and ρ are the Kähler form and Ricci form of g respectively and λ is the Einstein constant of g. On Kähler-Einstein manifolds we therefore have c1(J) = λ 2pi [ω] ∈ H2(M,R). A cohomology class α ∈ H2(M,R) has a sign if it contains a representative which has a sign, i.e. for which the associated symmetric (2, 0)-tensor has a sign. Cohomology classes may or may not have a sing. We write α < 0, α = 0 or α > 0 for α ∈ H2(M,R) in case α has a sign and the sign is negative, zero or positive respectively. We say that a complex vector bundle pi : E →M has a sign if its first Chern class c1(E) has a sign, and we say that the sign of E is positive, zero or negative if c1(E) is positive, zero or negative respectively. A metric g is Ricci-flat exactly when ρ(g) = 0, so if a manifold (M,J) admits a Ricci- flat metric which is Kähler with respect to J , then c1(J) = 2pii [ρ] = 0. Ricci-flat Kähler metrics can therefore only exist on manifolds with vanishing first Chern class. If (M,J) is compact and g is Kähler-Einstein with Einstein constant λ and c1(J) = 0, then this forces the Einstein constant of g to be zero as the volume of the metric is 0 6= dVg = ωnn! so ω cannot be exact. Hence, [ω] 6= 0 ∈ H2(M,R). But 0 = c1(J) = λ2pi [ω] which implies λ = 0. The next celebrated result about the existence of Kähler metrics with prescribed Ricci form on compact Kähler manifolds was conjectured by Eugenio Calabi in the 1950’s and proved by Shing-Tung Yau in the late 1970’s ([YAU78]). Theorem 2.5.6 (Calabi Conjecture). Let (M,J, g) be a compact complex Kähler manifold. Let ω be the Kähler form of g. For any ρ′ ∈ 2pic1(J) there exists a Kähler metric g′ with ρ′ as its Ricci form and with Kähler form ω′ satisfying [ω′] = [ω]. In the special case of vanishing first Chern class Theorem 2.5.6 can be used to construct Kähler-Einstein metrics with zero Einstein constant. 21 Chapter 2. Preliminaries Corollary 2.5.7. Let (M,J, g) be a compact complex Kähler manifold with c1(J) = 0 and let ω be the Kähler form of g. Then there exists a Ricci-flat Kähler metric in the Kähler class of ω. 2.6 Deformation theory of complex structures In this section, we give an introduction to the deformation theory of complex structures on compact complex manifolds. Deformation theory of complex structures is often formu- lated using the language of complex spaces. For our purpose, however, a smooth version will suffice. The material is borrowed from [KOD86] and [HUY05]. Two other relevant sources are [KOD-NIR-SPE58] and [KOD-SPE60]. A map pi : X → Y between topological spaces is proper if preimages of compact sets are compact. A smooth family of compact complex manifolds is defined as follows. Definition 2.6.1. Let X and S be complex manifolds and let pi : X → S be a proper holomorphic map. The fibres Xt = pi−1(t) are compact complex submanifolds of X and we say that pi : X → S is a smooth family of complex manifolds parametrized by S. Fixing 0 ∈ S and restricting pi : X → S to a germ around 0, the family pi : X → S from Definition 2.6.1 is a smooth family of deformations of the compact complex man- ifold M = X0. For such a family of deformations we can trivialize pi : X → S as a differentiable family. The fibres are all diffeomorphic and we can view X ' S ×M as a deformation of the complex structure on M = X0 parametrized by S. Elements of T0S are called infinitesimal complex deformations. Proposition 2.6.2. Let M be a compact complex manifold and let S be the parameter space of a versal family of deformations of M . Then there is a natural bijection between the space of infinitesimal complex deformations T0S and H1(M, TM). Next we define a particular class of compact complex manifolds which are well un- derstood when it comes to complex deformations. This will be convenient when we start looking for applications of our results in Chapter 4. Definition 2.6.3. A Calabi-Yau manifold is a compact complex Kähler manifold with triv- ial canonical bundle. 22 2.7. Koiso’s deformation theory on compact manifolds We say that a compact complex manifold M has unobstructed deformations if every infinitesimal deformation v ∈ H1(M, TM) integrates into a smooth curve of deforma- tions. If H2(M, TM) = 0, then M has unobstructed deformations. A special feature of a Calabi-Yau manifold M is that H2(M, TM) = 0, and they therefore have unobstructed deformations. In this thesis we will be studying properties of metrics and complex structures which are preserved under deformations. The next result by Kodaira will be important for our work. Theorem 2.6.4 (Kodaira). Let M be a compact complex Kähler manifold. If pi : X → S is a smooth family of deformation of M , then Xt is Kähler for all t ∈ S. The next lemma is a stability result concerning the first Chern class c1(J). Lemma 2.6.5. Let (M,J) be a compact complex manifold and assume that c1(J) = 0. Assume that M admits a smooth family of deformations pi : X → S. Then for t ∈ S small c1(Jt) = 0. Lemma 2.6.5 is a consequence of the fact that the first Chern class takes values in the image of H2(M,Z) in H2(M,R), and it must therefore be preserved for small deforma- tions. We say that a property which is preserved under small deformations is an open condition. 2.7 Koiso’s deformation theory on compact manifolds In this section, we give a review of Koiso’s deformation theory of Einstein metrics over compact complex manifolds. Let (M,J, g) be a compact complex Kähler-Einstein mani- fold with negative or vanishing first Chern class and with unobstructed complex deforma- tions. Koiso established in [KOI83] the remarkable fact that any Einstein deformation of g is Kähler, possibly with respect to a perturbed complex structure. He also showed that in a neighbourhood of a Kähler-Einstein structure the moduli space of Einstein structures is, up to an action of a finite group, a smooth manifold, and he found an expression for the dimension of it. We give an account of his results. For more details see [KOI83]. Alternatively see [BES87, Chapter 12] for an excellent review. 23 Chapter 2. Preliminaries We focus on the case of vanishing first Chern class. Koiso’s results also cover the case of negative first Chern class. The latter case is simpler than the former due to absence of any Hermitian infinitesimal Einstein deformations. Let (M, g) be a compact Ricci-flat manifold. We introduce an equivalence relation on the space of metricsM. Identify two metrics inM if one is the pull-back of the other by a diffeomorphism or if it is a rescaling of the other by a positive constant. This gives an equivalence relation g′ ∼ g if g′ = cφ∗g for some c > 0 and some φ ∈ D, where D = D(M) denote the group of diffeomorphisms from M to itself. The equivalence classes are called Riemannian structures and we denote the quotient space of Riemannian structuresM/ ∼ by M˜. If we denote byM1 the space of metrics with vol(g) = 1, then we could equivalently describe the quotient space as M1/D. Denote by Iso(M, g) the group of isometries from (M, g) to itself. Ebin produced in [EBI70] a slice Sg for the action of D onM containing g. This construction produces a homeomorphism U ⊂M1/D → Sg/Iso(M, g), (2.7) where U is a neighbourhood of [g]. This serves as a chart for the quotient space. If g is Ricci-flat, then the structure [g] is called a Ricci-flat structure. Denote the quotient space of Ricci-flat structures by R˜. . This is a subspace of M˜ and it is called the moduli space of Ricci-flat structures. Denote by Pg the subspace of Ricci-flat metrics in the slice Sg. It is called the premoduli space of Ricci-flat metrics. In Section 2.3 we introduced the operator δ∗g : Ω 1(M) → C∞(Sym2(T ∗M)). By [BES87, Lemma 1.60] it satisfies δ∗gη = 1 2 Lη#g. The L2-formal adjoint of δ∗g is δg : C∞(Sym2(T ∗M)) → Ω1(M). The operator δ∗g has finite dimensional kernel and closed image, so C∞(Sym2(T ∗M)) splits as a direct sum of the image of δ∗g and the kernel of δg. The tangent space ofM1 at g consist of symmetric 2-tensors h with ∫ M trgh dvolg = 0. Hence, TgM1 = Im(δ∗g)⊕ [ker(δg) ∩ TgM1] . Let gt be a smooth curve of Ricci-flat metrics with g0 = g and let h = ddtg(t)|t=0. The 24 2.7. Koiso’s deformation theory on compact manifolds linearization of Ric(gt) at g is 2 Ric′g(h) = ∇∗g∇h− 2δ∗δgh−∇d(trgh)− 2 ◦ R h (2.8) where ◦ R is the action of the Riemann curvature tensor on 2-tensors introduced in Section 2.3. Define by (g) the subspace of C∞(Sym2(T ∗M)) satisfying the three equations Ric′g(h) = 0 δgh = 0 ∫ M trg(h) dVg = 0. (2.9) Berger and Ebin simplified in [BER-EBI69] these equations to (∇∗g∇− 2 ◦ R)h = 0 δgh = 0 trg(h) = 0. (2.10) The first jet h = d dt gt|t=0 of the family of Ricci-flat deformations gt satisfies h ∈ (g), and elements of the space (g) is therefore called infinitesimal Ricci-flat deformations. An element h ∈ (g) is said to be integrable if there exists a smooth family of metrics through g with linearization h. The operator ∇∗∇ − 2 ◦R is elliptic, so (g) is finite dimensional. Koiso showed that there exists a finite dimensional submanifold Z ⊂ Sg with TgZ = (g) and with Pg as a real analytic subset. All elements h ∈ (g) therefore integrate into a smooth curve of metrics in Z through g. Without further assumptions, then elements of (g) need not integrate into smooth curves of Ricci-flat metrics through g. Assume that (M,J, g) is a compact complex Ricci-flat Kähler manifold. Symmet- ric 2-tensors h ∈ C∞(Sym2(T ∗M)) split as h = hH + hA, where hH is hermitian, i.e. hH(JX, JY ) = h(X, Y ), and hA is skew-hermitian, i.e. hA(JX, JY ) = −hA(X, Y ). From the tensor h we construct a new tensor h ◦ J(X, Y ) = h(X, JY ). Hermitian in- finitesimal Ricci-flat deformations can be identified with real differential 2-forms of type (1, 1) via the correspondence hH 7→ hH ◦ J . The equations (2.10) for (g)H translate to trg hH = (hH ◦ J, ω) δghH = −δg(hH ◦ J) ◦ J 25 Chapter 2. Preliminaries and the usual real Laplacian ∆ satisfies ∆(hH ◦ J) = (∇∗g∇− 2 ◦ R)hH ◦ J. Skew-hermitian infinitesimal Einstein deformations can be identified with infinitesimal complex deformations via g ◦ I = hA ◦ J . They satisfy the following equations trg hA = 0 δghA = −J ◦ (∂¯∗I) and the complex Laplacian ∆∂¯ = ∂¯∂¯∗ + ∂¯∗∂¯ satisfies the Weitzenböck formula g ◦ (∆∂¯I) = (∇∗∇− 2 ◦ R)hA ◦ J. The operator∇∗g∇−2 ◦ R preserves the Hermitian and skew-Hermitian type. This together with the above equations can be used to show that the Hermitian and skew-Hermitian splitting holds also for (g), i.e. for any h ∈ (g) both hH and hA belong to (g). Denote by (g)H and (g)A the vector subspaces of (g) of Hermitian and skew-Hermitian tensors respectively. If the first Chern class vanishes and g is a Ricci-flat Kähler metric, then the dimension of the space of infinitesimal Ricci-flat deformations is dim (g) = dim (g)H + dim (g)A with dim (g)H = dimH1,1(M,R)−1 and dim (g)A = 2 dimCH1(M, TM)− 2 dimCH 0,2(M, TM). Koiso then showed that if all infinitesimal Ricci-flat deformations integrate into a smooth curves of Ricci-flat metrics through g, then the premoduli space Pg of Ricci-flat metrics in the slice Sg is a smooth manifold. Assume that all infinitesimal complex deformations of (M,J) are integrable and de- note by J the smooth parameter space of complex deformations of J . Denote by V the vector bundle over J for which Vt = H1,1Jt (M,R) for each t ∈ J . We need to show that all elements of (g) integrate into smooth curves of Ricci-flat Kähler metrics through g. Take h ∈ (g). To find a smooth curve of Ricci-flat Kähler metrics with first jet h we do the following. The tensor h can be related to an infinitesimal complex deformation I ∈ H1(M, TM). By assumption it integrates into a smooth curve of complex structures Jt. From Theorem 2.6.4 we know that for small t there exists a smooth curve of metrics gt through g which are Kähler with respect to Jt. Let ωt be the Kähler form g ◦ Jt. It turns 26 2.7. Koiso’s deformation theory on compact manifolds out that tκt is an appropriate section of V . It provides a smooth curve of Kähler forms ωt + tκt with respect to Jt. Solutions to the complex Monge-Ampére equation, (ωt + tκt + i∂∂¯u) m − Aef (ωt + tκt)m = 0, (2.11) produces Ricci-flat Kähler metric g˜t of Jt in the class of [ωt + tκt]. Isolating f in (2.11) provides a map F ((Jt, κt), u) = f . This map has a surjective differential so we can apply the implicit function theorem to obtain a smooth map ψ from a neighbourhood of (0, 0) in V to the slice Sg whose image consists of Ricci-flat Kähler metrics. It turns out that the map ψ is surjective onto a neighbourhood of g in Sg, so Ricci-flat Kähler metrics span an entire neighbourhood of g in the slice Sg. The curve ψ(Jt, κt) produces a smooth curve of Ricci-flat Kähler metrics in Sg through g with first jet h. The findings of Koiso can be summarized in the following two theorems. Theorem 2.7.1. Let (M,J) be a compact complex manifold with c1(J) = 0. Let g be a Ricci-flat Käher metric on M and assume that all infinitesimal complex deformations are integrable. Then any Ricci-flat deformation of g is Kähler, possibly with respect to a perturbed complex structure. This can be used to get a local understanding of the moduli space or Ricci-flat struc- tures R˜. Theorem 2.7.2. Let M be as in Theorem 2.7.1. Then in a neighbourhood of the Ricci-flat Kähler structure [g] the moduli space of Ricci-flat structures is, up to an action of a finite group, a finite dimensional manifold of dimension dimH1,1R (M,J)− 1 + 2 dimCH1(M, TM)− 2 dimCH0,2(M,J). Theorem 2.7.2 can be deduced from Theorem 2.7.1 via the following argument. The map from (2.7) produces a chart from a neighbourhood of [g] in the moduli space of Ricci- flat structures R˜ onto Pg/Iso(M, g). Let I(M, g)0 be the identity component of Iso(M, g). When the scalar curvature vanishes, then the identity component I0g acts trivially on Pg. The isometry group Iso(M, g) is a compact Lie group and I0g is a normal subgroup. The quotient Iso(M, g)/I0g is therefore a finite group. A model for the moduli space R˜ in a neighbourhood of a Kähler structure is therefore given by Pg/(Iso(M, g)/I0g ). As Pg 27 Chapter 2. Preliminaries spans an entire neighbourhood of g in the manifold Z, then TgPg = (g). The dimension of Pg is therefore dim (g)H + dim (g)A. 28 Chapter 3 Orbifold Ricci-flat deformations 3.1 Introduction and results In this chapter, we generalize results by Koiso [KOI83] to orbifolds. In Section 2.7 we explained how Koiso in [KOI83] showed that if M is a compact complex manifold with vanishing first Chern class and with all infinitesimal complex deformations integrable, then Ricci-flat deformations of a Ricci-flat Kähler metrics on M are Kähler, possibly with respect to a perturbed complex structure. He also showed that the moduli space of Ricci- flat structures in a neighbourhood of a Ricci-flat Kähler structure is, up to the action of a finite group, a finite dimensional manifold. In this chapter we prove that a similar state- ment holds if you replace the compact manifold in Koiso’s results with a compact orbifold. We also prove Ebin’s slice theorem for compact orbifolds and the Calabi conjecture for compact Kähler orbifolds. An orbifold (Definition 3.2.3) is a generalization of a manifold. The difference is that the local model on an orbifold is more complicated than on a manifold. Instead of being locally homeomorphic to an open subset of Euclidean space, orbifolds are locally homeo- morphic to a quotient of an open subset of Euclidean space by an action of a finite group. This group may have non-trivial isotropy, which leads to the presence of singularities on an orbifold. Orbifolds may be considered a natural class of objects as they appear in various con- struction on smooth manifolds. For instance, as we explained in Section 2.7, the moduli space of Einstein structures on a compact Kähler-Einstein manifold with non-positive first 29 Chapter 3. Orbifold Ricci-flat deformations Chern class and all infinitesimal complex deformations integrable also carries the structure of an orbifold. For other constructions involving orbifolds see for instance [BOY-GAL08]. For a compact complex Kähler-Einstein orbifold with vanishing first Chern class, we prove that integrability of all infinitesimal complex deformations implies that all infinites- imal Ricci-flat deformations integrate into smooth curves of Ricci-flat deformations. We use this to prove that a Ricci-flat deformation of a Ricci-flat Kähler metric is Kähler, pos- sibly with respect to a perturbed complex structure. We also show that the moduli space of Ricci-flat structures is, up to an action of a finite group, a finite dimension manifold and we find an expression for the dimension of this moduli space in a neighbourhood of a Kähler structure. To generalize results from manifolds to orbifolds there are in general two approaches. One is to use that the set of regular points Vreg in an orbifold V is a smooth manifold and that Vreg is a dense open subset. This way one can extend result from manifolds to orbifolds by continuity. The drawback of this approach is that the space Vreg is not compact. The other approach is to use that an orbifold locally is a quotient of a manifold by a finite group Γ. A problem on an orbifold can therefore locally be lifted to a Γ-invariant problem on a manifold. The drawback to the second method is that it is local in nature. A number of the tools we wish to make use of require compactness of the base space, so we will mostly employ the second method. We now present the main results of this chapter. Proper definitions of the objects and operators involved will be given in the relevant sections. Let (V, J) be a complex orbifold and denote by c1(J)orb the orbifold first Chern class in H2(V,R). If the orbifold has no singularities, i.e. is a manifold, then c1(J)orb coincides with the usual first Chern class of the manifold and it takes values in the image of the integer cohomology in H2(V,R), however, in general c1(J)orb takes values in the image of the cohomology with rational coefficients in H2(V,R). The Calabi conjecture is an important result about the existence of a Kähler metric with a prescribed Ricci-form on a compact Kähler manifold. In Sec- tion 3.5 we adapt the proof of the Calabi conjecture on compact Kähler manifolds from [JOY00, Chapter 5] to compact Kähler orbifolds. Theorem 3.1.1. Let (V, J, g) be a compact Kähler orbifold with Kähler form ω. Then for any real closed (1, 1)-form ρ′ ∈ 2pic1(J)orb there exists a unique Kähler metric g′ on (V, J) such that its Ricci-form is ρ′ and its Kähler form ω′ is cohomologous to ω. 30 3.1. Introduction and results Let V be a compact orbifold and denote byDorb the group of orbifold diffeomorphisms on V . For a Riemannian metric g on V denote by Iso(V, g)orb the subgroup of Dorb of isometries from (V, g) to itself. Denote byMorb the space of orbifold Riemannian metrics on V . Similarly to the manifold case, we make use of the existence of a local slice inM for the action ofDorb. This is an orbifold version of a similar result for compact manifolds proved by Ebin [EBI70]. Denote the action of η ∈ Dorb on g ∈Morb by A(η, g) = η∗g. Theorem 3.1.2. Let V be a compact orbifold. For each g ∈Morb there exists a submani- fold S = Sg ⊆Morb containing g for which 1. If η ∈ Iso(V, g)orb then A(η, S) = S. 2. If η ∈ Dorb satisfies A(η, S) ∩ S 6= ∅ then η ∈ Iso(V, g)orb. 3. There exists a neighbourhood U ⊆ Dorb/Iso(V, g)orb around the identity coset and a local section χ : U → Dorb such that F : U × S →Morb : (η, g) 7→ A(χ(η), g) is a homeomorphism onto a neighbourhood of g ∈Morb. The following two theorems are generalizations of the results by Koiso from [KOI83] to compact orbifolds and are the main results of this chapter. The first result concerns the stability of the Kähler property for Ricci-flat deformations of a Ricci-flat Kähler metric. It is proved at the end of Section 3.7. Theorem 3.1.3. Let (V, J, g) be a connected compact complex orbifold with vanishing orbifold first Chern class and with all infinitesimal complex deformations integrable. Let g be a Ricci-flat Kähler metric on (V, J). Then small Ricci-flat deformations of g are Kähler, possibly with respect to a perturbed complex structure. Many results about compact manifolds have already been generalized to compact orb- ifolds, so Theorem 3.1.3 may come as less of a surprise to some. It has for instance previously been stated without proof by Boyer and Galicki in [BOY-GAL08, Theorem 5.5.5]. For a complex orbifold (V, J) denote by TV the sheaf of holomorphic sections of the holomorphic tangent bundle. The second main result of this chapter is about the dimension of the moduli space of Ricci-flat structures in a neighbourhood of a Ricci-flat Kähler structure. It is proved in Section 3.8. 31 Chapter 3. Orbifold Ricci-flat deformations Theorem 3.1.4. Assume the hypothesis of Theorem 3.1.3. In a neighbourhood of the Ricci- flat Kähler structure [g] the moduli space of Ricci-flat structures is, up to an action of a finite group, a finite dimensional manifold of dimension dimH1,1R (V, J)− 1 + 2 dimCH1(V, TV )− 2 dimCH0,2(V, J). The remaining part of Chapter 3 consists of seven sections. In Section 3.2, 3.3 and 3.4 we provide an introduction to orbifolds and various known results about them. In Section 3.5 we give a proof of the Calabi conjecture for compact Kähler orbifolds (Theorem 3.1.1). In Section 3.6 we prove Ebin’s slice theorem for orbifolds (Theorem 3.1.2). In Section 3.7 and 3.8 we prove Theorem 3.1.3 and 3.1.4 respectively. 3.2 Orbifolds Orbifolds were first introduced by Satake in [SAT56] and [SAT57], where he named them V -manifolds. They were later renamed orbifolds by Thurston [THU78]. An orbifold is a generalization of a manifold. Locally, it is modelled on a quotient of an open subset of Euclidean space by the action of a finite group. A good introduction to orbifolds can be found in [BOY-GAL08, Chapter 4]. In this section we present basic theory about orbifolds and orbifold bundles. The material is borrowed from the mentioned sources and [BAI56]. Definition 3.2.1. Let V be a topological space and let U¯ ⊂ V be an open subset. We say that a triple {U,Γ, φ} is a local uniformizing system or an (orbifold) chart for U¯ if for some n > 0 it satisfies i) U is a connected neighbourhood of 0 in Rn. ii) Γ ⊂ GL(n,R) is a finite subgroup, such that for all γ ∈ Γ, γ(U) ⊂ U and the dimension of the set fixed by γ is less than or equal to n− 2. iii) the map φ : U → U¯ is continuous and it satisfies φ(x) = φ(γ.x) for all γ ∈ Γ and induces a homeomorphism φΓ : U/Γ→ U¯ . Definition 3.2.2. Let {U,Γ, φ} and {U ′,Γ′, φ′} be local uniformizing systems. An injec- tion is a smooth linear injective map λ : U → U ′ satisfying φ′ ◦ λ = φ. 32 3.2. Orbifolds Definition 3.2.3. An orbifold atlas on a Hausdorff and second countable topological space X is a family of orbifold charts {Ui,Γi, φi} with X = ∪i φi(Ui) and such that for any {Ui,Γi, φi} and {Uj,Γj, φj} with some x ∈ φi(Ui) ∩ φj(Uj) there exist an orbifold chart {Uk,Γk, φk} with x ∈ φk(Uk) and injections λki : Uk → Ui and λkj : Uk → Uj . An atlas F is said to be a refinement of an orbifold atlas G if there exists an injection of every orb- ifold chart of F into some orbifold chart of G. Two orbifold atlases are equivalent if they have a common refinement. An orbifold is a Hausdorff and second countable topological space equipped with an equivalence class of orbifold atlases. We will use the symbol V for an orbifold. We denote the collection of all local uni- formizing systems on an orbifold V by FV and the collection of all injections by LV . Observe that for a chart {U,Γ, φ} each γ ∈ Γ is itself an injection. For a chart {U,Γ, φ} the isotropy subgroup of Γ at x ∈ U is Γx = {γ ∈ Γ | γ.x = x}. For a chart {U,Γ, φ}we say that x ∈ U is a singular point if Γx is non-trivial, i.e. |Γx| > 1. A singularity is a point p ∈ U¯ ⊂ V for which φ(x) = p for a singular point x ∈ U . We denote the set of all singularities on V by Vsing. Points in V \Vsing are called regular and we denote them by Vreg. The space Vreg is an open dense subset of V . For orbifolds V andW we define a local orbifold map to be a collection {fU}{U,Γ,φ}∈FV of maps such that for each {U,Γ, φ} ∈ FV there exists a {U∗,Γ∗, φ∗} ∈ FW with a map fU : U → U∗ satisfying that for each injection λ ∈ LV with λ : U → U ′ there exists an injection λ∗ ∈ LW with λ∗ : U∗ → U∗′ such that the following diagram commutes U fU // λ  U∗ λ∗  U ′ fU′ // U∗ ′ The local orbifold map {fU}FV gives rise to a map f : V → W satisfying f ◦φ = φ∗ ◦ fU for each chart {U,Γ, φ} ∈ FV (with the above notation). Such maps are called orbifold maps and we say they are of class Ck, Lpk, C ∞ etc. if each local orbifold map is. Definition 3.2.4. Let V andE be orbifolds, let F be a smooth manifold and let pi : E → V be a smooth map. Let G be a Lie group acting on F . We say (V,E, pi, F,G), or simply 33 Chapter 3. Orbifold Ricci-flat deformations pi : E → V , is an orbifold bundle if it satisfies the following conditions: i) There is a one-to-one correspondence FV ↔ FE such that if {U,Γ, φ} corresponds to {U∗,Γ∗, φ∗} then U∗ = U × F and if we denote the projection U∗ → U by piU∗ , then pi ◦ φ∗ = φ ◦ piU∗ . ii) Let {U,Γ, φ} ↔ {U∗,Γ∗, φ∗} and {U ′ ,Γ′ , φ′} ↔ {U∗′ ,Γ∗′ , φ∗′} with φ(U) ⊆ φ(U ′). Then φ∗(U∗) ⊆ φ∗′(U∗′) and there exists a one-to-one correspondence between in- jections λ : U → U ′ and λ∗ : U∗ → U∗′ such that for (p, q) ∈ U∗ = U × F we have λ∗(p, q) = (λ(p), σλ(p)q) where σλ : U → G is a smooth map satisfying σµλ(p) = σµ(λ(p))σλ(p) for injections λ : U → U ′ and µ : U ′ → U ′′. We call the collection (V,E, pi, F,G) for an orbifold bundle. A section of an orbifold bundle (V,E, pi, F,G) is an orbifold map {sU} from V to E such that each sU : U → U × F is a section in the usual sense and such that for each injection λ ∈ LV with λ : U → U ′ we have, using the notation from Definition 3.2.4, that sU ′ ◦ λ = λ∗ ◦ sU . We denote the space of for example smooth sections of an orbifold bundle E by C∞(E)orb. For an orbifold V we fix coordinates {u1, . . . , un} for each chart {U,Γ, φ}. Let F = Rn and let G = GL(n,R). For each injection λ : U → U ′ let σλ be the Jacobian matrix of λ at p, i.e. σλ(p) = ( ∂ u′i ◦ λ ∂uj ) . [SAT57, Theorem 1] then tells us that (V,E, pi,Rn,GL(n,R)) is an orbifold bundle. We call this bundle the tangent bundle of V . More generally, we can construct an (r, s)-orbifold tensor bundle by setting σλ(p) = ( ∂u′i ∂uj ) × . . .︸ ︷︷ ︸ r × ( ∂uj ∂u′i ) × . . .︸ ︷︷ ︸ s where× denotes the Kronecker product of matrices and where we set F = Rn(r+s) and let G = GL(n,R) operate on F as an (r, s)-tensor representation. We denote by T (r,s)V the (r, s)-tensor bundle on V . 34 3.2. Orbifolds For a chart {U,Γ, φ}we denote by Ωk(U) the usual differential k-forms and by Ωk(U)Γ the subset of differential k-forms ωU on U satisfying (ωU)p((p,X1), . . . , (p,Xk)) = (ωU)γ.p((γ.p, σγ(p)X1), . . . , (γ.p, σγ(p)Xk)), i.e. the differential k-forms ωU is invariant under the action of Γ. We say that ω = {ωU}FV is an orbifold differential k-form on V if for each chart {U,Γ, φ} the form ωU is in Ωk(U)Γ and for each injection λ : U → U ′ the form ωU satisfies (ωU)p(X1, . . . , Xk) = (ωU ′)λ(p)(σλ(p)X1, . . . , σλ(p)Xk). We denote the space of orbifold differential k-forms on V by Ωk(V )orb. For a (k, 0)-tensor ω = {ωU} the tensors Sym(ωU) and Alt(ωU) are Γ-invariant and compatible with injections, so they are well-defined operations on orbifolds. Similarly, the tensor product, wedge product and symmetric product are well-defined on orbifolds. The usual exterior differential operator d preserves Γ-invariance and is compatible with injections, so d : Ωp(U)Γ → Ωp+1(U)Γ lifts to define an exterior differential on Ω∗(V )orb. Orbifolds admit Riemannian metrics g and volume forms. A Riemannian metric on V is an element g = {gU}F(V ) ∈ C∞(Sym2(T ∗V ))orb such that for each orbifold chart {U,Γ, φ} the gU is a Γ-invariant Riemannian metric and such that each injection λ : {U,Γ, φ} → {U ′,Γ′, φ′} is an isometry, i.e. λ∗gU ′ = gU . The last statement is equivalent to a reduction of the structure group of V from GL(n,R) to O(n). We denote the volume form of g on V by dV orbg . An orbifold V is orientable if the coordinate systems of its charts can be chosen con- sistently such that det ( ∂u ′i◦λ ∂uj ) > 0 for each injection λ ∈ FV . This is equivalent to a reduction of the structure group to SO(n). Many differential geometric constructions are naturally compatible with the action of the isotropy groups, and the definition of orbifolds in general, for the following simple reason: For a chart {U,Γ, φ} let f be a smooth function on U . The tangent space (TpU)Γ at p consist of thoseX ∈ TpU which are invariant under the action of the JacobianDγp for all γ ∈ Γ. If f satisfies f(γ.p) = f(p) then D(f ◦ γ)p(X) = Dfγ.p ◦Dγp(X) = Dfp(X). For an orientable orbifold V define the integral ∫ V η of an n-form η in the following way. If the closure of the set {p ∈ V | ηp 6= 0} is contained in φ(U) for a chart {U,Γ, φ}, 35 Chapter 3. Orbifold Ricci-flat deformations set ∫ V η := 1 NΓ ∫ V ηU , where NΓ is the order of the isotropy group in Γ. In general, define the integral of an n-form η to be ∫ V η = ∑ i ∫ V fiη, where {fi} is a partition of unity satisfying that each fi has support inside φ(U) for some chart {U,Γ, φ}. Similarly, for a Riemannian orbifold (V, g) we define the integral of a smooth function f : V → R by ∫ V f dV orbg . We denote the volume of a Riemannian orbifold (V, g) by vol(g) orb. As remarked in [BOY-GAL08, p. 113], Stokes theorem holds on orbifolds. We denote by H∗dR(V ) the de Rham cohomology of Ω ∗(V )orb. Satake proved in [SAT56, Theorem 1] that the de Rham cohomology on an orbifold V computes the usual cohomology with real coefficients, i.e. for all 0 ≤ k ≤ dim(V ) we have HkdR(V ) ' Hk(V,R). In some way this shows that H∗dR(V ) is not a suitable cohomology theory for the study of orbifolds, as it does not ’see’ the isotropy. For the purpose of this project it will suffice though. For other approaches see [BOY-GAL08, p. 117-118] We will end our introduction to the basic concepts here. But many more differential geometric constructions generalize to orbifolds. For instance the construction of connec- tions, covariant derivatives and curvature generalizes to orbifolds. So does the existence of Levi-Civita covariant derivatives, the Bianchi identity and Hodge decomposition. See [BAI56] for details. 3.3 Tools from analysis In this section we introduce basic analysis on orbifolds. Let (V, g) be a compact oriented Riemannian orbifold. Denote by Lp(V )orb the space of integrable orbifold maps f : V → R with finite Lp-norm, ||f ||Lp = (∫ V |f |pdV orbg )1/p. Define the Sobolev spaceLpk(V ) orb to be the set of thoseLp-orbifold functions which are k- times weakly differentiable such that |∇rf | ∈ Lp(V )orb for 0 ≤ r ≤ k. We equipLpk(V )orb with the Sobolev norm ||f ||Lpk = (∑k i=0 ∫ V |∇if |pdV orbg )1/p . For k ≥ 0, denote by Ck(V )orb the space of bounded and continuous functions from V to R which have k-times continuous bounded derivatives. We equip it with the norm ||f ||Ck = ∑k i=0 supV |∇if |. Let α ∈ (0, 1). We say that a function V → R is Hölder continuous with exponent α if [f ]α = supx 6=y∈V |f(x)−f(y)| d(x,y)α < ∞. We define the Hölder space Ck,α(V )orb as the space of those Ck-functions for which [∇if ]α exists and is finite for each i ≤ k. The four constructions Lp(V )orb, Lpk(V ) orb, Ck(V )orb, and Ck,α(V )orb generalize to the space of 36 3.3. Tools from analysis sections of an orbifold vector bundle (V,E, pi, F,G) in the same way as on manifolds. We denote the corresponding spaces of sections by Lp(E)orb, Lpk(E) orb, Ck(E)orb, and Ck,α(E)orb . Theorem 3.3.1 (Sobolev Embedding Theorem). Let (V,E, pi, F,G) be an orbifold vector bundle over a compact orbifold V with dim(V ) = n. For all l > n 2 + s there exists a continuous linear embedding L2l (E) orb → Cs(E)orb. Theorem 3.3.2 (Kondrashov’s Theorem). Let (V, g) be a compact n-dimensional orbifold. Let k, l be integers with k ≥ l ≥ 0. Let q, r ≥ 1 be real numbers and let α ∈ (0, 1). If 1 q < 1 r + k − l n then the embedding Lqk(V ) orb → Lkl (V )orb is compact. If 1 q < k − l − α n then Lqk(V ) orb → C l,α(V )orb is compact. Also Ck,α(V )orb → Ck(V )orb is compact. We remark that while orbifolds admit singularities, then the spaces of sections of orb- ifold vector bundles do not. They are vector spaces just as on manifolds. Though not a very profound observation, then it is nevertheless central. It ensures that the analysis on the spaces of sections of orbifold bundles is very similar to that of manifolds. We see an immediate consequence of this in the next proposition. Proposition 3.3.3. Let (V, g) be a compact orbifold and let k ≥ 0, p ≥ 1 and α ∈ (0, 1), then L2k(E) orb is a Hilbert space. Lpk(E) orb, Ck(E)orb and Ck,α(E)orb are Banach spaces and C∞(E)orb is a Fréchet space. Proof. The spaces in question are vector spaces under the usual addition f + g = {fU + gU}, where {fU} and {gU} are the local orbifold maps of f and g respectively. First consider Ck(E)orb. Let fi = {(fi)U} be a Cauchy sequence in Ck(E)orb. For each l.u.s. {U,Γ, φ} the space Ck(U,E) is a Banach space so the sequence {(fi)U} converges to a section fU ∈ Ck(U,E). For γ ∈ Γ, the function ||(fi)U(x) − (fi)U(γ.x)||Ck is continuous in x, so ||fU(x) − fU(γ.x)||Ck = limi→∞ ||(fi)U(x) − (fi)U(γ.x)||Ck = 0. Hence fU ∈ Ck(U,E)Γ for each l.u.s. and each γ ∈ Γ, so f = {fU} ∈ Ck(E)orb. 37 Chapter 3. Orbifold Ricci-flat deformations The same argument shows that Ck,α(E)orb and L2k(E) orb are Banach spaces with their respective norms. Furthermore the norm on L2k(E) orb is generated by the inner product (f, g) = ∑ |α|≤k(∇αf,∇αg)L2(E)orb . Equipping C∞(E)orb with the family of semi-norms from each Ck(E)orb turns it into a Fréchet space. We define the linearization of a differential operator P on orbifolds in the same way as on manifolds. LuP (v) = lim t→0 P (u+ tv)− P (u) t . The same argument as in Proposition 3.3.3 shows that LuP is well defined on orbifolds. The next three results are central for the study of elliptic operators. The first one is elliptic regularity and the third one is the Fredholm alternative. Theorem 3.3.4. Let (V, g) be a compact orbifold and letE1, E2 be orbifold vector bundles over V of the same dimension. Let P be a smooth linear elliptic differential operator of order k from E1 to E2. Let α ∈ (0, 1) and let l ≥ 0. Assume that we have u ∈ L1(E1)orb and v ∈ L1(E2)orb such that Pu = v. If v ∈ C∞(E2)orb then u ∈ C∞(E1)orb. If v ∈ C l,α(E2)orb then u ∈ Ck+l,α(E1)orb and ||u||Ck+l,α ≤ C(||v||Cl,α + ||u||C0). for some C > 0 independent of v and w. Proof. Direct adaptation of the proof of [JOY00, Theorem 1.4.1]. Theorem 3.3.5. Let E1, E2 be orbifold vector bundles over a compact orbifold V , and let P be a smooth linear elliptic operator of order k from E1 to E2. Then P acts by P : C∞(E1)orb → C∞(E2)orb, P : Ck+l,α(E1)orb → C l,α(E2)orb and P : Lpk+l(E1)orb → Lpl (E2) orb. Then kernel ker(P ) of P is the same for all of these actions, and it is a finite- dimensional vector subspace of C∞(E1)orb. Proof. Direct adaptation of the proof of [JOY00, Theorem 1.5.1]. Theorem 3.3.6. Let (V, g) be a compact orbifold and letE1, E2 be orbifold vector bundles over V , equipped with metrics in the fibres, and P is a smooth linear elliptic operator of order k from E1 to E2. Let l ≥ 0 be an integer and let p > 1, and let α ∈ (0, 1). Then the 38 3.4. Complex orbifolds images of the maps P : Ck+l,α(E1)orb → C l,α(E2)orb and P : Lpk+l(E1)orb → Lpl (E2)orb are closed linear subspaces of C l,α(E2)orb and L p l (E2) orb respectively. If v ∈ C l,α(E2)orb then there exists u ∈ Ck+l,α(E1)orb with Pu = v if and only if v ⊥ ker(P ∗), and if one requires that u ⊥ ker(P ) then u is unique. Similarly, if v ∈ Lpl (E2)orb then there exists u ∈ Lpl+k(E1)orb with Pu = v if and only if v ⊥ ker(P ∗), and if u ⊥ ker(P ) then u is unique. Proof. Direct adaptation of the proof of [JOY00, Theorem 1.5.3]. 3.4 Complex orbifolds In this section, we introduce concepts related to complex orbifolds. The material is either borrowed from [BAI56], [BOY-GAL08, Chapter 4] or adapted to orbifolds from [HUY05, Chapter 6]. We define a complex orbifold (V, J) to be an even dimensional real orbifold of real dimension 2m with local charts {U,Γ, φ} where U ⊂ Cm and Γ is a finite subgroup of GL(m,C) such that JU is a complex structure on U compatible with Γ and J = {JU}F(V ) is compatible with injections. The construction of Dolbeault cohomology generalizes to orbifolds. For a l.u.s. {U,Γ, φ} denote by A∗JU (U)Γ the Γ-invariant complex differential forms on U . Denote by A∗J(V )orb the complex differential forms on (V, J). The (p, q)- Dolbeault cohomology group is denoted by Hp,q(V ). A Kähler metric g on (V, J) is an orbifold Riemannian metric for which the fundamental form ω(X, Y ) = g(JX, Y ) is d-closed. The presence of a Kähler metric on (V, J) is equivalent to a reduction of the structure group to U(m). Baily proved in [BAI56] the Hodge decomposition theorem for orbifolds. From this it follows that the ∂∂¯-Lemma extends to orbifolds. Lemma 3.4.1 (Global ∂∂¯-Lemma). Let (V, J, g) be a compact Kähler orbifold and let η be a smooth exact real (1, 1)-form on V . Then there exists an f ∈ C∞(V )orb such that η = i∂∂¯f . We can give the orbifold V the structure of a ringed space (V,OV ) over C in the following way. For x ∈ V take a chart {U,Γ, φ} for which x ∈ φ(U). The stalk Ox is isomorphic to the local ring of germs of Γ-invariant holomorphic functions on U . Let OV be the structure sheaf with these stalks. Following [MOE-PRO97, Section 2] we define 39 Chapter 3. Orbifold Ricci-flat deformations an orbifold sheaf, or orbisheaf, to be a sheaf F on the orbifold V satisfying i) for each chart {U,Γ, φ}, FU is a sheaf on U and ii) for each injection λ : {U,Γ, φ} → {U ′,Γ′, φ′} there exists an isomorphism of sheaves Fλ : FU → λ∗FU ′ . The construction of sheaf cohomology for orbisheaves goes through on orbifolds and for an orbisheaf F we denote by H∗(V,F) the orbifold sheaf cohomology of it. Denote by TV the orbisheaf of sections of the holomorphic tangent bundle T 1,0V . We define the structure orbisheaf OorbV of the orbifold V to be the orbisheaf defined by the structure sheaf OU for each chart {U,Γ, φ}. By [BOY-GAL08, Proposition 4.2.18] isomorphism classes of orbifold vector bundles and locally free orbisheaves are in a one-to-one correspondence. The canonical orbisheaf KorbV is defined as the orbisheaf det(ΩV ) = Ω m V . The Dolbeault cohomology H p,q(V ) computes the sheaf cohomology of the sheaf ΩpV , i.e. H p,q(V ) ' Hq(V,ΩpV ). On a complex manifold M the first Chern class of a complex vector bundle pi : E → M is a class [ i 2pi trC(F∇)] ∈ H2(M,R) for a covariant derivative ∇ on E. This class is independent of the choice of covariant derivative and is an invariant of the complex structure on E. The first Chern class of M is the first Chern class of the holomorphic tangent bundle T 1,0M . The class c1(M,J) is in the image of integer cohomology. This Chern-Weil construction of Chern classes goes through on orbifolds just as on manifolds, and we have Definition 3.4.2. Let (V, J) be a complex orbifold and pi : E → V a complex vector bundle. Let ∇ be a covariant derivative on E. Define the orbifold first Chern class of ∇ on E to be c1(E,∇)orb = [ i 2pi trC(F∇)] ∈ H2dR(V ) ' H2(V,R). Define the orbifold first Chern class of (V, J) to be c1(T 1,0V )orb. It only depends on the complex structure J , so we denote it by c1(J)orb. In [BOY-GAL08, Section 4.4] they explain the interplay between the orbifold first Chern class c1(J)orb and the singularities on a complex orbifold V . In this thesis we will only use c1(V )orb constructed via Chern-Weil theory so it is not essential for our work, but as many differential geometric constructions extend naturally to orbifolds, we thought it would be appropriate to at least briefly mention the relation outlined in [BOY-GAL08, Sec- tion 4.4]. It is an example of a generalization to orbifolds which requires some more care. 40 3.4. Complex orbifolds A Baily divisor is a collection of divisors {DU}F(U) such that for each chart {U,Γ, φ}, DU is a Cartier divisor on U satisfying i) if for each x ∈ V and each γ ∈ Γ, f ∈ Dγx then f ◦ γ ∈ Dx and ii) if λ : U → U ′ is an injection and f ∈ D′λx then f ◦ λ ∈ Dx. Here D is the divisor sheaf of the Baily divisor and Dx is the stalk of D at x. A branch divisor is a Weil divisor on V with coefficients in Q of the form ∑ ι ( 1− 1 mι ) Dι where the sum is taken over all Weil divisors Dι in Vsing and mα = gcd{|Γx|}x∈Dα is the ramification index of Dα. By [BOY-GAL08, Proposition 4.4.13] all branch divisors lift to a Baily divisor via φ∗Dα for each chart {U,Γ, φ}. Define a canonical divisor DV to be any divisor on V such that the line bundle ofDV ∩Vreg is the canonical bundleKVreg . If Vsing = ∅ the line bundle LDV of DV is the canonical bundle KV . If Vsing 6= ∅ then KV is in general not defined. By [BOY-GAL08, Proposition 4.4.15] the canonical divisor DorbV is related to the divisor DV viaDorbV = φ ∗DV + ∑ α ( 1 + 1 mα ) φ∗Dα, on each chart {U,Γ, φ}. The first Chern class of a Baily divisor is defined as the first Chern class of the corresponding complex line bundle. The first Chern class c1(J)orb is not in the image of the inclusion H∗(V,Z) → H∗(V,R), but instead corb1 (J) is in the image of H ∗(V,Q)→ H∗(V,R) ([BOY-GAL08, p. 120-121]) and by [BOY-GAL08, (4.4.2)] the orbifold first Chern class is related to the first Chern class c1(LDV) via c1(V ) orb = c1(LDV )− ∑ α ( 1− 1 mα ) c1(LDα). (3.1) It follows from (3.1) that if the orbifold V does not admit any branch divisors in its singular locus, then c1(V )orb coincide with c1(LDV ) and if Vsing = ∅ then c1(LDV ) recovers the usual first Chern class of the smooth manifold V . On a complex orbifold (V, J, g) the fundamental form of a J-invariant symmetric bi- linear tensor h is the real (1, 1)-form ψ(X, Y ) = h(JX, Y ). The fundamental form of the Ricci-tensor Ric(X, Y ) is the Ricci-form ρ(X, Y ) ∈ A1,1R (V, J)orb given by ρ(X, Y ) = Ric(JX, Y ). The proof of [HUY05, Proposition 4.A.11] generalizes in a straightforward manner to Kähler orbifolds, so the Ricci-form ρ satisfies ρ(X, Y ) = i trC(F∇). Hence, ρ ∈ 2pic1(J)orb. A real (1, 1)-form α ∈ A1,1R (V, J)orb is positive, vanishing or negative if the corresponding real symmetric J-invariant form a is positive definite, vanishing or neg- ative definite respectively. A class in H2dR(V,R) is positive, vanishing or negative if it can be represented by a positive, vanishing or negative 2-form respectively. Calabi [CAL54] conjectured in 1954 that to any closed real (1, 1)-form ρ in the first Chern class of a com- 41 Chapter 3. Orbifold Ricci-flat deformations pact Kähler manifold (M,J, g) there exists a unique Kähler metric with ρ as its Ricci-form and whose Kähler form is cohomologous to the Kähler form of g. We provide a gener- alization of this result to compact Kähler orbifolds. The Calabi conjecture for orbifolds is Theorem 3.4.3 (Calabi Conjecture, orbifolds). Let (V, J, g) be a compact Kähler orbifold with Kähler form ω. Then for any closed real (1, 1)-form ρ with [ρ] = 2pic1(J)orb there ex- ists a unique Kähler metric g′ such that its Kähler form ω′ satisfies [ω′] = [ω] ∈ H2(V,R) and ρ′ is the Ricci-form of g′. The claim that the proof of Theorem 3.4.3 goes through largely unchanged on orb- ifolds has been made both in [JOY00, Theorem 6.5.6] and in [BOY-GAL08, Theorem 5.2.2]. While this might be clear to Joyce and Boyer-Galicki, we feel that it is nevertheless worthwhile to actually give a proof. We have included a proof of Theorem 3.4.3 in Sec- tion 3.5. We follow the proof of the Calabi conjecture for manifolds outlined in [JOY00, Chapter 5]. The continuity method works, as predicted by Joyce, and the theorem can be proved by locally viewing the orbifold as a quotient of a manifold by a finite group Γ and locally lifting the problem to a Γ-invariant problem on a manifold. We check that all local constructions preserve the Γ-invariance and glue to global constructions. Deformation theory of complex structures Deformation theory of complex structures on complex orbifolds can be defined similarly to the deformation theory for complex structures on manifolds we introduced in Section 2.6. For our purpose, the parameter space of deformations will always be smooth, so we can make do with a simplified smooth version of the deformation theory. Definition 3.4.4. Let X be a complex orbifold and S a smooth manifold with a proper holomorphic map pi : X → S. The fibres Xt = pi−1(t) are compact complex suborbifolds of X and we say that pi : X → S is a smooth family of complex orbifolds parametrized by S. If we fix 0 ∈ S and restrict the family pi : X → S from Definition 3.4.4 to a germ around 0, then it can be viewed as a smooth family of deformations of the compact com- plex orbifold V = X0. This family of deformations can be trivialized as a differentiable 42 3.4. Complex orbifolds family X ' S × V . In this way Xt is diffeomorphic to X0 for all t and pi : X → S can in- stead be viewed as a family of deformations of the complex structure J on (V, J). Kodaira and Spencer proved in [KOD-SPE60, Theorem 15] that for a compact Kähler manifold (M,J), small deformations are Kähler. This was generalized to orbifolds by El Kacimi Alaoui in [KAC88]. Theorem 3.4.5 (El Kacimi Alaoui). Let (V, J, g) be a compact Kähler orbifold and assume that there exists a smooth family of deformations pi : X → S of V . Then it admits a smooth family of compatible Kähler metrics gt. Next we introduce infinitesimal complex deformations. A complex structure J satisfies the two equations J2 = −Id and N(J) = 0, where N(J) is the Nijenhuis tensor, so the linearization I = d dt Jt|t=0 ∈ C∞(TV ⊗ T ∗V )orb satisfies the two equations 0 = d dt (−Id)|t=0 = ddtJ2t |t=0 = IJ + JI and 0 = N ′J(I) = 12J ◦ ∂¯I . For X ∈ T 1,0V we have JIX = −IJX = −iIX so IX ∈ T 0,1V and I ∈ A0,1(T 1,0V )orb. But the tensor I also satisfies ∂¯I = 0, so it is an element of H1(V, TV ). Tensor fields I ∈ C∞(TX ⊗ T ∗X) satisfying the two equations IJ + JI = 0 and N ′J(I) = 0 and which are not of the form LXJ are called (essential) infinitesimal complex deformations and we denote the space of such deformations by ICD(J)orb. We say that an infinitesimal complex deformation I is integrable if there exists a smooth curve of deformation Jt with J0 = J for which I = d dt Jt|t=0. An orbifold V is said to have unobstructed deformations if all infinitesimal complex deformations integrate into a smooth curve of deformations. Proposition 3.4.6. Let (V, J) be a compact complex orbifold. Then there is a natu- ral bijection between the space of infinitesimal complex deformations ICD(J)orb and H1(V, TV ). If H2(V, TV ) = 0 then V has unobstructed deformations. Proof. adaptation to orbifolds of [KOD-NIR-SPE58, Theorem, p. 452]. For a compact orbifold (V, J) and a smooth family of complex deformations Jt, the orbifold first Chern class c1(Jt)orb is stable for small deformations. This is because the denominator mα for each branch divisor is stable under small deformations, so c1(Jt)orb, despite taking values in the image of H2(V,Q) in H2(V,R), actually only make integer value jumps. Definition 3.4.7. A Calabi-Yau orbifold is a compact Kähler orbifold with trivial canoni- cal bundle. 43 Chapter 3. Orbifold Ricci-flat deformations A Calabi-Yau orbifold V has structure group in SU(m) and vanishing orbifold first Chern class. It also satisfies H2(V, TV ) = 0, so all infinitesimal complex deformations are integrable. This follows by adapting [HUY05, Prpoisition 6.1.11] to orbifolds. 3.5 Calabi conjecture Calabi [CAL54] conjectured in 1954 that for any closed real (1, 1)-form ρ′ in 2pic1(J) of a compact Kähler manifold (M,J, g) there exists a unique Kähler metric g′ with ρ′ as its Ricci-form and whose Kähler form is cohomologous to the Kähler form of g. The uniqueness part of the theorem was proved by Calabi himself. The existence part of the theorem was proved later by Yau [YAU78]. This is a fundamental existence result about Kähler metrics with prescribed Ricci-forms on Kähler manifolds. In particular the result tells us that on any Kähler manifold with vanishing first Chern class there exists a Ricci- flat Kähler metric. Joyce claimed in [JOY00, section 6.5] that the proof can be generalized to compact Kähler orbifolds by viewing an orbifold locally as a quotient of a manifold by a finite group Γ. Following this approach we adapt the proof of the Calabi conjecture for compact Kähler manifolds presented in [JOY00, chapter 6] to compact Kähler orbifolds using the orbifold first Chern class c1(J)orb. The Calabi Conjecture for orbifolds was stated in Section 3.1, but we restate it here for the convenient of the reader. We prove Theorem 3.5.1. Let (V, J, g) be a compact Kähler orbifold with Kähler form ω. Then for any closed real (1, 1)-form ρ′ with [ρ′] = 2pic1(J)orb there exists a unique Kähler metric g′ such that its Kähler form ω′ satisfies [ω′] = [ω] ∈ H2(V,R) and ρ′ is the Ricci-form of g′. The fist step of the proof is to reformulate Theorem 3.5.1 as an existence result for solutions to a complex Monge-Ampére equation. The reformulation hinges on the ∂∂¯- Lemma (Lemma 3.4.1) and ωm being a volume form on V . Both hold on orbifolds and we get the following reformulation of Theorem 3.5.1. Theorem 3.5.2. Let (V, J, g) be a compact Kähler orbifold with Kähler form ω. Then for any f ∈ C3,α(V )orb there exists A > 0 and φ ∈ C5,α(V )orb satisfying the following two conditions i) ∫ V φ dV orbg = 0, 44 3.5. Calabi conjecture ii) (ω + ddcφ)m = Aefωm. In a local holomorphic coordinate chart {U,Γ, φ} with coordinates {z1, . . . , zm} the equation (ω + ddcφ)m = Aefωm becomes det ( gαβ¯ + ∂2φ ∂zα∂z¯β¯ ) = Aef det(gαβ¯). (3.2) This equation is a complex Monge-Ampére equation. It is a non-linear partial differential equation of second order in φ. Note that it follows from part ii) that ω + ddcφ is a positive definite real (1, 1)-form. For the rest of this section let (V, J, g) be a compact complex Kähler orbifold of complex dimension m and let ω be the Kähler form of g. We use the notation C = C(X, . . . , Z) to express that a constant C only depends on the parameters X, . . . , Z. Definition 3.5.3. Fix α ∈ (0, 1) and f ∈ C3,α(V )orb. Define S to be the set of those t ∈ [0, 1] for which there exists a φ ∈ C5,α(V )orb with ∫ V φ dV orbg = 0 and an A > 0 satisfying the equation (ω + ddcφ)m = Aetfωm. The proof of Theorem 3.5.2 is based on the continuity method. The idea is to show that 1 ∈ S. The space S is non-empty as for t = 0 the function 0 ∈ C5,α(V )orb is a solution to (3.2). We show that S = [0, 1] by showing that it is both open and closed in [0, 1]. To show that S is open we use the inverse mapping theorem for Banach spaces. Let X ⊆ C5,α(V )orb be the vector space consisting of those φ for which ∫ V φ dV orbg = 0 and let U = {φ ∈ X |ω + ddcφ is a positive (1, 1)-form}. Suppose φ ∈ U and a ∈ R, then there exists a unique real function f on V such that (ω + ddcφ)m = ea+fωm on V , and as φ ∈ C5,α(V )orb, then f ∈ C3,α(V )orb. Define a function F : U × R→ C3,α(V )orb : (φ, a) 7→ f, where (ω + ddcφ)m = ea+fωm. The expression of the map F in local holomorphic coor- dinates {z1, . . . , zm} is F (φ, a) = log det ( gαβ¯ + ∂2φ ∂zα∂z¯β¯ ) − log det(gαβ¯)− a = f. The expression of the Laplacian on a Kähler orbifold is the same as on a Kähler manifold, i.e. ∆φ = −gαβ¯∂α∂¯β¯φ. The Jacobi formula also holds on orbifolds. Denote by gφ the 45 Chapter 3. Orbifold Ricci-flat deformations metric gαβ¯ + ∂2φ ∂zα∂z¯β¯ . The linearization of F in (φ, a) is the map DF(φ,a) : X × R → C3,α(V )orb given by DF(φ,a)(u, b) = L(φ,a)F (φ+ tu, a+ tb) = d dt F (φ+ tu, a+ tb)|t=0 = d dt log det(gφ+tu)|t=0 − d dt (a− tb)|t=0 = 1 det(gφ+tu) d dt det(gφ+tu)|t=0 − b (3.3) = 1 det(gφ+tu) det(gφ+tu)tr((gφ+tu)−1 d dt gφ+tu)|t=0 − b = tr((gφ+tu)−1 d dt ∂α∂¯β¯(tu))|t=0 − b = tr((gφ)−1∂α∂¯β¯u)− b = −∆gφu− b. To show that (u, b) 7→ −∆gφu− b is surjective onto C3,α(V )orb take v ∈ C3,α(V )orb. The operator ∆gφ is elliptic, so solutions in C 5,α(V )orb to ∆gφu − b = v exist by the orbifold version of Theorem 2.1.8 if (v + b) ⊥ ker(∆∗gφ). The Laplace operator is self-adjoint, so ∆∗gφ(v) = e f∆gφ(e −fv), and so ker(∆∗gφ) is the constant multiples of e −f . We can always choose b ∈ R such that (v+b) ⊥ e−f . A solution (u, b) to−∆gφu−b = v therefore always exists. DF |(φ,a) is therefore an invertible linear map. It is furthermore a homeomorphism, so the inverse function theorem for Banach spaces (Theorem 2.1.9) applies in the same way as in the proof of [JOY00, Theorem C3]. This allows us to conclude. Theorem 3.5.4. Fix α ∈ (0, 1) and suppose that f ′ ∈ C3,α(V )orb and φ′ ∈ C5,α(V )orb and A′ > 0 satisfy the equations∫ V φ′ dV orbg = 0 and (ω + dd cφ′)m = A′ef ′ ωm. Then for every f ∈ C3,α(V )orb with ||f − f ′||C3,α sufficiently small, there exist φ ∈ C5,α(V )orb and A > 0 such that∫ V φ dV orbg = 0 and (ω + dd cφ)m = Aefωm. (3.4) 46 3.5. Calabi conjecture Corollary 3.5.5. The set S is open in [0, 1]. Proof. Let t0 ∈ S, i.e. ∃φ ∈ C5,α(V )orb and A > 0 such that (ω + ddcφ)m = Aet0fωm. If we take t ∈ [0, 1] sufficiently close to t0, then ||tf − t0f ||C3,α = |t − t0| ||f ||C3,α is arbitrarily small, so by Theorem 3.5.4 there exist φ ∈ C5,α(V )orb and A > 0 such that (ω + ddcφ)m = Aetfωm. Hence t ∈ S. The set S is therefore open. To show that S is closed we prove that the limit point of any convergent sequence in S is in S. Yau used some hard a priori third order norm estimates to show this. The orbifold version of this is summarized in the next theorem. Theorem 3.5.6. Let Q1 ≥ 0. Then there exist Q2, Q3, Q4 ≥ 0 depending only on V, J, g and Q1 such that the following holds: Suppose f ∈ C3(V )orb, φ ∈ C5(V )orb and A > 0 satisfy the equations ||f ||C3 ≤ Q1, ∫ V φ dV orbg = 0, and (ω − ddcφ)m = Aefωm. Then ||φ||C0 ≤ Q2, ||ddcφ||C0 ≤ Q3 and ||∇ddcφ||C0 ≤ Q4. Proof. To prove the zero’th order estimate the idea is to find an Lk-bound on φ for each k and use this to bound the C0-norm of φ. The Sobolev embedding theorem (Theorem 3.3.1), Stokes theorem and Hölder’s inequality are used to find constants C4, Q2 > 0 depending only on V , g and Q1 such that ||φ||Lk ≤ Q2(C4k)−m/k for all k ≥ 2. The theorems involved generalize to orbifolds and so does the properties of the constants C4 and Q2. The function φ is continuous and V is compact, so ||φ||C0 = limk→∞ ||φ||Lk . Hence ||φ||C0 is bounded by Q2 as limk→∞ ||φ||Lk ≤ limk→∞Q2(C4k)−m/k = Q2. The second order estimate ||ddcφ||C0 ≤ Q3 is based on a pointwise bound |ddcφ|2g ≤ 2m+ 2(m−∆φ)2 depending only on m and ||∆φ||C0 and a C0-bound on ∆φ depending only on m, Q1, Q2 and g. Both bounds use a number of local inequalities involving ∇φ, ∆φ and Riemann curvature tensor R, which all have identical expressions on orbifolds and manifolds. The estimates in [JOY00, section 5.4] therefore generalize to orbifolds. The third order estimate ||∇ddcφ||C0 ≤ Q4 for a positive constant Q4 depending only on V, J, g and Q1 is a lengthy calculation, but uses only similar methods to the previous estimates and therefore also generalizes to orbifolds. The C0-bound of ∆φ is used to produce a C0-bound on a non-negative function S ∈ C∞(V )orb given by 4S2 = |∇ddc|2g′ . 47 Chapter 3. Orbifold Ricci-flat deformations From the relation |∇ddcφ|2g′ ≤ C3/2|∇ddcφ|2g′ , where C is a constant depending only on V , J , g and Q1, and the C0-bound on S, we get the bound Q4 on ||∇ddc||C0 . Theorem 3.5.7. Let Q1, Q2, Q3, Q4 ≥ 0 and α ∈ (0, 1). Then there exists Q5 ≥ 0 depending only on V, J, g,Q1, . . . , Q4 and α, such that the following holds. Suppose f ∈ C3,α(V )orb, φ ∈ C5(V )orb and A > 0 satisfy (ω + ddcφ)m = Aefωm and the inequalities ||f ||C3,α ≤ Q1, ||φ||C0 ≤ Q2, ||ddcφ||C0 ≤ Q3 and ||∇ddcφ||C0 ≤ Q4. Then φ ∈ C5,α(V )orb and ||φ||C5,α ≤ Q5. Also, if f ∈ Ck,α(V )orb for some k ≥ 3 then φ ∈ Ck+2,α(V )orb, and if f ∈ C∞(V )orb then φ ∈ C∞(V )orb. The proof of Theorem 3.5.7 is based on the inductive process known as bootstrapping. The first part of the proof is to generalize three regularity results with Schauder type norm bounds to the orbifold setting. The first lemma is an application of Theorem 3.3.4. Lemma 3.5.8. Let k ≥ 0 and α ∈ (0, 1). Then there exists a positive constant E1 = E1(k, α, V, g) such that if φ ∈ C2(V )orb satisfies ∆φ = f for some f ∈ Ck,α(V )orb, then φ ∈ Ck+2,α(V )orb and ||φ||Ck+2,α ≤ E1 (||∆φ||Ck,α + ||φ||C0). Lemma 3.5.9. Let α ∈ (0, 1). Then there exists a constant E2 = E2(α, V, g, ||g′ab||C0 , ||g′ab||C0,α) > 0 such that if φ ∈ C2(V )orb satisfies ∆′φ = f for some f ∈ C0(V )orb, then φ ∈ C1,α(V )orb and ||φ||C1,α ≤ E2 (||∆′φ||C0 + ||φ||C0). Proof. Let f ∈ C0(V )orb and assume we have a solution φ ∈ C2(V )orb to ∆′φ = f . V is compact, so we can choose two finite collections of charts {B1(0),Γ1,i, φ1,i} and {B2(0),Γ2,i, φ2,i} covering V such that for each i they satisfy φ1,i(B1(0)) ⊂ φ2,i(B2(0)). Remember that Ck(B2(0))Γ ⊂ Ck(B2(0)) for all k, so the solution to ∆′φ = f locally satisfies φ ∈ C2(B2(0)). For each i Theorem 1.4.3 of [JOY00] tells us that φ|B1(0) ∈ C1,α(B1(0)) and that there exists Di = Di(m,α) > 0 such that the C1,α-norm of φ|B1(0) satisfies ||φ|B1(0)||C1,α < Di(||∆′φ||C0 + ||φ||C0). But φ|B1(0) is Γ-invariant, and therefore 48 3.5. Calabi conjecture φ|B1(0) ∈ C1,α(B1(0))Γ. Setting E2 = supi{Di} gives us the desired constant for the Schauder estimate for ∆′ on V . Lemma 3.5.10. Let k ≥ 0 and α ∈ (0, 1), then there exists a constant E3 = E3(k, V, α, g, ||g′ab||C0 , ||g′ab||C0,α) > 0 such that if φ ∈ C2(V )orb satisfies ∆′φ = f for f ∈ Ck,α(V )orb, then φ ∈ Ck+2,α(V )orb and ||φ||Ck+2,α ≤ E3 (||∆′φ||Ck,α + ||φ||C0). Proof. The proof is similar to that of Lemma 3.5.9 except that we use equation (1.14) instead of (1.13) from Theorem 1.4.3 of [JOY00]. Proof of Theorem 3.5.7. This is proved via the inductive process known as bootstrapping. It only uses already established results on orbifolds and norm bound estimates similar to what we have already seen, so it generalizes to orbifolds. We summarize the argument. In the proof of Theorem 3.5.6 we established a bound on ||∆φ||C0 . From Lemma 3.5.8 and Lemma 3.5.9 it then follows that φ ∈ C3,α(V )orb and that ||φ||C3,α is bounded by a constant depending only on V, g, J,Q1, . . . , Q4 and α. Lemma 3.5.8 and Lemma 3.5.10 then show that if f ∈ Ck,α(V )orb then φ ∈ Ck+2,α(V )orb. An inductive argument together with Lemma 3.5.8 applied to ∆φ shows that ∆φ ∈ Ck,α(V )orb. From this we get an a priori bound on ||φ||Ck+2,α depending only on V, g, J,Q1, . . . , Q4 and α. As f ∈ Ck,α(V )orb implies φ ∈ Ck+2,α(V )orb, f smooth implies φ smooth. Corollary 3.5.11. The set S is closed in [0, 1]. Proof. Consider a sequence {tj}∞j=0 ⊂ S with limit t′ ∈ [0, 1]. By definition there are sequences {φj}∞j=0 and Aj > 0 satisfying i) and ii) of Theorem 3.5.2. By Theorem 3.5.6 and 3.5.7 the sequence {φj}∞j=0 is bounded in C5,α(V )orb. By Kondraschov’s Theorem for orbifolds (Theorem 3.3.2) there exists a convergent subsequence {φjk}∞k=0 with limit φ′ ∈ C5(V )orb which satisfies i) and ii) of Theorem 3.5.2. From Theorem 3.5.6, and 3.5.7 it follows that the limit is indeed in C5,α(V )orb. The set S therefore contains its limit points and it is therefore closed in [0, 1]. This concludes the proof of Theorem 3.5.2. It remains to be shown that the metric found in Theorem 3.5.1 is indeed unique. 49 Chapter 3. Orbifold Ricci-flat deformations Theorem 3.5.12. Let (V, J) be a compact complex orbifold and g a Kähler metric with Kähler form ω. Let f ∈ C1(V )orb. Then there is at most one function φ ∈ C3(V )orb satisfying ∫ V φ dV orbg = 0 and (ω + dd cφ)m = Aefωm on V for A > 0. Proof. This is an application of Stokes’ theorem with a straight forward generalization to orbifolds and we therefore omit the proof. See [JOY00, Theorem C4] for details. 3.6 Slice construction In this section, we construct a slice for the action of the diffeomorphism group in the space of metrics on a compact orbifold. This is a generalization of a similar slice construction for compact manifolds made by Ebin [EBI70]. A slice around a point is a subset containing the point such that each orbit passing through the subset has exactly one representative in it. Slices are usually constructed by exponentiating the orthogonal complement of the tangent space of the orbits. We call the equation that has this orthogonal complement as its kernel the slice equation. We will follow the same line of arguments as Ebin but work equivariantly on the orb- ifold charts and carefully check that all constructions carry over to orbifolds. The appli- cation of the slice construction we have in mind does not require a specific slice, which allows us to simplify the argument by Ebin considerably by avoiding a number of technical arguments about smoothness of particular bundles. The fact that Proposition 3.3.3 holds on orbifolds as well as manifolds is not a profound statement, but is nevertheless a central ingredient that makes the generalization of Ebin’s construction from compact manifolds to compact orbifolds work. Let (V, g) be a compact orbifold and denote byMorb the space of smooth (orbifold) Riemannian metrics on V . It is an open subset of C∞(Sym2(T ∗V ))orb. In Section 3.3 we gave Ck(Sym2(T ∗V ))orb the topology of uniform convergence and it is a Banach space in this norm. The space of smooth sections is the Fréchet space C∞(Sym2(T ∗V ))orb = ∩∞k=0Ck(Sym2(T ∗V ))orb. Denote by (·, ·)⊗ig the inner product on Sym2(T ∗V ) ⊗ (T ∗V )⊗i induced by the metric g. The space L2k(Sym 2(T ∗V ))orb is a Hilbert space with respect to the inner product (η, γ)L2k = ∑k i=0 ∫ V (∇iηp,∇iγp)⊗ig dV orbg . Denote by CkMorb the space 50 3.6. Slice construction of Ck (orbifold) Riemannian metrics in Ck(Sym2(T ∗V ))orb and define Morbk = L2k(Sym2(T ∗V ))orb ∩ C0Morb. Lemma 3.6.1. Let (V, g) be a compact orbifold of dimension n and let k > n/2. Then Morbk is a Banach manifold inside L2k(Sym2(T ∗V ))orb. Proof. The space C0(Sym2(T ∗V ))orb is a Banach space by Proposition 3.3.3. The space C0Morb is open in C0(Sym2(T ∗V ))orb. Theorem 3.3.1 tells us that for k > n/2 the map L2k(Sym 2(T ∗V ))orb → C0(Sym2(T ∗V ))orb is continuous, so L2k(Sym2(T ∗V ))orb is open in C0(Sym2(T ∗V ))orb. The spaceMorbs is therefore the intersection of two open sets in C0(Sym2(T ∗V ))orb. Consider the product orbifold F = V ×V as a trivial fibre bundle over V with orbifold fibres V . Ck sections of this bundle are exactly the Ck orbifold maps from V to V . We equip the space ofCk-sections of this bundle, Ck(F )orb, with the topology of uniform con- vergence. The space of Ck-diffeomorphisms from V to V can alternatively be described as CkDorb = {f ∈ Ck(F )orb | f has an inverse and f−1 ∈ Ck(F )orb}. Define Dorbk = L2k(F )orb ∩ C1Dorb. Lemma 3.6.2. For k > n/2 + 1 the space Dorbk is a Hilbert manifold. Proof. Theorem 3.3.1 applies to sections of the bundle F = V × V . The inclusion L2k(F ) orb → C1(F )orb is therefore continuous and L2k(F )orb ∩ C1(F )orb is an open subset in L2k(F ) orb. Proposition 3.3.3 applies to L2k(F ) orb as well, so Dorbk is an open subset in a Hilbert space. The group Dorbk+1 acts onMorbk by pull-back. We denote this action by A : Dorbk+1 ×Morbk →Morbk : (η, g) = η∗g. 51 Chapter 3. Orbifold Ricci-flat deformations The space of (smooth) diffeomorphisms Dorb acts on Morb in the same way. We also denote that action by A. For (V, g) compact and k ≥ n 2 , the Banach manifoldMorbk has tangent space L2k(Sym 2(T ∗V ))orb. The inner product onMorbk that we get from the L2k- inner product (η, γ)g L2k is invariant under the Dorbk+1-action. The argument is identical to the manifold case. See [EBI70, p.20-21] for details. We denote by Ig the isometry group {η ∈ Dorb | η∗g = g} of the metric g ∈ Morb. Before proving Theorem 3.1.2 we start by producing a slice for the action of Dorbk+1 onMorbk for k > n/2 + 2. Theorem 3.6.3. Let V be a compact orbifold and let k > n/2 + 2. For each g ∈ Morb there exists a submanifold S = Sg ⊆Morbk containing g for which i) If η ∈ Ig then A(η, S) = S. ii) If η ∈ Dorbk+1 satisfies A(η, S) ∩ S 6= ∅ then η ∈ Ig. iii) There exists a neighbourhood U ⊆ Dorbk+1/Ig around the identity coset and a local section χ : U → Dorbk+1 such that F : U × S →Morbk : (η, g) 7→ A(χ(η), g) is a homeomorphism onto a neighbourhood of g inMorbk . The construction of the cross-section χ is identical to the one for manifolds. We refer the reader to [EBI70, Proposition 5.10] for details. Lemma 3.6.4. The projection map pi : Dorbk+1 → Dorbk+1/Ig admits a smooth local cross section at any η ∈ Dorbk+1/Ig. Denote by Okg the orbit (Dorbk+1)∗g = (Dorbk+1/Ig)∗g. Denote by ν the normal bundle to T (Okg ) in T (Morbk )|Okg with respect to the restricted metric fromMorbk and let exp be the exponential map on the manifoldMorbk with respect to the inner product given above. We are now ready to prove Theorem 3.6.3. Proof of Theorem 3.6.3. let U ′ ⊂ Okg be an open subset such that it has a section χ : U ′ → Dorbk+1. For ′ > 0 let V ′ = {v ∈ νg | ||v||g ≤ (′)2} and W ′ = {d(η∗)v | v ∈ V ′ and η ∈ χ(U ′)}. We can choose ′ > 0 such that exp |W ′ is a diffeomorphism onto its image and exp(W ′) ∩ Okg = U ′. For some δ > 0 we have B2δ(g) ⊂ exp(W ′). Now choose U ⊂ U ′ 52 3.6. Slice construction and  < ′ and let V and W be defined as before but with  and U in place of ′ and U ′. Let  and U be chosen such that exp(W ) ⊂ Bδ(g). Define S = exp(V ). Claim 3.6.5. The set S has the three properties of a slice. For the first property, let η ∈ Ig and x ∈ S with x = exp(v) for some v ∈ V . Then η satisfies η∗g = g so η is also an isometry for the inner product onMorbk . Hence ||η∗v||g = ||v||g so η∗v ∈ V . For any isometry φ ∈ Ig the exponential map satisfies φ ◦ expp(v) = expφ(p)(dφp(v)) and η∗ is linear so dη∗ = η∗. Hence, η∗ expg(v) = expη∗g(dη ∗(v)) = expη∗g(η ∗v). It follows that η∗x = η∗ exp(v) = exp(η∗v) ∈ S. For the second property, let η ∈ Dorbk+1 and assume we have x, y ∈ S with η∗(x) = y. Denote by d(x, y) the distance function onMorbk given by the metric onMorbk introduced above. We have to show that η ∈ Ig. First η∗x ∈ S and S ⊂ W so d(η∗x, g) < δ. η∗ is an isometry, so d(η∗x, η∗g) = d(x, g) < δ. This gives us, d(g, η∗g) ≤ d(g, y) + d(y, η∗g) = d(g, y) + d(η∗x, η∗g) = δ + d(x, g) < 2δ. So η∗g ∈ exp(W ′). The exponential map exp |W ′ is a diffeomorphism onto a neighbour- hood of g in Morbk and satisfies that for x, y ∈ S with y = η∗x both a, b ∈ W and exp(b) = y = η∗x = η∗ exp(a) = exp(η∗a). By injectivity of exp we have b = η∗a, and as a, b ∈ V ′, the definition of W ′ tells us that η ∈ Ig as η does not move a out of the V ′. For the third property, let U ⊆ Okg , χ : U → Dorbk+1 and W be given as above. As exp |W is a diffeomorphism onto a neighbourhood of g, then F : U × S → Morbk : (u, x) 7→ A(χ(u), x) is a bijection onto exp(W ) and is continuous. Now, let z ∈ exp(W ) and let pi : ν → Okg be the bundle projection map, then we can express the inverse of F as follows F−1(z) = (pi ◦ exp−1(z) , A((χ ◦ pi ◦ exp−1(z))−1, z) ). (3.5) This map is continuous so F is a homeomorphism onto a neighbourhood of g in Morbk . This concludes the proof of Claim 3.6.5. For the convenience of the reader we restate Theorem 3.1.2 here. It is a smooth version 53 Chapter 3. Orbifold Ricci-flat deformations of Theorem 3.6.3. Theorem 3.6.6. For each g ∈ Morb there exists a submanifold S = Sg ⊆ Morb contain- ing g for which i) If η ∈ Ig then A(η, S) = S. ii) If η ∈ Dorb satisfies A(η, S) ∩ S 6= ∅ then η ∈ Ig. iii) There exists a neighbourhood U ⊆ Dorb/Ig around the identity coset and a local section χ : U → Dorb such that F : U × S →Morb : (η, g) 7→ A(χ(η), g) is a homeomorphism onto a neighbourhood of g ∈Morb. Proof. The sets S and U from Theorem 3.6.3 we rename Sk and Uk respectively and use S and U to denote S = Sk ∩Morb and U = Uk ∩ Dorb. We show that S has the three properties of a slice. For the first property, let η ∈ Ig. The metric η∗g′ is smooth for each g′ ∈ S. In particular A(η, S) = S. For the second property, any η ∈ Dorbk+1 satisfies the second property of a slice by Theorem 3.6.3. In particular, any η ∈ Dorb for which A(η, S) ∩ S 6= ∅ satisfies η ∈ Ig. For the third property, the map χ defined in Theorem 3.6.3 maps U to Dorb and gives a local cross section χ : U → Dorb : aIg 7→ a. Denote by ν the normal bundle to the tangent bundle of the orbit (Dorb/Ig)∗g inside T (Morb). Denote by V ⊂ νg the subset satisfying exp(V ) = S. Let W = {d(η∗)y | y ∈ V, η ∈ χ(U)}. The exponential map is a local diffeomorphism and the map F satisfies F (U ×S) = exp(W )∩Morb, so it is continuous and bijective onto a neighbourhood of g inMorb. From Equation (3.5) it follows that F−1 is continuous. The map F is therefore a homeomorphism onto its image. This concludes the proof. For a compact orbifold (V, g) the set S constructed above provides a slice in Morb through g for the action of Dorb/Ig, i.e. S is homeomorphic to a neighbourhood of [g] inMorb/(Dorb/Ig). The map S × D/Ig → Morb : (s, ηIg) 7→ χ(ηIg)∗s constructed in Theorem 3.6.6 induces a homeomorphism S/Ig → Morb/Dorb onto a neighbourhood of 54 3.7. Ricci-flat deformations [g]. This construction restricts to the volume 1-metrics and produces a homeomorphism (S ∩Morb1 )/Ig →Morb1 /Dorb onto a neighbourhood of [g]. So we have, Corollary 3.6.7. Let (V, g) be a compact orbifold, then S/Ig is homeomorphic to a neigh- bourhood of the structure [g] in Morb/Dorb. This construction restricts to a slice (S ∩ Morb1 )/Ig for the action of Dorb onMorb1 . 3.7 Ricci-flat deformations In this section, we study Ricci-flat deformations and infinitesimal Ricci-flat deformations of Ricci-flat metrics. At the end of this section we give a proof of Theorem 3.1.3, which is the first of the two main results of Chapter 3. Let (V, g) be a compact orbifold. A smooth curve of deformations of a metric g is a smooth curve g : (−, ) → C∞(Sym2(T ∗V ))orb : t 7→ gt with g0 = g. Any such family of deformations can be written as gt = g+ht for a smooth curve ht ∈ C∞(Sym2(T ∗V ))orb with h0 = 0. To keep notation simple we denote both the individual deformations and the smooth curve of deformations by gt. Positive definiteness is an open condition, so for a deformation gt with |ht|g < 1 small enough, the deformation gt is positive definite. In the following we will often be discussing small deformations gt, i.e. deformations gt for which  > 0 is sufficiently small, or equivalently that |ht|g is sufficiently small for the given context. A smooth curve of Einstein deformations of an Einstein metric g is a smooth curve of deformations gt such that gt is Einstein for each t. The Einstein operator E introduced in Definition 2.6 generalizes to a map E :Morb1 → C∞(Sym2(T ∗V ))orb : g 7→ Ric(g)− Tg n g, where Tg = ∫ V sg dV orb g is the total scalar curvature of g. An orbifold metric g is Einstein exactly when E(g) = 0. One difference between the operator E on an orbifold and on a manifold is the order of isotropy appearing as a denominator in the total scalar curvature on orbifolds as explained in Section 3.2. We will be working with Ricci-flat metrics, so this difference will not play a role for our purpose. The operators d,∇g, δg, δ∗g ,∆L and ◦ R as defined in Section 2.3 generalize naturally to orbifolds following the constructions in 55 Chapter 3. Orbifold Ricci-flat deformations Section 3.2. We will often omit the subscript g from these operators when there is no risk of confusion. From now on assume that the compact orbifold (V, g) is Ricci-flat. The operators involved have the same expressions on manifolds and orbifolds and the expression for the linearized Einstein operator is the same on manifolds and orbifolds. Let gt be a smooth curve of deformations of g and let h ∈ C∞(Sym2(T ∗V ))orb be the first jet h = d dt gt|t=0 = d dt (g + ht)|t=0 = ddtht|t=0. From Equation 2.8 it follows that the linearization of E(gt) is d dt E(gt)|t=0 = Ric′g(h) = 1 2 ∆L(h)− δ∗gδgh− 1 2 ∇gd(trgh) (3.6) = ( 1 2 ∇∗∇− ◦R ) h− δ∗δh− 1 2 ∇d(trgh). (3.7) Assume that the metric g has volume 1. By Theorem 3.6.6 there exists a slice Sg ⊂Morb1 through g. Denote by P = Pg the set of all Ricci-flat metrics in S = Sg. It is called the premoduli space of Ricci-flat metrics. We equip it with the subspace topology. Definition 3.7.1. Let (V, g) be a compact Ricci-flat orbifold. Define the space (g)orb ⊂ C∞(Sym2(T ∗V ))orb by (g)orb = {h ∈ C∞(Sym2(T ∗V ))orb |h satisfies i), ii) and iii)} where i) Ric′g(h) = 0 ii) δh = 0 iii) ∫ V trgh = 0. (3.8) The first jet of a smooth curve of Ricci-flat deformations gt of g belongs to the space (g)orb. Without further assumptions the space (g)orb might contain elements that are not first jets of Ricci-flat deformations of g. Later we will impose sufficient conditions on V and g such that every element of (g)orb is indeed a first jet of a Ricci-flat deformation of g. With this in mind we name (g)orb the space of infinitesimal Ricci-flat deformations. Lemma 3.7.2. Let (V, g) be a compact orbifold. A tensor field h ∈ C∞(Sym2(T ∗V ))orb 56 3.7. Ricci-flat deformations belongs to (g)orb if and only if it satisfies (∇∗∇− 2 ◦R)h = 0 δh = 0 trgh = 0. (3.9) Proof. (3.9) ⇒ (3.8): It is immediate that the equations in (3.9) imply conditions i), ii) and iii) of Definition 3.7.1. (3.8) ⇒ (3.9): Denote by Hess(f) the (2, 0)-tensor ∇gd(f). To see that i), ii) and iii) of Definition 3.7.1 imply equations (3.9) first observe that δh = 0 and Ric′g(h) = 0 simplifies equation (3.6) to 0 = Ric′g(h) = 1 2 (∆Lh−Hess(trgh)). Remember that trg(∆Lh) = ∆(trgh) and trg(Hess(f)) = −∆f . If we take trace on both sides of the equation ∆Lh = Hess(trgh), then [BER-EBI69, p. 388-389] gives us ∆(trgh) = trg(Hess(trgh)) = −∆(trgh). The function trgh is therefore harmonic. The only harmonic functions on V , however, are the constant functions, so ∫ V trgh = 0 implies trgh = 0. From Equation (3.7) it follows that (∇∗∇− 2 ◦R)h = 0. Lemma 3.7.3. Let (V, g) be a compact Ricci-flat orbifold. The space (g)orb is finite dimensional. Proof. The space (g)orb is in the kernel of the elliptic operator ∇∗∇ − 2 ◦R and it is therefore finite dimensional by Theorem 3.3.5. Theorem 3.7.4. Let (V, g) be a compact Ricci-flat orbifold. Then there exists a finite dimensional real analytic submanifold Z ⊂ Sg with TgZ = (g)orb and with Pg as a real analytic subset. Proof. Fix k > n + 2 and let Skg ⊂ Morbk be the slice generated by Theorem 3.6.3 and let Sk1 be the subset of volume 1 metrics. The elliptic operator F = 2Ric ′ g + 2δ ∗δ = ∆L − ∇dtrg : L2k(Sym2(T ∗V ))orb → L2k−2(Sym2(T ∗V ))orb is Fredholm, so it has closed image. By the same argument as in [BES87, Lemma 12.48] F (TgSk1 ) is closed as well. The operator Ric satisfies Ric′g(TgS k 1 ) = F (TgS k 1 ) and it is real analytic according to 57 Chapter 3. Orbifold Ricci-flat deformations [DeT-KAZ81]. The space L2k(Sym 2(T ∗V ))orb is a Hilbert space by Proposition 3.3.3 and we can therefore apply Theorem 2.1.11 to conclude that there exists a neighbourhood U of g in Sk1 such that U ∩ Pg is a real analytic subset of a real analytic submanifold Z ⊂ U whose tangent space TgZ coincides with ker(Ric′g) ∩ Tg(Sk) = 2k(g)orb. This is true for every k > n+ 2. In particular, it is true for the slice S1. By Lemma 3.7.3 the space (g)orb is finite dimensional. For the rest of this section, let (V, J) be a compact complex orbifold of complex di- mension m. For h ∈ C∞(Sym2(T ∗V ))orb define the tensor h ◦ J(X, Y ) = h(X, JY ). A tensor field h ∈ C∞(Sym2(T ∗V ))orb is said to be Hermitian if h(JX, JY ) = h(X, Y ) and skew-Hermitian if h(JX, JY ) = −h(X, Y ). Any h ∈ C∞(Sym2(T ∗V ))orb can be written as h = hA + hH with hH Hermitian and hA skew-Hermitian. For a smooth curve of complex deformations Jt, I = ddtJt|t=0 ∈ C∞(TX ⊗ T ∗X) satisfies −Id = J2t and N(Jt) = 0, where N(Jt) is the Nijenhuis tensor of Jt. Dif- ferentiating −Id = J2t gives us 0 = IJ + JI , and differentiating N(Jt) = 0 gives 0 = N ′J(I) = 1 2 J ◦ ∂¯I . The tensor g ◦ I satisfies g ◦ I(JX, JY ) = g(JX, IJY ) = −g(JX, JIY ) = −g ◦ I(X, Y ), so it is skew-Hermitian. We say that I ∈ ICD(J)orb is symmetric or anti-symmetric if g ◦ I is symmetric or anti-symmetric respectively. Denote by Is and Ia the symmetric part and skew-symmetric part of I respectively. Both Ia and Is belong to ICD(J)orb. Denote by ICD(J)orbS and ICD(J) orb A the subspace of symmetric and skew-symmetric infinitesimal complex deformations respectively. For a Hermitian symmetric tensor hH , the tensor ψ = hH ◦ J satisfies ψ(X, Y ) = hH(X, JY ) = hH(JX, J 2Y ) = −hH(Y, JX) = −ψ(Y,X), so it is a real differential 2- form. Also, Jψ(X, Y ) = ψ(JX, JY ) = hH(JX, J2Y ) = −h(JX, Y ) = −h(J2X, JY ) = h(X, JY ) = i1−1ψ(X, Y ), so ψ is of type (1, 1). Hence ψ = hH ◦ J ∈ A1,1R (V, J)orb. It will be convenient to rewrite the equations from (3.9) for a Hermitian tensor and a skew-Hermitian tensor respectively. We start with the Hermitian case. Let hH ∈ C∞(Sym2(T ∗V )) be a Hermitian tensor and let ψ = hH ◦ J . The description of Käh- ler metrics from Remark 2.5.2 holds also on orbifolds, so the metric g is Kähler with respect to the complex structure J exactly if∇J = 0, or equivalently if J∇YX = ∇Y JX for all X, Y ∈ TpV and for all p ∈ V . Letting {Yi}2mi=1 be an orthonormal frame and using 58 3.7. Ricci-flat deformations this relation gives us −(d∗ψ) ◦ J(X) = −(∇∗ψ)(JX) = − 2m∑ i=1 ∇Yiψ(Yi, JX) = − 2m∑ i=1 Yi(ψ(Yi, JX))− ψ(∇YiYi, JX)− ψ(Yi,∇YiJX) = − 2m∑ i=1 Yi(hH(Yi, J 2X))− hH(∇YiYi, J2X)− hH(Yi, J∇YiJX) = 2m∑ i=1 Yi(hH(Yi, X))− hH(∇YiYi, X)− hH(Yi,∇YiX) = ∇∗hH(X) = δghH(X) with the conclusion δghH = −(d∗ψ) ◦ J. (3.10) Let ω be the Kähler form of g with respect to J . As for manifolds, a local calculation shows that trghH = (ψ, ω)g. (3.11) Similarly to [BES87, (12.92’)], there is a Weitzenböck formula for Hermitian tensors on the Ricci-flat orbifold (V, g) ∆ψ = (∇∗∇− 2 ◦R)hH ◦ J. (3.12) A skew-Hermitian symmetric tensor hA can be identified with an infinitesimal complex deformation in the following way. Let I ∈ End(TV )orb be the real endomorphism satis- fying hA ◦ J = g ◦ I. (3.13) 59 Chapter 3. Orbifold Ricci-flat deformations This endomorphism I satisfies g(X, IJY ) = hA(X, J 2Y ) = −hA(X, Y ) = hA(JX, JY ) = g(JX, IY ) = −g(JX,−IY ) = −g(JX, J2IY ) = −g(X, JIY ), (3.14) which translates to g(X, (IJ + JI)Y ) = 0 and I therefore anti-commutes with J . This implies, for X ∈ T 0,1V , that J(IX) = −IJX = −I(−iX) = iIX , so IX ∈ T 1,0V and I : T 0,1V → T 1,0V . I may therefore be regarded as an element of A0,1(T 1,0V )orb. The tensor field ∂¯∗I ∈ C∞(T 1,0V )orb is, by [BES87, (12.93)], related to δghA ∈ C∞(T ∗V )orb via δghA = −J ◦ ∂¯∗I. (3.15) In particular, δghA = 0 exactly when I is ∂¯∗-closed, i.e. when I determines a class in H1(V, TV ). We need a version of the Weitzenböck formula for the complex Laplacian ∆∂¯ . The Weitzenböck formula is essentially proved by expanding the definition of ∆∂¯ = ∂¯∗∂¯ − ∂¯∂¯∗. The expression of ∂¯ and ∂¯∗ and the way they act on tensors is unchanged on orbifolds. The involved operators ∇,∇∗ and ◦R also have the same expressions on orbifolds as they had on manifold. The Weitzenböck decomposition on manifolds found in [BES87, equation (12.93’)] therefore extends to orbifolds. We have (∇∗∇− 2 ◦R)hA ◦ J = g ◦∆∂¯I. (3.16) Hence (∇∗∇− 2 ◦R)hA = 0 exactly when I ∈ A0,1(T 1,0V )orb is in the kernel of ∆∂¯ . Proposition 3.7.5. Let (V, J, g) be a compact Ricci-flat Kähler orbifold and let h ∈ (g)orb be decomposed as h = hH + hA. Then hH , hA ∈ (g)orb. Proof. We show that hA ∈ (g)orb. It then follows that hH = h− hA ∈ (g)orb as (g)orb is a vector space. We have 0 = (∇∗∇− 2 ◦R)h = (∇∗∇− 2 ◦ R)hH + (∇∗∇− 2 ◦ R)hA. The operator (∇∗∇− 2 ◦R) preserves type so the skew-Hermitian tensor (∇∗∇− 2 ◦ R)hA must vanish. The Weitzenböck formula (3.16) gives us ∆∂¯I = 0, which in turn implies 60 3.7. Ricci-flat deformations ∂¯I = 0. It then follows from (3.15) that δghA = 0. The tensor hA is skew-hermitian, so its trace is necessarily zero. The tensor hA satisfies the three equations in (3.9), so hA ∈ (g)orb. For the compact Kähler orbifold (V, J, g) let Jt be a smooth curve of deformations of J and let gt be a corresponding smooth curve of deformations of g such that each gt is Kähler with respect to Jt. Denote by h the tensor field ddtgt|t=0 and by I the tensor field d dt Jt|t=0. Identify the (1, 1)-tensor I = I ij with the (2, 0)-tensor g ◦ I and denote by Iij the components of g ◦ I . According to [KOI83, section 9] the tensor fields h and I are related via 2ihαβ + (Iαβ + Iβα) = 0, (3.17) 2i(Dαhβγ¯ −Dβhαγ¯) = Dγ¯(Iαβ − Iβα), (3.18) where we use Greek indices as explained in the preliminaries. We say that a symmetric (2, 0)-tensor h and an infinitesimal complex deformation I satisfying (3.17) and (3.18) are Kähler related. The space (g)orb can be written as (g)orb = (g)orbH ⊕ (g)orbA , where (g)orbH and (g)orbA denote the subspaces of (g) orb of Hermitian and skew-Hermitian tensor fields respectively. In the following two subsections the subspaces (g)orbH and (g) orb A will be studied separately. Hermitian symmetric 2-tensors Let (V, J, g) be a compact Ricci-flat Kähler orbifold. We explained above that for a Her- mitian symmetric (2, 0)-tensor h, the tensor h◦J is a real differential 2-form of type (1, 1). For h ∈ (g)orbH the Weitzenböck formula (3.12) gives us ∆(h ◦ J)(X, Y ) = (∇∗∇− 2 ◦R)h(X, JY ) = 0. This way we have identified elements of (g)orbH with elements of H1,1R (V, J). Denote by ω the Kähler form of g. We saw in (3.11) that trgh = (h ◦ J, ω)g. The tensor h is trace-free, so the form h ◦ J is orthogonal to ω and therefore so is [h ◦ J ] to [ω]. We explained in Section 3.2 that for compact V we have H1,1R (V, J) ' H1,1R (V, J). The map 61 Chapter 3. Orbifold Ricci-flat deformations (g)orbH → H1,1R (V, J)/R · ω : h 7→ h ◦ J is linear. If h1 ◦ J = h2 ◦ J then h1 = h2 so the map is injective. To see that the map is surjective take ψ ∈ H1,1R (V, J)/R · ω and define h(X, Y ) = ψ(JX, Y ). Differential forms are skew-symmetric, so h is symmetric. The form ψ satisfies ψ(JX, JY ) = i1−1ψ(X, Y ), so h is Hermitian. By (3.11) we that trgh = (ψ, ω)g = 0. The form ψ is harmonic, so (3.10) implies δh = 0. From (3.12) it follows that (∇∗∇ − 2 ◦R)h = 0. Hence, h ∈ (g)orbH . The map is an isomorphism of vector spaces and we have shown the next proposition. Proposition 3.7.6. Let (V, J, g) be a compact Ricci-flat Kähler orbifold and let ω be the Kähler form of g. Then the map (g)orbH −→ H1,1R (V, J)/R · ω h 7→ h ◦ J is an isomorphism and dim (g)orbH = dimH 1,1 R (V, J)− 1. Lemma 3.7.7. All h ∈ (g)orb Kähler related to 0 are Hermitian. Proof. Take h ∈ (g)orb Kähler related to 0. Equation 3.17 becomes hαβ = 0 for all α, β. Expanding the expression hαβ , we see that hαβ = − i4((h◦J)ab+(h◦J)ba). The tensor h◦J is therefore skew-symmetric. We obtain h from ψ = h ◦ J via h(X, Y ) = −ψ(X, JY ). Hence, h(JX, JY ) = −ψ(JX, J2Y ) = ψ(J2Y, JX) = h(J2Y, J2X) = h(X, Y ), so h is Hermitian. Skew-Hermitian symmetric 2-tensors Let (V, J, g) be a compact Ricci-flat Kähler orbifold with volume 1. From the above remarks about anti-Hermitian tensors hA ∈ (g)orbA we know that an endomorphism I satisfying hA ◦ J = g ◦ I defines a ∂¯-closed element I ∈ A0,1(T 1,0V )orb. Also, g ◦ I(X, Y ) = hA ◦ J(X, Y ) = hA(X, JY ) = −hA(JX, J2Y ) = hA(JX, Y ) = hA(Y, JX) = hA ◦ J(Y,X) (3.19) = g ◦ I(Y,X) 62 3.7. Ricci-flat deformations so g ◦ I is symmetric. Furthermore, g ◦ I(JX, JY ) = g(JX, IJY ) = −g(JX, JIY ) = −g(X, IY ) = −g ◦ I(X, Y ). So the tensor g ◦ I is skew-Hermitian. Proposition 3.7.8. Let (V, J, g) be a compact Ricci-flat Kähler orbifold. Then the map (g)orbA −→ ICD(J)orbS h 7−→ I where h ◦ J = g ◦ I , is an isomorphism onto the ∂¯-harmonic elements. Proof. We showed above that any h ∈ (g)orbA produces a symmetric and ∂¯-harmonic infinitesimal complex deformation I ∈ ICD(J)orbS via the correspondence h ◦ J = g ◦ I . To see that (g)orbA surjects onto the space of harmonic elements in ICD(J) orb S , take I be a ∂¯-harmonic element of ICD(J)orbS . The inverse map is I 7→ h, where h is determined by h(X, Y ) = −g(JX, IY ). It is indeed an inverse, as h ◦ J(X, Y ) = h(X, JY ) = −g(JX, IJY ) = g(JX, JIY ) = g ◦ I(X, Y ). The tensor h is well-defined on orbifolds, as both g and I are well-defined on orbifolds and the operations defining the inverse h are well-defined on orbifolds. We check that the inverse h belongs to (g)orbA . The tensor h is symmetric as h(X, Y ) = −g(JX, IY ) = −g ◦ I(Y, JX) = −g ◦ I(JY, J2X) = −g(JY, IX), and h is skew-Hermitian as h(JX, JY ) = −g(J2X, IJY ) = g(J2X, JIY ) = g(JX, IY ) = −h(X, Y ). (3.20) The tensor h is δ-closed by (3.15) and it is trace-free as it is skew-Hermitian. The tensor I is ∂¯-harmonic, so (∇∗∇− 2 ◦R)h = 0 by the Weitzenböck formula (3.16). Corollary 3.7.9. Let (V, J, g) be a compact Ricci-flat Kähler orbifold. Then dim (g)orbA = 2 dimCH 1(V, TV )− 2 dimCH0,2(V, J). 63 Chapter 3. Orbifold Ricci-flat deformations Proof. First we show that I ∈ ICD(J)orbA is harmonic exactly if g ◦ I is harmonic. By the Weitzenböck formula (3.16), the form g ◦ I satisfies (∇∗∇− 2 ◦R)(g ◦ I) = 0 exactly if I is ∂¯-harmonic. If g ◦ I is anti-symmetric then ◦R (g ◦ I) = 0, see [KOI83, Lemma 7.1] for details. Hence∇∗∇(g◦I) = 0. This is equivalent to g◦I being parallel, i.e. ∇(g◦I) = 0. Being parallel implies being harmonic as d(g ◦ I) is the skew-symmetric part of ∇(g ◦ I) and g ◦ I parallel implies ∗g ◦ I parallel. Hence I ∈ ICD(J)orbA is harmonic exactly if g ◦ I is. That I ∈ A0,1(T 1,0V )orb is anti-symmetric means that g ◦ I is a differential 2-form and using g to identify T 1,0V with ∧0,1V it follows that ICD(J)orbA ' H0,2(V, J). Following the construction in Section 3.4 the subspace of ∂¯-harmonic forms in ICD(J)orb is isomorphic to H1(V, TV ). Corollary 3.7.10. Each h ∈ (g)orbA is Kähler related to some g ◦ I for an element I ∈ ICD(J)orbS and this relation is equivalent to the relation h ◦ J = g ◦ I . Proof. Let h ∈ (g)orbA . Then h ◦ J = g ◦ I for an element I ∈ ICD(J)orbS by Proposition 3.7.8. The relation between h and g ◦ I can alternatively be phrased as hαβ = −i(g ◦ I)αβ which, as g ◦ I is symmetric, is equivalent to Equation 3.17. Both sides of Equation 3.18 vanish. We are now ready to prove Theorem 3.1.3. Proof of Theorem 3.1.3. Let (V, J, g) be a compact complex orbifold with vanishing first Chern class. Assume that all infinitesimal complex deformations are integrable and let g be a Ricci-flat Kähler metric with respect to J . Denote by J the parameter space of nearby complex structures on V and let 0 ∈ J correspond to J0 = J . It follows from the hypothesis that the parameter space J is smooth in a neighbourhood of 0 and it follows from Theorem 3.4.5 that for every t ∈ J the deformed complex structure Jt admits a compatible Kähler metric gt. Consider the vector bundle V over J with fibre Vt at t ∈ J the space of real differential 2-forms which are harmonic with respect to gt, of type (1, 1) with respect to Jt and orthogonal to ωt with respect to the metric induced on H1,1R (V, Jt) by gt. The sign of the first Chern class is stable for small deformations, so c1(Jt)orb vanishes. Take h ∈ (g)orbA and let I ∈ ICD(J)orbS be Kähler related to h. All infinitesimal complex deformations are integrable, so there is a smooth curve of complex structure Jt with I = d dt Jt|t=0. Let gt be a smooth curve of Kähler metrics with respect to Jt and denote 64 3.7. Ricci-flat deformations by ωt the Kähler form of gt. Set κ = (g′0− h) ◦ J and consider a smooth curve (t, κt) ∈ V of Hermitian tensors κt with κ0 = κ. For sufficiently small t the tensor ω˜t = ωt − tκt is a Hermitian positive definite closed real (1, 1)-form with respect to Jt. Denote by g˜t the Kähler metric of ω˜t. A solution u ∈ C5(V )orb to the complex Monge-Ampére equation (ωt − tκt + i∂∂¯u)n − Aef (ωt − tκt)n = 0 (3.21) produces a C3-real (1, 1)-form in the class of ωt − tκt which is Ricci-flat and Kähler with respect to Jt. We already have a solution u = 0 to this equation for t = 0. Let X be the vector subspace of C5,α(V )orb consisting of those u for which ∫ V u dV orbg˜t = 0. For each t the metric g˜t has Ricci-form ρ˜t ∈ 2pic1(Jt) = 0. We can therefore find a smooth curve ft ∈ C∞(V )orb satisfying ρ˜t = i∂∂¯ft. Following Section 3.5 we define a map F : V ×X × R→ C3,α(V )orb by F ((t, κt), u, a) = log det ( (g˜t)αβ¯ + ∂2u ∂zα∂z¯β¯ ) − log det((g˜t)αβ¯)− ft − a. For the sake of readability we abbreviate the metric ((g˜t)αβ¯ + ∂2u ∂zα∂z¯β¯ ) by ((gˆt)αβ¯). In (3.3) we calculated DF(t,κt,u,a)|X×R(h, b) = −∆gˆh− b, and we showed that DF(0,0)|X×R is a linear homeomorphism. So the implicit function theorem (Theorem 2.1.10) applies and there exists a smooth map ψ : U → X × R from a neighbourhood U of (0, 0) in V to a neighbourhood of (0, 0) in X × R such that for each (Jt, κt) ∈ U the metric gˆt = ψ(Jt, κt) corresponding to the Kähler form ωˆt = ωt − tκt + i∂∂¯u is Ricci-flat and Kähler with respect to the complex structure Jt. From [DeT-KAZ81, Theorem 6.1] it follows that C2 Ricci-flat Kähler metrics are actually smooth, so ψ produces smooth metrics. We can restrict ψ to a subset U˜ of U such that ψ|U˜ only takes values in Sg. Elements in the image of ψ are Ricci-flat, so ψ(U˜) is contained in the premoduli space Pg. From Theorem 3.7.4 we know that dψ(T(0,0)U˜) ⊂ (g)orb. The rest of the proof consists in showing that ψ is a submersion onto Pg. Take the h ∈ (g)orbA from before which is Kähler related to I = ddtJt|t=0. For the Kähler form 65 Chapter 3. Orbifold Ricci-flat deformations ω˜t = ωt − tκt, the form ω˜′0 − h ◦ J satisfies ω˜′0 − h ◦ J = d dt (g˜t ◦ Jt)|t=0 − h ◦ J = d dt (g˜t)|t=0 ◦ J + g˜ ◦ I − h ◦ J = d dt (gt − tφt)|t=0 ◦ J + g˜ ◦ I − h ◦ J = (g′0 − 1 · φ0 − 0 · φ′0) ◦ J + g˜ ◦ I − h ◦ J = g˜ ◦ I + (g′0 − h) ◦ J − φ0 ◦ J = g˜ ◦ I + (g′0 − h) ◦ J − κ = g˜ ◦ I + 0 = g˜ ◦ Ia + g˜ ◦ Is, where we denoted by φt the tensor satisfying κt = φt ◦ Jt. The tensor g˜ ◦ Is is symmetric and the tensor g˜ ◦ Ia satisfies J(g˜ ◦ Ia)(X, Y ) = g˜ ◦ Ia(JX, JY ) = g˜(JX, IaJY ) = −g˜(JX, JIaY ) = −g˜(X, IaY ) = −g˜ ◦ Ia(X, Y ) so the 2-form g˜ ◦ Ia is not of type (1, 1). Hence [ω˜′0 − h ◦ J ] = [g˜ ◦ I] = 0 in H1,1(V, J). It is now clear why we added the form κ in (3.21). The class [ω˜′0− h ◦ J ] would otherwise not have been cohomologous to zero. Now, ωˆ′0 = d dt (ω˜t + i∂∂¯ut)|t=0 = ω˜′0 + i∂∂¯u′0, so [ωˆ′0] = [ω˜ ′ 0]. Hence [ωˆ ′ 0 − h ◦ J ] = 0. The symmetric tensors gˆ′0 and h are both Kähler related to I = J ′ 0, so gˆ ′ 0 − h is Kähler related to 0. It is therefore Hermitian by Lemma 3.7.7. For the Hermitian tensor gˆ′0 − h ∈ (g)orb the form (gˆ′0 − h) ◦ J is harmonic by the Weitzenböck formula (3.12) since ∆((gˆ′0 − h) ◦ J) = (∇∗∇− 2 ◦ R)(gˆ ′ 0 − h) ◦ J = 0. 66 3.8. Moduli space of Ricci-flat structures Now, 0 = [ωˆ′0 − h ◦ J ] = [(gˆ′0 − h) ◦ J + gˆ ◦ I] = [(gˆ′0 − h) ◦ J ] + [gˆ ◦ I] = [(gˆ′0 − h) ◦ J ]. A harmonic form cohomologous to zero is zero, so (gˆ′0 − h) ◦ J = 0, but this means that gˆ′0 = h. Hence, the differential of the map ψ is surjective and ψ is therefore a submersion. This concludes the proof. 3.8 Moduli space of Ricci-flat structures In this section, we study the moduli space of Ricci-flat structures. We prove that a neigh- bourhood of a Kähler structure is, up to an action of a finite group, a finite dimensional manifold and we find an expression for its dimension in terms of cohomology groups. This is the content of Theorem 3.1.4. Let (V, J, g) be a compact complex orbifold. Introduce the following equivalence re- lation ∼ onMorb. Two Riemannian metrics g and g′ are equivalent if for some φ ∈ Dorb and some c > 0 they satisfy g′ = c φ∗g. Equivalence classes are called Riemannian structures and the quotient spaceMorb/ ∼ is equipped with the quotient topology. We denote the quotient space by M˜orb. Volume scales as vol(cg)orb = √ cn vol(g)orb, so the quotient space can equivalently be express as Morb1 /Dorb. A Ricci-flat structure is a Riemannian structure containing a Ricci-flat metric. Denote by M˜orbR the subspace of Ricci-flat structures in M˜orb. A diffeomorphism φ ∈ Dorb is an isometry φ : (V, φ∗g)→ (V, g) and it therefore satisfies Ric(φ∗g) = φ∗Ric(g). From [BES87, Theorem 1.159] we know that Ric(efg) = Ric(g) + (2− n)(∇df − df ⊗ df) + (∆f − (n− 2)||df ||2)g. Rescaling by a constant c therefore satisfies Ric(cg) = Ric(g). Denote by Rorb the sub- space of Morb of Ricci-flat structures. The space M˜orbR can therefore equivalently be 67 Chapter 3. Orbifold Ricci-flat deformations expressed asRorb/ ∼, which we therefore denote by R˜orb. This space is called the moduli space of Ricci-flat structures. For an orbifold (V, g) denote by Iso(V, g)orb the group of isometries. The identity com- ponent of Iso(V, g)orb is the connected component of the identity. We denote it by (Iorbg ) 0. Myers and Steenrod showed in [MYE-STE39] that the isometry group of a Riemannian manifold is a Lie group and that if the manifold is compact, then so is the isometry group. This was generalized to orbifolds by Bagaev and Zhukova in [BAG-ZHU07, Corollary 1]. Theorem 3.8.1. Let (V, g) be a compact orbifold. Then Iso(V, g)orb is a compact Lie group. Proposition 3.8.2. Let (V, g) be a compact Ricci-flat orbifold. Then the identity com- ponent (Iorbg ) 0 of the isometry group Iso(V, g)orb acts trivially on the premoduli space of Ricci-flat metrics Pg. Proof. From Theorem 3.1.2(2) it follows that any φ ∈ Dorb satisfying φ∗Sg ∩ Sg 6= ∅ belongs to Iso(V, g). Hence Iso(V, g′) ⊂ Iso(V, g). In particular (Iorbg′ )0 ⊂ (Iorbg )0. Similar to the case of compact Ricci-flat manifolds ([BES87, Theorem 1.84]), dim Iso(V, g) = b1(V,R) = dim Iso(V, g′), so the connected Lie groups (Iorbg′ )0 and (Iorbg )0 coincide. Any η ∈ (Iorbg )0 therefore acts trivially on Pg. Corollary 3.8.3. Let (V, g) be a compact Ricci-flat orbifold. Then a neighbourhood of [g] in R˜orb is homeomorphic to Pg/(Iso(V, g)orb/(Iorbg )0), where Iso(V, g)orb/(Iorbg )0 is a finite group acting on the premoduli space Pg. Proof. The identity component (Iorbg ) 0 is a normal subgroup of the compact Lie group Iso(V, g)orb. The quotient group Iso(V, g)orb/(Iorbg ) 0, called the group of components, is therefore a finite group. A consequence of Theorem 3.1.2 and Proposition 3.8.2 is that a neighbourhood of [g] in the moduli space R˜orb is locally homeomorphic to the quotient Pg/Iso(V, g)orb = Pg/(Iso(V, g)orb/(Iorbg ) 0). Let (V, J, g) be a compact complex orbifold. Define a Kähler structure on V to be a Riemannian structure containing a Kähler metric. Proof of Theorem 3.1.4. Let (V, J, g) be a compact complex orbifold with vanishing first Chern class and all infinitesimal complex deformations integrable and let g be a Ricci-flat 68 3.8. Moduli space of Ricci-flat structures Kähler metric on (V, J). By Theorem 3.1.2 there exists a slice Sg ⊂ Morb1 for the action of Dorb. From Theorem 3.7.4 we know that there exists a finite dimensional manifold Z ⊂ Sg with TgZ = (g)orb and Pg as a real analytic subset. From Theorem 3.1.3 we know that all infinitesimal Ricci-flat deformations integrate into smooth curves of Ricci- flat deformations through g, so Pg spans an entire neighbourhood of g in Z and therefore satisfies TgPg = (g)orb. By Corollary 3.6.7 and Corollary 3.8.3 a neighbourhood U ⊂ R˜orb of [g] is homeomorphic to Pg/(Iso(V, g)orb/(Iorbg )0). To compute the dimension of R˜orb at [g] we have to compute the dimension of Pg at g, which is dim (g)orb. From Corollary 3.7.6 and Corollary 3.7.9 it follows that dim((g)orb) = dim (g)orbH + dim (g) orb A = dimH1,1R (V, J)− 1 + 2 dimCH1(V, TV )− 2 dimCH0,2(V, J). This concludes the proof. 69 Chapter 4 Examples: Orbifold K3 surfaces In this section, we provide examples of compact complex orbifolds with vanishing first Chern class and a Ricci-flat Kähler metric for which all infinitesimal complex defor- mations are integrable. We use Theorem 3.1.4 to calculate the dimension of the mod- uli space of Ricci-flat structures on these orbifolds. The examples are hypersurfaces in weighted projective spaces given by weighted homogeneous polynomials. For more de- tails about these spaces see for instance [BOY-GAL08, Chapter 4], [JOY00, Chapter 6], [CAN-LYN90] and [DOL82]. Definition 4.0.4 (Weighted projective space). Let m ≥ 1 be an integer and let Cm+1 have coordinates (z0, . . . , zm). Let a = (a0, . . . , am) be a tuple of integers satisfying a0, . . . , am ≥ 1 and di = gcd(a0, . . . , ai−1, ai+1, . . . , am) = 1 for i = 0, . . . ,m. The weighted C∗-action, denoted by C∗(a), on Cm+1\{0} is defined by u.(z0, . . . , zm) = (u a0z0, . . . , u amzm). We define the weighted projective space CPma0,...,am to be (C m+1\{0})/C∗(a). Note that if all weights are equal to 1 then a weighted projective space is the ordi- nary complex projective space. In the next example we construct an orbifold structure on CPma0,...,am explicitly. The example is borrowed from [BOY-GAL08, p. 134]. Example 4.0.5. Let CPma0,...,am be the weighted projective space and fix an integer i with 0 ≤ i ≤ m. We construct the local uniformizing system {Ui,Γi, φi} explicitly. Consider the set Ui = {(z0, . . . , zm) ∈ Cm+1 | zi = 0} and let Γi = {z ∈ C∗ | zai = 1} be the 70 ai’th roots of unity. The finite group Γi acts on Ui via the weighted C∗(a0, . . . , am)-action γ.(z0, . . . , zm) = (γ a0z0, . . . , γ amzm). Coordinates on Ui are y = (y0, . . . , yˆi, . . . , ym), where yˆi means that yi has been removed. We can coverCPma0,...,am with coordinate patches of the form U¯i = {[z0, . . . , zm] ∈ CPma0,...,am | zi 6= 0} with 0 ≤ i ≤ m. Define the map φi : Ui → U¯i : y 7→ yai = (yai0 , . . . , yˆiai , . . . , yaim). It satisfies φ(γ.y) = φ(y) for all γ ∈ Γi, so it descends to a map φi : Ui/Γi → U¯i. The C∗(a0, . . . , am)-orbits of the space (Cm+1\{0})\{zi = 0} are in bijection with the points in Ui/Γi via the map φi. For each i the tuple {Ui,Γi, φi} is therefore a local uniformizing system. We can extend the family of l.u.s.’s for the open cover ∪mi=0U¯i to an atlas by including charts of the form {Ui0 ∩ · · · ∩ Uik ,Γi0...ik , φi0...ik} where Γi0...ik = Zgcd(ai0 ...aik ) and the map φi0...ik : Ui0 ∩ · · · ∩ Uik → U˜i0 ∩ · · · ∩ U˜ik is given by φi0...ik(yi0...ik) = y gcd(ai0 ,...,aik ) i0...ik . Injections can be created as follows. There is an open set Uij ⊂ Ui ∩ Uj such that the injection map λij,i : Uij → Ui is λij,i(yij) = ytij where t = gcd(ai,aj)ai . This satisfies the condition φi ◦ λij,i = φij and is well-defined with the action of Γi on Ui. We call this the standard orbifold structure on CPma0,...,am . We will always assume that CP m a0,...,am is equipped with this orbifold structure. The next remark tells us how to find orbifold singularities in weighted projective spaces and what the isotropy groups at their singularities look like. Remark 4.0.6. Let z = [z0, . . . , zm] ∈ CPma0,...,am and let {zi1 , . . . , zir} be a collection of those zi’s which equal zero. Let U¯z = {[z1, . . . , zm] ∈ CPma0,...,am | zis = 0 for s = 1, . . . , r} and let d be the greatest common divisor of those ai for which zi 6= 0. Let {Uz,Γz, φz} be the l.u.s. corresponding to U¯z. If d = 1 then z = [z0, . . . , zm] is a regular point. If d > 1 then [z0, . . . , zm] is a singular point with isotropy group Zd and U¯z is homeomorphic to Uz/Zd. Definition 4.0.7. Let f(z0, . . . , zm) ∈ C[z0, . . . , zm] be a non-zero polynomial. We say that f is weighted homogeneous of degree d if f(ua0z0, . . . , u amzm) = u df(z0, . . . , zm) ∀u, z0, . . . , zm ∈ C. LetCPma0,...,am be a weighted projective space. We say that a subvariety V ⊆ CPma0,...,am is weighted homogeneous if it is the common zero locus of a collection of weighted ho- mogeneous polynomials. A subvariety V ⊂ CPma0,...,am of a single weighted homogeneous 71 Chapter 4. Examples: Orbifold K3 surfaces polynomial f is called a (weighted homogeneous) hypersurface and we denote it by Vf . For a weighted homogeneous polynomial f ∈ C[z0, . . . , zm] we say that the hypersurface Vf ⊂ CPma0,...,am is well-formed if for all 0 ≤ i < j ≤ m the greatest common divisor dij = gcd(a0, . . . , aˆi, . . . , aˆj, . . . , am) divides the degree of f . We say that Vf is quasi- smooth of dimension m if the partial derivatives of f do not vanish simultaneously. In the next lemma we summarize two results from [BOY-GAL08, p. 140-141]. Lemma 4.0.8. Let CPma0,...,am and let f(z0, . . . , zm) ∈ C[z0, . . . , zm] be a weighted ho- mogeneous polynomial and let Vf ⊆ CPma0,...,am be a quasi-smooth hypersurface defined by f , then the orbifold structure on CPma0,...,am induces on Vf the structure of an orbifold where all isotropy groups are cyclic. Furthermore, Vf is well-formed if and only if it has no branch divisors. The weighted projective space CPma0,...,am admits a generalization of the Fubini-Study metric. As explained in Example 4.0.5 the weighted projective space is covered by open sets of the form U¯i = {[z0, . . . , zm] ∈ CPma0,...,am | zi 6= 0} with 0 ≤ i ≤ m. The orbifold Fubini-Study metric is locally on U¯i given by ωi = i 2pi ∂∂¯ log ( m∑ l=0 ∣∣∣∣zailzali ∣∣∣∣2 ) ∈ A1,1(U¯i). It patches to a global form and defined a Kähler metric on CPma0,...,am . The following result from [JOY00, section 6.7] explains why well-formed and quasi-smooth hypersurfaces in weighted projective spaces provide applications of the main results of Chapter 3. Theorem 4.0.9. Let CPma0,...,am be a weighted projective space and let f ∈ C[z0, . . . , zm] be a weighted homogeneous polynomial of degree d for which the hypersurface Vf in CPma0,...,am defined by f is well-formed and quasi-smooth. Then Vf has trivial canonical bundle if and only if d = a0 + · · ·+ am. The hypersurface Vf from Theorem 4.0.9 inherits a Kähler metric from the ambi- ent weighed projective space CPma0,...,am , so it is a Calabi-Yau orbifold. This implies that H2(V, TV ) = 0, so all infinitesimal complex deformations are integrable. Also, TV ' Ωm−1V , so H1(V, TV ) ' H1(V,Ωm−1V ) and the dimension of H1(V, TV ) is the Hodge number hm−1,1. The arguments are similar to the smooth case. See e.g. [HUY05, section 6.1 and 6.2] for details. Next we give an example of an orbifold K3 surface 72 Example 4.0.10. In the weighted projective space CP31,1,2,4 we consider the degree 8 weighted homogeneous polynomial f(x, y, z, w) = x8+y8+z4+w2. The greatest common divisors dij ∈ {1, 2} of the weights (1, 1, 2, 4) all divide 8, so f is well-formed. The poly- nomial f is known as a Brieskorn-Pham polynomial. Such polynomials are quasi-smooth. The hypersurface Vf is an orbifold of complex dimension 2 with isotropy Z2 singularities in [0, 0, z1, z2]. The degree of f equals the sum of the weights (1, 1, 2, 4) so by Theorem 4.0.9 the hypersurface Vf is a Calabi-Yau orbifold. All infinitesimal complex deforma- tions are also integrable on Vf , it has vanishing first Chern class and it admits a Ricci-flat Kähler metric. The orbifold Vf therefore satisfies the conditions of Theorem 3.1.3 and we can use Theorem 3.1.4 to compute the dimension of its moduli space of Ricci-flat struc- tures in a neighbourhood of a Ricci-flat Kähler structure. It is explained in [BOY-GAL08, Appendix B.1] that the hypersurface Vf is an orbifold K3 surface with Hodge diamond 1 0 0 1 18 1 0 0 1 We have TVf ' Ω2−1Vf so H1(Vf , TVf ) ' H1(Vf ,Ω1Vf ) and so dim (g)orb = h1,1(Vf )− 1 + 2h1,1(Vf )− 2h0,2(Vf ) = 18− 1 + 2 · 18− 2 = 51. For orbifold K3 surfaces V the expression for the dimension of the moduli space of Ricci-flat structures from Theorem 3.1.4 simplifies to dim (g)orb = 3h1,1(V )− 3 = 3 b2(V )− 9. (4.1) Boyer-Galicki reproduced in [BOY-GAL08, Appendix B.1] a list by Miles Reid from 1979 of 95 orbifoldK3 surfaces arising as hypersurfaces inCP3a0,a1,a2,a3 with varying Betti 73 Chapter 4. Examples: Orbifold K3 surfaces numbers. All orbifolds on the list satisfy the assumptions of Theorem 3.1.4. To compute the dimension of the moduli space of Ricci-flat structures for these orbifold K3 surfaces we simply need to input the relevant value for h1,1 in the expression (4.1). We calculate the dimension of the moduli space of a few of the orbifold K3 surfaces listed in [BOY-GAL08, Appendix B.1]. Table 4.1: Orbifold K3 hypersurfaces Ambient space Degree Polynomial dim (g)orb CP31,1,4,6 12 x12 + y12 + z3 + w2 54 CP31,1,2,4 8 x8 + y8 + z4 + w2 51 CP31,1,2,2 6 x6 + y6 + z3 + w3 48 CP31,2,2,5 10 x10 + y5 + z5 + w2 42 CP31,2,6,9 18 x18 + y9 + z3 + w2 42 CP31,2,3,6 12 x12 + y6 + z4 + w2 39 CP31,3,8,12 24 x24 + y8 + z3 + w2 36 CP31,3,4,4 12 x12 + y4 + z3 + w3 30 CP31,4,5,10 20 x20 + y5 + z4 + w2 30 CP31,6,14,21 42 x42 + y7 + z3 + w2 30 CP32,3,3,4 12 x6 + y4 + z4 + w3 24 CP32,3,10,15 30 x15 + y10 + z3 + w2 24 74 Chapter 5 ALE Ricci-flat deformations 5.1 Introduction and results In this chapter, we extend results by Koiso from [KOI83] (see Section 2.7 for a review) about Ricci-flat deformations of Ricci-flat Kähler metrics on compact manifolds to a class of complete non-compact manifolds known as asymptotically locally Euclidean(ALE) manifolds. We show that ALE Ricci-flat deformations of ALE Ricci-flat Kähler metrics are Kähler, possibly with respect to a perturbed complex structure, and we show that the moduli space of ALE Ricci-flat structures is, up to the action of a finite group, a finite dimensional manifold and we find an expression for its dimension. ALE metrics have the following model. Let G ⊂ SO(n) be a finite subgroup acting freely on Rn\{0} and let h0 be the standard Euclidean metric on Rn. h0 is invariant under G so it descends to a metric on (Rn\{0})/G, which we also denote by h0. An ALE manifold (X, g) is a non-compact Riemannian manifold with one end such that outside a compact set it is diffeomorphic to (Rn\BR(0))/G and such that the metric g approximates the flat metric h0 towards infinity. To define ALE metrics let ∇ be the Levi-Civita covariant derivative of h0 and let r : Rn/G → [0,∞) be the radius function on (Rn\{0})/G given by h0. We use the big O-notation f(x) = O(rk) to express that there exist constants R,C > 0 such that |f(x)| ≤ Crk for all |x| > R. Definition 5.1.1. A Riemannian manifold (X, g) of real dimension n is an asymptotically locally Euclidean (ALE) manifold asymptotic to Rn/G of order τ > 0 if there exists a 75 Chapter 5. ALE Ricci-flat deformations compact set K ⊂ X , and a map pi : X\K → Rn/G such that for some R > 0 the map pi : X\K −→ (Rn\BR(0))/G is a diffeomorphism and such that the push-forward metric pi∗g satisfies ∇k(pi∗g − h0) = O(r−τ−k) on {z ∈ Rn/G | r(z) > 0} for all k ≥ 0. We denote this by saying that (X, g) is ALEτ . Whenever we say that (X, g) is an ALE manifold asymptotic to Rn/G it is implicitly assumed that X is a manifold of real dimension n. We say that (X, g) is ALE if it is ALEτ for some τ > 0. The map pi in Definition 5.1.1 is called an asymptotic coordinate system for X . Note that if a metric g is ALEτ for some τ > 0, then it is ALEτ ′ for all 0 < τ ′ < τ . We introduce radius functions on ALE manifolds. Definition 5.1.2. Let (X, g) be an ALEτ manifold asymptotic to Rn/G. We say that a smooth function ρ : X → [1,∞) is a radius function on X if, given an asymptotic coordi- nate system pi : X\K −→ (Rn\BR(0))/G, it satisfies ∇k(pi∗ρ− r) = O(r1−τ−k) on {z ∈ Rn/G | r(z) > R} (5.1) for all k ≥ 0. We now present the main results of this chapter. Proper definitions of the objects and operators involved will be given in the relevant sections. We face two immediate challenges when attempting to generalize Koiso’s results to ALE manifolds (X, g). The first one is that we made use of the Fredholm alternative for elliptic operators involving the linearization of the Einstein operator and the linearization of the Complex Monge-Ampère equation in the compact case. Elliptic operators on Sobolev spaces and Hölder spaces over a non-compact base manifold need not be Fredholm in general. To overcome this we use weighted versions of Ck,α(X) and Lpk(X). The second problem is that we do not have a non-compact Kodaira-Spencer type result about stability of the Kähler property for deformations of the complex structure. We will solve this by working with ALE manifolds X that arise as the complement of a smooth divisor D in a compact manifold X¯ and use deformations of the pair (X¯,D) to deform the complex structure on X . 76 5.1. Introduction and results An ALE manifold (X, g) always admits a radius function ρ and it can be used to con- struct weighted versions of the Banach spaces Ck(X), Ck,α(X) and Lpk(X). By weight- ing the usual norms by powers ρ−β we get Banach spaces Ckβ(X), C k,α β (X) and L p k,β(X), which in turn allows us to define the space C∞β (X). Weighted Sobolev and Hölder spaces are convenient for the study of ALE metrics as for instance a function f ∈ Ck,αβ (X) sat- isfies f = O(ρβ) and ∇pf = O(ρβ−l) for all l ≤ k. Another important aspect of the weighted spaces is that the Laplace operator is Fredholm if the weight β is sufficiently small (See Section 5.2). Let (X, g) be an ALEτ manifold asymptotic to Rn/G of order τ > 0. Denote by MALEτ the space of ALEτ metrics (Section 5.4) and denote by Dτ (Definition 5.4.2) the group of diffeomorphisms on X generated by a neighbourhood of the identity map given by vector fields of small C1-norm and appropriate control at infinity. Dτ+1 acts onMALEτ by pull-back and it introduces an equivalence relation on it. Elements of the quotient spaceMALEτ/Dτ+1 are called structures. For each g the tangent space Tg(MALEτ ) can be identified with the space C∞−τ (Sym 2(T ∗X)). As in Section 2.3 we denote by δ∗g the symmetric part of the covariant derivative ∇g restricted to symmetric tensors and by δg the formal adjoint of δ∗g . This makes sense for all τ > n 2 as there is an L2-inner product on the space C∞−τ (Sym 2(T ∗X)) then. To study the local structure of the quotient space MALEτ/Dτ+1 we use a slice in MALEτ for the action of Dτ+1. In the compact case the three equations Ric′gh = 0, δgh = 0 and ∫ X trgh dVg = 0 were used to define the space of infinitesimal Ricci-flat deformations (g). We will replace the two slice equations δgh = 0 and ∫ X trgh dVg = 0 with the Bianchi equation (2δg + d trg)h = 0. We do this because ALE manifolds are non-compact manifolds with infinite volume. In Section 5.4 we produce a slice Sτ through g inMALEτ for the action of Dτ+1 by exponentiating a neighbourhood of 0 in the space of solutions to the equation (2δg + d trg)h = 0 in C∞−τ (Sym 2(T ∗X)). On the set of solutions to the equation (2δg + d trg)h = 0 in C∞−τ (Sym 2(T ∗X)) the linearization of the Einstein equation is∇∗∇−2 ◦R. In Section 5.5 we introduce the space of infinitesimal Ricci-flat deformations ∞−τ (g). It is the subspace of C ∞ −τ (Sym 2(T ∗X)) defined by the two equations (∇∗∇ − 2 ◦R)h = 0 and (2δg + d trg)h = 0. For suitable τ the kernel of the elliptic operator∇∗∇− 2 ◦R: C∞−τ (Sym2(T ∗X))→ C∞−τ−2(Sym2(T ∗X)) is finite dimensional (Lemma 5.5.2), so the space ∞−τ (g) ⊂ ker(∇∗∇ − 2 ◦ R) is finite dimensional. 77 Chapter 5. ALE Ricci-flat deformations The moduli space of Ricci-flat structures is the quotient space RALEτ/Dτ+1, where RALEτ denotes the Ricci-flat metrics in MALEτ . We denote by Pg the subset of Sτ of Ricci-flat metrics and call it the premoduli space of Ricci-flat metrics. Pg is a slice through g for the action of Dτ+1 on the space of ALEτ Ricci-flat metrics. The real analytic im- plicit function theorem can be applied in this context as well and gives us the following understanding of the premoduli space of Ricci-flat metrics in the slice Sτ . Theorem 5.1.3. Let (X, g) be an ALEτ Ricci-flat manifold asymptotic to Rn/G of order τ > n 2 . In the slice Sτ through g there exists a finite dimensional real analytic submanifold Z ⊂ Sτ with TgZ = ∞−τ (g) and Z contains Pg as a real analytic subset. A complex ALE manifolds (Definition 5.3.3) is a complex manifold asymptotic to Cm/G, where G is some finite subgroup of U(m) acting freely on Cm\{0}. The next proposition tells us that a deformation of a Ricci-flat Kähler metric splits into a deforma- tion of the complex structure and a deformation of a Kähler form. Proposition 5.1.4. Let (X, J, g) be an ALEτ Ricci-flat Kähler manifold asymptotic to Cm/G of order τ > m. For h ∈ ∞−τ (g) both its Hermitian part and its skew-Hermitian part lie in ∞−τ (g). The next two theorems are the main results of this chapter. The first theorem tells us that ALE Ricci-flat deformations of ALE Ricci-flat Kähler metrics are Kähler and the second theorem tells us what the moduli space of Ricci-flat structures looks like in a neigh- bourhood of an ALE Ricci-flat Kähler structure. We will consider ALE Ricci-flat Kähler manifolds X that arise as the complement of a particular divisor D inside a compact Käh- ler manifold X¯ as we wish to make use of the deformation theory by Kawamata for pairs (X¯,D) to deform the complex structure on X and prove a Kodaira-Spencer type result for deformations of the complex structure on X . Theorem 5.1.5. Let (X, J, g) be an ALE2m Ricci-flat Kähler manifold asymptotic toCm/G and assume that X = X¯\D for a compact Kähler manifold (X¯, J¯ , g¯) with a smooth and ample divisor D satisfying KX¯ = −βLD for some β ≥ 1. Also assume that all in- finitesimal deformations of the pair (X¯,D) are integrable. Then small ALE2m−1 Ricci-flat deformations of g are Kähler, possibly with respect to a perturbed complex structure. The motivation for considering ALE2m manifolds in Theorem 5.1.5 is that the low dimensional examples by Eguchi-Hanson (Examples 6.0.10) and Kronheimer ([KRO89-1] 78 5.1. Introduction and results and [KRO89-2]) as well as the higher dimensional exambles by Calabi(Examples 6.0.11) all have order 2m. Another reason for the order 2m in Theorem 5.1.5 is that we make use of results from [JOY00, Chapter 8] about ALE2m manifolds, but also because the order 2m is optimal in a certain sense ([JOY00, Section 8.2.1]). A solution to ∆u = f for f ∈ Ck,αβ (X) for some β ∈ (−2m,−2) satisfies u ∈ Ck+2,αβ+2 (X), but if β ∈ (−2m− 1,−2m) then u ∈ Ck+2,α−2m (X). The reason for the order 2m−1 is that the cohomology of differential forms with weight 2m − 1 is related to the cohomology of the underlying manifold as explained in [JOY00, Theorem 8.4.1]. A consequence of Theorem 5.1.5 is that under the hypothesis of the theorem all in- finitesimal ALE2m−1 Ricci-flat deformations integrate into curves of ALE2m−1 Ricci-flat metrics in the premoduli space Pg. Pg is almost a slice for the action of D2m on the space of ALE2m−1 Ricci-flat metrics, and it can be used to study the local structure of the moduli spaceRALE2m−1/D2m. Theorem 5.1.6. Assume the hypothesis of Theorem 5.1.5 and assume that g is not identi- cally flat. Then the premoduli space of ALE2m−1 Ricci-flat metrics is, in a neighbourhood of a Kähler structure, a smooth manifold and the moduli space of Ricci-flat structures is locally, up to an action of a finite group, a finite dimensional manifold of dimension dimH1,1R (X, J)− 1 + 2 dimCH1(X, TX). (5.2) The hypothesis that g is not flat in Theorem 5.9.1 is used only in Lemma 5.9.1 to ensure that the isometry group acts as a finite group on the premoduli space. Some might wonder why there is not a term representing the anti-symmetric infinitesimal complex deforma- tions in (5.2). This is because ALE Kähler manifolds are crepant resolutions of Cm/G and therefore have H0,2(X, J) = 0. Computing the cohomology groups in Theorem 5.1.6 could be done via the compactly supported cohomology as explained in Section 5.3, but could potentially also be done by relating the cohomology groups of Theorem 5.1.6 to the cohomology groups of X¯ and D. This is for instance done in the asymptotically cylin- drical case in [KOV06]. In Example 6.0.12 we compute the cohomology groups from Theorem 5.1.6 for the pair (CPm,CPm−1). Throughout Chapter 5 most spaces of sections of tensor bundles will have a subscript attached to them describing the order of growth of the sections. We have strived to use τ as a positive index related to an ALEτ metric and to use the index β as a negative or 79 Chapter 5. ALE Ricci-flat deformations general index for discussions of tensors more generally. This chapter is organised in the following way. In Section 5.2 and Section 5.3 we pro- vide an introduction to ALE manifolds and various known results about them. In Section 5.4 we produce a slice for the action of diffeomorphisms. In Section 5.5 we introduce ALE Ricci-flat deformations of metrics and study the corresponding space of infinitesimal deformations. In Section 5.6 we set up the theory for deformations of a pair (X¯,D). In Section 5.7 we discuss stability results for the deformations of the pair (X¯,D). In Section 5.8 and Section 5.9 we prove Theorem 5.1.5 and Theorem 5.1.6 respectively. In Section 5.10 we discuss a recent preprint on the same topic as this chapter. In Section 5.11 we discuss the possibility of extending the results advertised in this introduction from ALE manifolds to asymptotically conical manifolds. 5.2 Tools from analysis In this section, we introduce weighted Sobolev spaces and weighted Hölder spaces and provide a number of known results about them. In Section 2.1 we introduced Sobolev spaces and Hölder spaces. Elliptic operators on Sobolev and Hölder spaces over compact manifolds have a range of useful properties. Elliptic regularity and the Fredholm alter- native, just to mention a few. If we work with weighted versions of Sobolev spaces and Hölder spaces over non-compact manifolds, then, as we shall see in this section, we can recover some of these desirable properties. Let (X, g) be an ALEτ manifold asymptotic to Rn/G and let ρ be a radius function on X . Definition 5.2.1 (Weighted Sobolev spaces). Let p, k ∈ Z≥0 and let β ∈ R. Define the weighted Sobolev space Lpk,β(X) to be the set of function f : X → R which are locally integrable, k times weakly differentiable and for which the norm ||f ||Lpk,β = ( k∑ j=0 ∫ X |ρj−β∇jf |pρ−ndVg )1/p is finite. With this norm Lpk,β(X) is a Banach space and L 2 k,β(X) is a Hilbert space. Definition 5.2.2 (Weighted Ck spaces and Hölder spaces). Let k ∈ Z≥0 and β ∈ R. Define Ckβ(X) to be the space of continuous functions f : X → R with k times continuous 80 5.2. Tools from analysis derivatives and such that ρj−β|∇jf |g is bounded for each j = 0, . . . , k. Equip Ckβ(X) with the norm ||f ||Ckβ = k∑ j=0 sup x∈X |ρj−β∇jf |g. For a tensor field T on X and α, γ ∈ R define [T ]α,γ = sup x∈X,d(x,y)<δ(g) [ min(ρ(x), ρ(y))−γ · |T (x)− T (y)|g d(x, y)α ] where δ(g) is the injectivity radius of g and d(x, y) is the distance between x and y using g. |T (x)− T (y)| is interpreted using parallel translation along a unique geodesic joining x and y. For β ∈ R, k ∈ Z≥0 and α ∈ (0, 1) define the weighted Hölder space Ck,αβ (X) to be the set of functions f ∈ Ckβ(X) for which the norm ||f ||Ck,αβ = ||f ||Ckβ + [∇ kf ]α,β−k−α is finite. Define C∞β (X) to be the intersection ∩k≥0Ckβ(X). Note that both Ckβ(X) and Ck,αβ (X) are Banach spaces, but C ∞ β (X) is not. Observe that functions f ∈ Ck,αβ (X) satisfy f = O(ρβ) and ∇jf = O(ρβ−j) for all j = 0, . . . , k. The next Kondraschov type theorem is taken from [JOY00, Theorem 8.3.3], which in turn is an ALE version of a result by Chaljub-Simon and Choquet-Bruhat. Theorem 5.2.3. Let (X, g) be an ALEn manifold asymptotic to Rn/G and let k ≥ 0, α ∈ (0, 1) and β, γ ∈ R with β < γ, then the embedding Ck,αβ (X)→ Ckγ (X) is compact. Similarly we borrow a Sobolev Embedding type theorem from [LEE-PAR87, Lemma 9.1] Lemma 5.2.4 (Sobolev Embedding). Let (X, g) be an ALEn manifold asymptotic toRn/G. Suppose p > 1 and l − k − α > n p . For each  > 0 there are continuous embeddings C l,αβ−(X) ⊂ Lpl,β(X) ⊂ Ck,αβ (X). In particular, if f ∈ Lpl,β with l > np then f = O(ρβ). 81 Chapter 5. ALE Ricci-flat deformations Remark 5.2.5. Let (X, g) be an ALE manifold asymptotic to Rn/G. We wish to determine for which β and k there is an L2-inner product on Ck,αβ (X). On L 2 k,β(X) there is the inner product (η, γ)L2k,β := k∑ j=0 ∫ X (ρj−β∇jη, ρj−β∇jγ)gρ−ndVg. The radius function ρ satisfies ρ ≥ 1, so if we assume that β < −n 2 , then ρ−2β−n > 1. Hence, (η, η)L2k,β ≥ ∫ X |ρ−βη|2gρ−ndVg = ∫ X |η|2gρ−2β−ndVg ≥ ∫ X |η|2gdVg = (η, η)L2 . This shows that for β < −n 2 and k > 0 there is an L2-inner product (·, ·)L2 on L2k,β(X). If we furthermore assume that k > n 2 + 1, then Lemma 5.2.4 ensures that Ck,αβ (X) ⊂ L2k,β+(X). Hence, for k > n 2 + 1 and β < −n 2 there is an L2-inner product on Ck,αβ (X). This construction extends to general tensor bundles over X . We remark that the order of the ALE metric did not appear in Remark 5.2.5. The order of the metric has an effect on which functions belong to Ck,αβ (X), but it does not affect whether or not the L2k,β-inner product bounds the L 2-inner product. The differential operators introduced in Section 2.3 act on weighted spaces in the fol- lowing way. The exterior derivative d satisfies d : Ck,αβ (∧rT ∗X) → Ck−1,αβ−1 (∧r+1T ∗X). The operator d∗g = (−1)n(r+1)+1 ∗ d∗ acts as d∗ : Ck,αβ (∧rT ∗X) → Ck−1,αβ−1 (∧r−1T ∗X). The Levi-Civita covariant derivative ∇ satisfies ∇ : Ck,αβ (T (r,s)X)→ Ck−1,αβ−1 (T (r−1,s)X). The formal adjoint operator ∇∗g satisfies ∇∗ : Ck,αβ (T (r,s)X) → Ck−1,αβ−1 (T (r−1,s)X). The Riemann curvature tensor operator ( ◦ R η)(X1, X2) = ∑n i=1 η(R(X, ei)X2, ei) satisfies◦ R: C k,α β (Sym 2(T ∗X))→ Ck−2,αβ−2 (Sym2(T ∗X)). Lemma 5.2.6 (Integration by parts). Let (X, g) be an orientable complete Riemannian manifolds with C2 Riemann curvature tensor. Let k > n 2 + 2, α ∈ (0, 1) and β < −n 2 . Then for the L2-inner product on Ck,αβ (∧pT ∗X) we have for any η ∈ Ck,αβ (∧pT ∗X) and γ ∈ Ck,αβ (∧p+1T ∗X), (dη, γ)L2 = (η, d ∗γ)L2 82 5.2. Tools from analysis Proof. As β < −n 2 and k > n 2 + 2 Remark 5.2.5 ensures the L2-inner product is well- defined on both Ck,αβ (∧pT ∗X) and Ck−1,αβ−1 (∧pT ∗X). The proof is now a direct conse- quence of the version of Stokes theorem for complete metrics by [GAF54]. First observe that, as in the compact case,∫ X d(η ∧ ∗γ) = (dη, γ)L2 − (η, d∗γ)L2 . From [GAF54, Lemma 2] we know that |η ∧∗γ|g ≤ |η|g |γ|g and as β < −2 both |η|g and |γ|g are bounded. [GAF54, Theorem] then tells us that ∫ X d(η ∧ ∗γ) = 0. A similar argument to Lemma 5.2.6 can be used to show that (∇η, γ)L2 = (η,∇∗γ)L2 for Ck,αβ -sections of general tensor bundles T (r,s)X for β < −n 2 and k > n 2 + 2. The next theorem is taken from [JOY00, Theorem 8.3.6] and is a generalization of results for asymptotically Euclidean manifolds by Chaljub-Simon and Choquet-Bruhat. Theorem 5.2.7. Let (X, g) be an ALEn manifold asympotic to Rn/G for n > 2, and ρ a radius function on X . Let k ≥ 0 be an integer and α ∈ (0, 1). Then • Let β ∈ (−n,−2). Then there exists C > 0 such that for each f ∈ Ck,αβ (X) there is a unique u ∈ Ck+2,αβ+2 (X) with ∆u = f , which satisfies ||u||Ck+2,αβ+2 ≤ C||f ||Ck,αβ . • Let β ∈ (−n− 1,−n). Then there exist C1, C2 > 0 such that for each f ∈ Ck,αβ (X) there is a unique u ∈ Ck+2,α−n+2 (X) with ∆u = f . Moreover u = Aρ−n+2 + v, where A = |G| (n− 2)Ωn−1 ∫ X f dVg and v ∈ Ck+2,αβ+2 (X), satisfies |A| ≤ C1||f ||C0β and ||v||Ck+2,αβ+2 ≤ C2||f ||Ck,αβ . Here Ωn−1 is the volume of the unit sphere Sn−1 in Rn. The next result about Fredholm properties of the Laplaian on ALE manifolds is re- lated to Theorem 5.2.7 but comes from a different source. We have borrowed it from [CON-HEI13, Theorem 2.9]. They borrowed it from the PhD thesis of Stephen Marshall, who in turn borrowed it from [LOC-McO85]. Theorem 5.2.8. Let (X, g) be an ALEτ manifold asymptotic to Rn/G of order τ > 0. Let k ≥ 0 be an integer and let α ∈ (0, 1). Then the operator ∆ : Ck+2,αβ+2 (X)→ Ck,αβ (X) has finite dimensional kernel and closed range (i.e. is Fredholm) if β+2 /∈ {−n+2−N0}∪N0. 83 Chapter 5. ALE Ricci-flat deformations Weights in P = {−n + 2 − N0} ∪ N0 are called exceptional and weights in R\P are called not exceptional. We finish this section with a basic observation about ALE metrics. Lemma 5.2.9. Let (X, g) be an ALEτ manifold of order τ > 0. Then the metric g is complete. Proof. Let {xi}i∈Z≥0 be a Cauchy sequence on (X, g) and let X be equipped with a radius function ρ. We denote by h0 the flat metric on (Rn\BR(0))/G and by r its radius function. If {ρ(xi)}i∈Z≥0 is bounded by some R > 0 then {xi} is contained in the compact set K ∩ ρ−1([1, R]), so it must converge. If {ρ(xi)}i∈Z≥0 is unbounded, then for all  > 0 we can find N > 0 and R0 > 0 such that ∀n,m ≥ N we have dg(xn, xm) < 2 and R0r(pi∗xn)−τ < 2 . The metric g is ALEτ , so dh0(pi∗xn, pi∗xm) = dg(xn, xm) + O(r −τ ) <  2 +  2 = . The sequence {pi∗xi}i∈Z≥0 is therefore Cauchy with respect to h0. The metric h0 is complete, so it converges to some y ∈ (Rn\BR(0))/G. The sequence {xi}i∈Z≥0 therefore converges to x = pi∗y ∈ X . 5.3 ALE differential geometry In this section, we introduce various differential geometric constructions on ALE mani- folds. For an introduction to differential geometry on ALE manifolds we refer the reader to [JOY00, Chapter 8], where the material presented in this section is taken from. ALE de Rham Cohomology Let (X, g) be an ALE manifold asymptotic to Rn/G. We denote by H∗(X) the usual de Rham cohomology of X and we denote by H∗c (X) the de Rham cohomology of com- pactly supported forms on X . ALE manifolds can be regarded as compact manifolds with boundary Sn−1/G. Viewing an ALE manifold this way, the long exact sequence · · · → Hkc (X,R)→ Hk(X,R)→ Hk(Sn−1/G,R)→ Hk+1c (X,R)→ . . . can be used to get the following expression for the cohomology of X . H0(X,R) = R Hk(X,R) = Hkc (X,R) for 0 < k < n Hn(X,R) = 0. 84 5.3. ALE differential geometry On Weighted Hölder spaces the Hodge Laplacian ∆ = dd∗ + d∗d satisfies ∆ : Ck+2,αβ+2 (∧rT ∗X)→ Ck,αβ (∧rT ∗X). Elements in the kernel of ∆ are called harmonic forms. We denote the space of d-closed and d∗-closed r-forms by (Hk,αβ )r(X) = {η ∈ Ck,αβ (∧rT ∗X) | dη = 0 and d∗η = 0}. We denote by Hrβ(X) the space of smooth d-closed and d∗-closed r-forms with weight β. On a closed manifold ∆g-harmonic forms are closed and co-closed. On non-compact manifolds this need not be true. But, as the next lemma will show, the result can be recovered over a non-compact base for weighted spaces with suitable weights β. Proposition 5.3.1. Let (X, g) be an ALE manifold asymptotic to Rn/G. Let k > n 2 + 3, α ∈ (0, 1) and β < −n 2 . Then any ∆-harmonic form η ∈ Ck,αβ (∧rT ∗X) is d-closed and d∗-closed. Proof. Since k > n 2 +3 and β < −n 2 , Remark 5.2.5 says that there areL2-inner products on Ck,αβ (∧rT ∗X) and Ck−2,αβ−2 (∧rT ∗X). By Lemma 5.2.6 the operator d∗ is the formal adjoint of d with respect to the L2-inner product on Ck−2,αβ−2 (∧rT ∗X), so the usual argument from compact manifolds applies, i.e. for η harmonic, 0 = (∆η, η)L2 = ||d∗η||2L2 + ||dη||2L2 . Hence dη = 0 and d∗η = 0. The next result is the Hodge decomposition theorem for ALE manifolds. It is taken from [JOY00, Theorem 8.4.1]. Theorem 5.3.2. Let (X, g) be an ALEn manifold asymptotic to Rn/G for some n > 2. Then H0−n+1(X) = 0 = Hn−n+1(X) and the map Hr−n+1(X) → Hr(X,R) : η 7→ [η] induce isomorphisms Hr−n+1(X) ' Hr(X,R) ' Hrc (X,R) for 0 < r < n. Suppose that −n+ 1 ≤ β < −n/2. Then C∞β (∧rT ∗X) = Hr−n+1(X)⊗ d [ C∞β+1(∧r−1T ∗X) ]⊕ d∗ [C∞β+1(∧r+1T ∗X)] where the summands are L2-orthogonal. 85 Chapter 5. ALE Ricci-flat deformations ALE Kähler metrics The model for a complex ALE manifold of complex dimension m is (Cm/G, h0), where G is a finite subgroup of U(m) acting freely on Cm\{0} and h0 is the standard Hermitian metric on Cm. h0 is invariant under the action of G, so it descends to a Hermitian metric on Cm/G, which we also denote by h0. Denote by r the radius function defined by h0. Definition 5.3.3. Let (X, g) be a non-compact complex manifold of dimension m. We say that (X, g) is a complex ALEτ manifold asymptotic to Cm/G of order τ > 0 if there exists a compact subset K ⊂ X , an R > 0 and asymptotic coordinates(a diffeomorphism) pi : X\K → (Cm\BR(0))/G such that ∇k(pi∗g − h0) = O(r−τ−k) and ∇k(pi∗J − J0) = O(r−τ−k) on {z ∈ Cm/G | r(z) > R} for all k ≥ 0. Whenever we say that (X, g) is a complex ALE manifold asymptotic to Cm/G it is implied that X has complex dimension m. For the purpose of this project a resolution (X, pi) of a singularity Cm/G is a nonsingular manifold X and a proper birational map pi : X → Cm/G. It follows from [JOY00, Section 8.9] that an ALE2m Kähler manifold (X, pi) is birational to a deformation of Cm/G. However, for m ≥ 3 the Schlessinger Rigidity Theorem ([JOY00, Theorem 6.4.8]) asserts thatCm/G is rigid, so (X, pi) is in fact a resolution of Cm/G. It follows from [JOY00, Proposition 8.2.1] and [JOY00, Theorem 8.2.4] that a resolution (X, pi) of Cm/G for a finite subgroup G ⊂ U(m) acting freely on Cm\{0} with an ALE2m Ricci-flat Kähler metric g is in fact a crepant resolution, i.e. KX ' pi∗KCm/G, and G = SU(m). By [JOY00, Theorem 8.4.3] such a crepant resolution satisfies H2,0(X) = H0,2(X) = 0. By [JOY00, Proposition 8.4.5] ALE2m Kähler manifolds admit in each Kähler class an ALE2m Kähler metric which is flat, i.e. g = pi∗h0, outside a compact set. There are weighted Banach spaces of (p, q)-forms Llk,β(∧p,qX) and Ck,αβ (∧p,qX) on ALE Kähler manifolds and we have the following splitting theorem borrow from [JOY00, Theorem 8.4.2] Theorem 5.3.4. Let (X, J, g) be an ALE2m Kähler manifold asymptotic to Cm/G. Define Hp,q(X) = {η ∈ C∞−2m+1(∧p,qX) | dη = d∗η = 0}. Then Hp,q(X) is a finite dimensional vector space, and the map from Hp,q(X) to Hp+q(X,C) defined by η 7→ [η] is injective. Define Hp,q(X) to be the image of this map. Then Hk(X,C) = ⊕kj=0Hj,k−j(X) for 0 < k < 2m. 86 5.4. Slice construction Lemma 5.3.5. Let (X, J, g) be a complex ALE Kähler manifold asymptotic to Cm/G. Let k > 2m+ 3 be an integer and let β < −m. Then any ∆∂¯-harmonic form in Ck,αβ (∧0,rX) is ∂¯- and ∂¯∗-closed. Proof. Let η be a real form in ker(∆∂¯) ∩ Ck,αβ (∧0,rT ∗X). On Kähler manifolds ∆∂¯η = 1 2 ∆gη = 0, and so for η ∈ ker(∆∂¯) ∩ Ck,αβ (∧0,r(T ∗X), Lemma 5.3.1 implies that dη = 0 and d∗η = 0. On a complex manifold d = ∂ + ∂¯ and d∗ = ∂∗ + ∂¯∗, so ∂¯η = 0 and ∂¯∗η = 0. We borrow a global ∂∂¯-lemma from [JOY00, Theorem 8.4.4] Theorem 5.3.6. Let m > 1 and let (X, J, g) be an ALE2m Kähler manifold asymptotic to Cm/G. Let β < −m and suppose that η ∈ C∞β (∧1,1R X) is a closed real (1, 1)-form and [η] = 0 in H2(X,R). Then there exists a unique real function u ∈ C∞β+2(X) with η = ∂∂¯u. 5.4 Slice construction In this section, we construct a slice for the action of a group of diffeomorphisms on the space of ALE metrics on an ALE manifold. A slice through a point is a subset containing the point such that for each orbit passing through the subset exactly one element of the orbit lies in the subset. It is used as a chart on the orbit space. Koiso used in [KOI83] the two equations δgh = 0 and ∫ trgh = 0 to construct a slice for the action of the group of diffeomorphisms on the volume 1 metrics on a compact manifold. ALE manifolds have infinite volume, so it will be convenient to replace to two equations δgh = 0 and ∫ trgh = 0 with the single equation (2δg + d trg)h = 0. This slice equation has been used before by various people, e.g. Biquard([BIQ00]), Kovalev([KOV06]) and Nordström([NOR08]). The spaceMALEτ Let (X, g) be an ALEτ manifold asymptotic to Rn/G or order τ > 0. Denote byM the collection of complete Riemannian metrics in C∞(Sym2(T ∗X)). We give it the subspace topology. ALEτ metrics belong to M by Lemma 5.2.9. Denote byMALE andMALEτ the subspaces ofM of ALE and ALEτ metrics respectively and equip them both with a subspace topology. 87 Chapter 5. ALE Ricci-flat deformations Let ρ be a radius function on the ALEτ manifold X and let pi be the asymptotic co- ordinate system. Take β ∈ R. A function f ∈ C∞(X) satisfies f = O(ρβ) exactly if pi∗f = O((pi∗ρ)β), which happen exactly if pi∗f = O(rβ). Also |∇kpi∗gpi∗f −∇kh0pi∗f |h0 = O(r−τ−k). Hence f ∈ C∞−τ (X) exactly if |∇kh0pi∗f |h0 = O(r−τ−k). The same holds for tensors. If h ∈ C∞(Sym2(T ∗X)) with |h|g < 1 satisfies |∇kh0h|h0 = O(r−τ−k) then g + h is ALEτ . The tangent space toMALEτ at g is therefore described as Tg(MALEτ ) = C∞−τ (Sym 2(T ∗X)). Similarly, for any two ALEτ metrics g′, g′′ the difference h = g′ − g′′ is an element of C∞−τ (Sym 2(T ∗X)). We say that a deformation gt of g is small if |ht|g, where ht = gt − g, is sufficiently small. Lemma 5.4.1. Let (X, g) be an ALEτ manifold asymptotic to Rn/G of order τ > 0. Then for any τ ′ ≤ τ , small ALEτ ′ deformations of g are ALEτ ′ . This follows from observing that for an ALEτ metric g a deformation g + h of g with h ∈ C∞−τ ′(Sym2(T ∗X)) satisfies ∇k(pi∗(g + h) − h0) = ∇k(pi∗g − h0) + ∇k(pi∗h) = O(r−τ−k) +O(r−τ ′−k) = O(r−τ ′−k). We define the space Mk,τ = L2k,−τ (Sym2(T ∗X)) ∩ MALEτ . This definition makes sense as any g′ ∈ MALEτ can be written as g + h for some h ∈ L2k,−τ (Sym2(T ∗X)) and g′ satisfies ∇gg′ = ∇g(h − g) = ∇gh. Hence ||g′||L2k,−τ = ||h||L2k,−τ . The tangent space TgMk,τ is naturally identified with L2k,−τ (Sym2(T ∗X)). Also define Mk,αALEτ and MkALEτ to be the ALEτ metrics in Ck,α−τ (Sym2(T ∗X)) and Ck−τ (Sym2(T ∗X)) respectively. Same argument as forMk,τ shows that these definitions make sense. The tangent space TgMk,αALEτ is naturally identified with Ck,α−τ (Sym2(T ∗X)) and the tangent space TgMkALEτ is naturally identified with Ck−τ (Sym 2(T ∗X)). Diffeomorphisms Let (X, g) be an ALEτ manifold asymptotic toRn/G. Denote by δ the injectivity radius of X . Note that for ALE manifolds the injectivity radius is always strictly positive. Denote by D = D(X) the group of diffeomorphisms fromX to itself and equip it with the supremum norm ||φ||∞ = supx∈X distg(x, φ(x)). We say that φ ∈ D is given by V ∈ C∞(TX), with ||V ||C1 sufficiently small, if φ(p) = expp(Vp) for each p ∈ X . Definition 5.4.2. Let (X, g) be an ALEτ manifold asymptotic to Rn/G or order τ > 0. Let δ > 0 be the injectivity radius of X . Denote by Dτ the group of diffeomorphisms 88 5.4. Slice construction generated by the neighbourhood of the identity map defined by those diffeomorphisms φ which are given by some V ∈ C∞−τ (TX) with |V |g ≤ δ. A similar group of diffeomorphisms has been used by Kovalev in [KOV06]. The weight of the generating vector field V of a diffeomorphism φ ∈ Dτ+1 in Definition 5.4.2 ensures that φ satisfies |∇kh0(pi∗(φ∗ − Id∗)g′)|h0 = O(r−τ−k) for all g′ ∈ MALEτ and all k ≥ 0. This gives us the next lemma. Lemma 5.4.3. Let (X, g) be an ALEτ manifold asymptotic to Rn/G of order τ > 0. For any φ ∈ Dτ+1 and any g′ ∈MALEτ the metric φ∗g′ isMALEτ . Proof. For all k ≥ 0 the metric φ∗g′ satisfies |∇kh0(pi∗φ∗g′ − h0)|h0 = |∇k(pi∗φ∗g′)−∇kh0| ≤ |∇k(pi∗φ∗g′)−∇k(pi∗g′)|+ |∇k(pi∗g′)−∇kh0| = |∇k(pi∗(φ∗ − Id∗)g′)|+ |∇k(pi∗g′ − h0)| = O(r−τ−k) +O(r−τ−k) = O(r−τ−k). The metric φ∗g′ is therefore ALEτ . We explained in Section 2.3 that a 1-parameter flow φt of diffeomorphisms with in- finitesimal generator X ∈ C∞(TX) satisfies LXg = ddt(φ∗tg)|t=0. If the flow φt belongs to Dτ+1, then Tg((Dτ+1)∗g) ⊂ Tg(MALEτ ). Define δ∗ : C∞β (Sym k(T ∗X)) → C∞β−1(Symk+1(T ∗X)) : η 7→ Sym ◦ ∇|Symk(T ∗X)), where β ∈ R. For β < −n 2 , δ∗ admits an L2-formal adjoint δ given by δ = ∇∗|Symk+1(T ∗X). This is similar to the construction in Section 2.3. From [BES87, Lemma 1.60] we get the expression δ∗η = 1 2 Lη#g for a 1-form η ∈ C∞β (T ∗X), where β < −n2 and η# is the dual vector field. The tangent space of the orbit (Dτ+1)∗g at g can therefore be described as Tg((Dτ+1)∗g) = Im(δ∗) ∩ Tg(MALEτ ). Slice equation Proposition 5.4.4. Let (X, g) be an ALEτ Ricci-flat manifold asymptotic toRn/G of order τ > 0. Let k > n 2 + 3, α ∈ (0, 1) and β < −n 2 . Then for η ∈ Ck,αβ (∧1T ∗X), ∆η = ∇∗∇η. Also ∆η = 0 exactly if∇η = 0. 89 Chapter 5. ALE Ricci-flat deformations Proof. The first statement follows immediately from the Weitzenböck formula for 1- forms, (2.4). The second statement follows from integration by parts, Lemma 5.2.6, as (∆η, η)L2 = (∇∗∇η, η)L2 = (∇η,∇η)L2 = ||∇η||2. Proposition 5.4.5. Let (X, g) be an ALEτ Ricci-flat manifold asymptotic toRn/G of order τ > 0. Let k > n 2 + 3, α ∈ (0, 1) and β < −n 2 . Then on the space Ck,αβ (∧1T ∗X) it holds that ∆ = (2δg + d trg)δ∗. Proof. The symmetric part of∇ : Ck,αβ (∧1T ∗X)→ Ck−1,αβ−1 (∧2T ∗X) is δ∗ and on 1-forms the anti-symmetric part is 1 2 d. So for α ∈ Ck,αβ (∧1T ∗X) we have δ∗α = ∇α − 12dα for a 1-form α. Also, trg ◦ δ∗g = d∗. Using Proposition 5.4.4 we get (2δ + d trg)δ∗ = 2∇∗∇−∇∗d+ d trgδ∗ = 2∇∗∇− d∗d− dd∗ = ∆g. Proposition 5.4.6. Let (X, g) be an ALEτ Ricci-flat manifold asymptotic toRn/G of order τ > 0. Let k > n 2 + 3, α ∈ (0, 1) and β < −n 2 . Then Ck,αβ (Sym 2(T ∗X)) = Im(δ∗g)⊕ ker(2δg + d trg), where δ∗g : C k+1,α β+1 (∧1T ∗X)→ Ck,αβ (Sym2(T ∗X)) and 2δg + d trg : Ck,αβ (Sym2(T ∗X))→ Ck−1,αβ−1 (∧1T ∗X). Proof. Ricci-flatness of g and the weight β allows us to apply Proposition 5.4.4, so har- monic 1-forms in Ck+1,αβ+1 (∧1T ∗X) are parallel and closed. On such 1-forms δ∗ = ∇− 12d. For a harmonic form α ∈ Ck+1,αβ+1 (∧1T ∗X) we therefore have δ∗α = (∇− 1 2 d)α = ∇α− 1 2 dα = 0− 0 = 0, which by contraposition implies that if an α ∈ Ck+1,αβ+1 (∧1T ∗X) satisfies δ∗α 6= 0 then ∆gα 6= 0. Using the relation ∆ = (2δ + d trg)δ∗ on 1-forms from Proposition 5.4.5 we conclude that non-zero elements in Im(δ∗) are not in the kernel of 2δ + d trg. Take h ∈ Ck,αβ (Sym2(T ∗X)). Then (2δg + d trg)h ∈ Ck−1,αβ−1 (X). Theorem 5.2.7 then provides a solution u to the equation ∆u = (2δg + d trg)h. Now, (2δg + d trg)(δ∗gu− h) = ∆u − (dδg + d trg)h = ∆u − ∆u = 0, so the element δ∗gu − h satisfies δ∗gu − h ∈ 90 5.4. Slice construction ker(2δg + d trg). Hence we have arrived at the desired decomposed h = δ∗gu+ (δ ∗ gu− h), where h has been decomposed into a sum of an element from Im(δ∗g) and and element from ker(2δg + d trg). For an ALEτ manifold (X, g) asymptotic to Rn/G of order τ > n2 we call the equation (2δg + d trg)h = 0 for the slice equation for the action of the diffeomorphism group Dτ+1 as solutions to it in Ck,α−τ (Sym 2(T ∗X)) are orthogonal to the tangent space of the orbit (Dτ+1)∗g at g. In the next subsection we will see that exponentiating a neighbourhood of 0 in ker(2δ + d trg) ⊂ C∞−τ (Sym2(T ∗X)) produces a slice through g in MALEτ for the action of Dτ+1. Slice construction Let (X, g) be an ALEτ manifold asymptotic to Rn/G of order τ > 0. In this subsection, we produce a slice for the action of Dτ+1 on the space MALEτ . Denote by Dk,ατ the subgroup of Dτ consisting of those diffeomorphisms which are Ck,α. Denote byMk,αALEτ the spaceMALEτ ∩ Ck,α(Sym2(T ∗X)). We have the following action Dk+1,ατ+1 ×Mk,αALEτ →Mk,αALEτ : (φ, g′) 7→ φ∗g′. Denote by Ik,ατ (g) the isometry group {η ∈ Dk,ατ | η∗g = g}. Theorem 5.4.7 (Slice theorem, Ck,ατ version). Let (X, g) be an ALEτ manifold asymptotic to Rn/G of order τ > n 2 and let k > n 2 + 3. Then there exists a subset Sk,ατ ⊂ Mk,αALEτ through g satisfying i) If η ∈ Ik+1,ατ+1 (g) then η∗(Sk,ατ ) = Sk,ατ . ii) If η ∈ Dk+1,ατ+1 satisfies η∗Sk,ατ ∩ Sk,ατ 6= ∅ then η ∈ Ik+1,ατ+1 (g). iii) There exists a local cross section χ : Dk+1,ατ+1 /Ik+1,ατ+1 (g)→ Dk+1,ατ+1 defined in a neigh- bourhood U of the identity coset such that F : U × Sk,ατ →Mk,αALEτ : (u, t) 7→ χ(u)∗t is a homeomorphism onto a neighbourhood of g. 91 Chapter 5. ALE Ricci-flat deformations Proof. The construction of the local cross section χ in [EBI70, Proposition 5.10] general- izes to ALEτ manifolds to produce a local cross section χ : Dk+1,ατ+1 /Ik+1,ατ+1 (g) → Dk+1,ατ+1 . Denote by Ok+1,ατ+1 (g) the orbit (Dk+1,ατ+1 )∗g = (Dk+1,ατ+1 /Ik+1,ατ+1 (g))∗g. It is a submanifold of the Banach manifold Mk,αALEτ . Denote by ν the smooth normal bundle to the tangent bundle of Ok+1,ατ+1 (g). By Proposition 5.4.6 fibres are given by νg′ = ker(2δ∗g′ + 2dtrg′) ∩ Ck,α−τ (Sym 2(T ∗X)). Let U ′ ⊂ Dk+1,ατ+1 /Ik+1,ατ+1 (g) be a neighbourhood of the identity coset small enough that it admits a cross section χ : U ′ → Dk+1,ατ+1 and let ′ > 0 and U ′ be chosen such that the following is satisfied. Let V ′ = {v ∈ νg | ||v||g ≤ (′)2} and let W ′ = {d(η∗)v | v ∈ V ′ and η ∈ χ(U ′)} be such that exp |W ′ is a diffeomorphism onto its image and exp(W ′)∩ Ok+1,ατ+1 (g) = U ′. For some δ > 0 we have B2δ(g) ⊂ exp(W ′). Now choose U ⊂ U ′ and  < ′ such that exp(W ) ⊂ Bδ(g). Define Sk,ατ = exp(V ). The set Sk,ατ has the three properties of a slice. The argument is similar to the orbifold case. See the proof of Claim 3.6.5 for details. Theorem 5.4.8 (Slice theorem, C∞τ version). If we replace Dk+1,ατ+1 by Dτ+1 andMk,αALEτ byMALEτ in the hypothesis of Theorem 5.4.7, then there exists a slice Sτ ⊂MALEτ with the three properties of a slice outlines in Theorem 5.4.7. Proof. Generalizing the proof of Theorem 5.4.7 to the smooth case is similar to the proof of [EBI70, Theorem 7.4]. The details of the proof of Theorem 3.6.6 applies to the case of ALEτ manifolds. From Theorem 5.4.8 it follows that Corollary 5.4.9. Let (X, g) be an ALEτ manifold asymptotic to Rn/G of order τ > n2 . Then the slice Sτ ⊂ MALEτ through g constructed in Theorem 5.4.8 produces a homeo- morphism Sτ/Iτ+1(g)→ U ⊂MALEτ/Dτ+1 where U is a neighbourhood of the identity coset [g]. Denote by Pg ⊂ Sτ the subspace of Ricci-flat metrics in Sτ and call it the premod- uli space of Ricci-flat metrics. Denote by RALEτ the subspace of MALEτ of Ricci-flat metrics. 92 5.5. ALE Ricci-flat deformations Corollary 5.4.10. Let (X, g) be an ALEτ Ricci-flat manifold asymptotic to Rn/G of order τ > n 2 , then the homeomorphism of Proposition 5.4.9 restricts to a homeomorphism Pg/Iτ+1(g)→ U ⊂ RALEτ/Dτ+1. 5.5 ALE Ricci-flat deformations In this section we study ALEτ Ricci-flat deformations on a non-compact manifold X . We introduce the space of infinitesimal Ricci-flat deformations and study the Hermitian and skew-Hermitian components of it. Let (X, g) be a Riemannian manifold. A family of deformations of g is a smooth curve (−, ) → C∞(Sym2(T ∗X)) : t 7→ gt with g0 = g. The deformation gt can be written as gt = g + ht for ht ∈ C∞(Sym2(T ∗X)). We say that a family of deformations is ALEτ , Einstein, Ricci-flat, Kähler etc. if each deformation is ALEτ , Einstein, Ricci-flat, Kähler respectively. Infinitesimal Ricci-flat deformations Let (X, g) be an ALEτ Ricci-flat manifold asymptotic to Rn/G of order τ > n2 . In the last section we constructed for k > n 2 + 3 a slice Sk,ατ ⊂ MALEτ for the pull-back action by diffeomorphisms. Consider an ALEτ Ricci-flat deformation gt of smooth Ricci-flat metrics of g. Then h = d dt gt|t=0 ∈ C∞−τ (Sym2(T ∗X)) satisfies Ric′g(h) = 0. Assuming that the curve gt belongs to Sk,ατ , then h belongs to TgS k,α τ and satisfies the equation (2δg+ d trg)h = 0. This inspires the next definition. Definition 5.5.1 (Infinitesimal Ricci-flat deformations). Let (X, g) be an ALEτ Ricci-flat manifold asymptotic to Rn/G of order τ > 0. For an integer k > 2 and α ∈ (0, 1) define the space k,α−τ (g) ⊂ Ck,α−τ (Sym2(T ∗X)) by the equations Ric′g(h) = 0, (2δg + d trg)h = 0. (5.3) Similarly define ∞−τ (g) to the be solutions to (5.3) in C ∞ −τ (Sym 2(T ∗X)). We call ∞−τ (g) the space of infinitesimal Ricci-flat deformations of g. 93 Chapter 5. ALE Ricci-flat deformations Lemma 5.5.2. Let (X, g) be an ALEτ Ricci-flat manifold asymptotic to Rn/G of order τ > 0. Let k > n 2 + 3 and α ∈ (0, 1). Then dim k,α−τ (g) <∞. Proof. Towards infinity the curvature R approximates zero, so the elliptic operator ∆L = ∇∗∇−2 ◦R approximates∇∗∇, and ∆ via the Weitzenböck formula. Choose an  ∈ (0, 1) such that−τ+ is not exceptional and denote by L2k,−τ+ the space L2k,−τ+(Sym2(T ∗X)). It then follows from [CAN81, Theorem 7.4] that ∆L|L2k,−τ+ has a finite dimensional ker- nel. From the inclusion k,α−τ (g) ⊂ Ck,α−τ (Sym2(T ∗X)) ⊂ L2k,−τ+(Sym2(T ∗X)) it then follows that dim k,α−τ (g) <∞. We remark that the question about the finite dimensionality of k,α−τ (g) in Lemma 5.5.2 is independent of k, α and τ , provided k and τ are sufficiently large. The space ∞−τ (g) is contained in k,α−τ (g) so it follows from Lemma 5.5.2 that ∞−τ (g) is finite dimensional. For the convenience of the reader we restate Theorem 5.1.3. Theorem 5.5.3. Let (X, g) be an ALEτ Ricci-flat manifold asymptotic toRn/G, for τ > n2 . Let Sτ be the slice through g in MALEτ for the action of the group of diffeomorphisms Dτ+1. Then there exists a finite dimensional real analytic submanifold Z ⊂ Sτ with TgZ =  ∞ −τ (g) and Z containing Pg as a real analytic subset. Proof. Let k ≥ n 2 + 3 and let Sk,ατ be a slice through g in Mk,αALEτ . Restrict the Ricci operator Ric to this slice, i.e Ric : Sk,ατ ⊂ Mk,αALEτ → Ck−2,α−τ−2 (Sym2(T ∗X)). Choose an  ∈ (0, 1) such that−τ + is not exceptional. The construction of a slice in Theorem 5.4.7 could equally well have been proved for Sobolev spaces instead of Hölder spaces. We can therefore consider the slice S2k,−τ+ of L 2 k,−τ+-metrics though g. The elliptic operator F = 2Ric′g + 2δ ∗δ = ∇∗∇ − 2 ◦R −∇d trg on the Hilbert space L2k,−τ+(Sym2(T ∗X)) has closed range by [CAN81, Theorem 7.4] and it follows from a similar argument to [BES87, Lemma 12.48] that F (TgS2k,−τ+) is closed as well. The operator Ric satisfies Ric′g(TgS 2 k,−τ+) = F (TgS 2 k,−τ+) and it is real analytic by [DeT-KAZ81]. We can there- fore apply Theorem 2.1.11 to conclude that there exists a neighbourhood U of g in S2k,−τ+ such that U ∩ (Pg)2k,−τ+ is a real analytic subset of a real analytic submanifold Z2k,−τ+ ⊂ U whose tangent space TgZ2k,−τ+ coincides with ker(Ric ′ g)∩ TgS2k,−τ+ = 2k,−τ+(g). In- tersecting this byCk,α−τ (Sym 2(T ∗X)) gives us a real analytic submanifold Zk,α−τ of Sk,ατ with TgZ k,α −τ =  k,α −τ (g) which contains U ∩ (Pg)k,α−τ as a real analytic subset. By Lemma 5.5.2 the space k,α−τ (g) is finite dimensional. This holds for arbitrarily large k. In particular, it holds for the slice Sτ . 94 5.5. ALE Ricci-flat deformations Let (X, g) be a Ricci-flat ALEτ manifold asymptotic to Rn/G of order τ > n2 . From Theorem 5.5.3 we know that every h ∈ ∞−τ (g) integrates into a curve of ALEτ metrics through g in the slice Sτ . Similarly to the compact case, we wish to identify conditions on X and g for which we can be sure that any h ∈ ∞−τ (g) integrates into a curve of ALEτ Ricci-flat metrics through g. Note that if each h ∈ ∞−τ (g) is the infinitesimal deformation of a curve inMALEτ through g of Ricci-flat metrics, then the premoduli space Pg of ALEτ Ricci-flat metric spans an entire neighbourhood of g in Sτ . In proving Theorem 5.1.5 we will be working with Ricci-flat Kähler metrics which are of class at least C2, so they are actually smooth by [DeT-KAZ81, Theorem 6.1]. We will therefore not limit ourselves by restricting our attention to ∞−τ (g) rather than bigger spaces like k,α−τ (g). From now on let (X, J, g) be an ALEτ Ricci-flat Kähler manifold asymptotic to Cm/G for a finite subgroup G ⊂ U(m) of order τ > m. Any symmetric (2, 0)-tensor h ∈ C∞(Sym2(T ∗X)) splits as h = hH + hA, where hH is Hermitian symmetric and hA is skew-Hermitian symmetric, that is hH(JX, JY ) = hH(X, Y ) and hA(JX, JY ) = −hA(X, Y ). The defining equations of ∞−τ (g) for Hermitian and skew-Hermitian tensors respectively can be reformulated in a convenient way. For appropriately chosen weights this is similar to the compact case (and the orbifold case), so we keep it short to minimize repetition. Note that if h = O(r−τ ), then h ◦ J = O(r−τ ) as h ◦ J(X) = (h ◦ J ◦ pi1,0 + h ◦ J ◦ pi0,1)(X1,0 +X0,1) = h ◦ J |T 1,0X(X1,0) + h ◦ J |T 0,1X(X0,1) = ih(X1,0) + (−i)h(X0,1) = O(r−τ ) +O(r−τ ) = O(r−τ ). The complex structure J cannot belong to C∞−τ (TX ⊗ T ∗X) for any τ > 0, as J satisfies (J2)ij = J i kJ k j = δij . If |J ij| → 0 at any rate, then so does |J ikJkj| which contradicts the definition of J . For a smooth curve of complex structures Jt let I = ddtJt|t=0 ∈ C∞(TX ⊗ T ∗X). Differentiating −Id = J2t with respect to t shows that I satisfies IJ + JI = 0 and differentiating N(Jt) = 0 gives us 0 = N ′J(I) = 1 2 J ◦ ∂¯I , where N is the Nijenhuis tensor of J . The tensor g ◦ I is skew-Hermitian as g ◦ I(JX, JY ) = g(JX, IJY ) = −g(JX, JIY ) = −g ◦ I(X, Y ). 95 Chapter 5. ALE Ricci-flat deformations Definition 5.5.4. Let (X, J, g) be a complex ALEτ manifold asymptotic to Cm/G of order τ > m. We define the space of infinitesimal complex deformations ICD∞−τ (J) to be the subspace of C∞−τ (TX⊗T ∗X) consisting of those I which satisfy ∂¯I = 0 and IJ+JI = 0. We say that I is symmetric or anti-symmetric if g ◦ I is symmetric or anti-symmetric respectively. Denote by Is and Ia the symmetric and anti-symmetric part of I respec- tively and by ICD∞−τ (J)S and ICD ∞ −τ (J)A the symmetric and anti-symmetric subspaces of ICD∞−τ (J) respectively. For a Hermitian symmetric tensor hH the tensor ψ = hH ◦ J satisfies ψ(X, Y ) = −ψ(Y,X), so it is a real differential 2-form of order−τ . Also, Jψ(X, Y ) = i1−1ψ(X, Y ), so ψ = hH ◦J is a differential form of type (1, 1). The same calculation as in (3.10) shows that for g Kähler, δghH = −(d∗ψ) ◦ J. (5.4) Let ω be the Kähler form of g with respect to J . A local calculation shows that trghH = (ψ, ω)g. (5.5) By [BES87, (12.92’)] The Weitzenböck formula for Hermitian tensors is ∆ψ = (∇∗∇− 2 ◦R)hH ◦ J. (5.6) A skew-Hermitian symmetric tensor hA can be identified with a real endomorphism I ∈ End(TX) via hA ◦ J = g ◦ I. (5.7) The same calculation as in (3.14) shows that the endomorphism I from (5.7) satisfies the relation g(X, IJY ) = −g(X, JIY ), so g(X, (IJ + JI)Y ) = 0, i.e I anti-commutes with J . This implies, for X ∈ T 0,1X , that J(IX) = −IJX = −I(−iX) = iIX , so IX ∈ T 1,0X and I : T 0,1X → T 1,0X . I may therefore be regarded as an element of A0,1(T 1,0X). The metric g is bounded but does not decay to zero and the form h ◦ J satisfies h◦J = O(r−τ ). Locally (hA)ikJkj = gikIkj . Hence each Ikj satisfies Ikj = O(r−τ ). So we can think of I as an element of A0,1−τ (T 1,0X). The tensor field ∂¯∗I ∈ C∞−τ−1(TX ⊗ 96 5.5. ALE Ricci-flat deformations T ∗X) is related to δghA ∈ C∞−τ−1(T ∗X) via δghA = −J ◦ ∂¯∗I. (5.8) In particular, δghA = 0 exactly when I is ∂¯∗-closed. From [BES87, (12.93’)] we get a Weitzenböck formula for the complex Laplacian ∆∂¯ , (∇∗∇− 2 ◦R)hA ◦ J = g ◦∆∂¯I. (5.9) So (∇∗∇ − 2 ◦R)hA = 0 exactly when I ∈ A0,1−τ (T 1,0X) is in the kernel of ∆∂¯ . For the convenience of the reader we restate Proposition 5.1.4. Proposition 5.5.5. Let (X, g) be an ALEτ Ricci-flat Kähler manifold asymptotic to Cm/G of order τ > m. For an infinitesimal Ricci-flat deformation h ∈ ∞−τ (g) both its Hermitian part and its skew-Hermitian part lie in ∞−τ (g). Proof. Let h = hH +hA be the decomposition of h into its Hermitian and skew-Hermitian parts. We show that hH ∈ ∞−τ (g). This suffices, as ∞−τ (g) is a vector space, so hA = h − hH ∈ ∞−τ (g). Expanding the definition of ∇∗∇ − 2 ◦ R and using that on a Kähler manifold ∇XJY = J∇XY , a computation shows that (∇∗∇− 2 ◦ R)hA is anti-Hermitian and (∇∗∇−2 ◦R)hH is Hermitian. The decomposition of symmetric tensors into Hermitian and anti-Hermitian ones is orthogonal at all points and∇∗∇− 2 ◦R is linear. The tensor h satisfies (∇∗∇− 2 ◦R)h = 0, which implies (∇∗∇− 2 ◦R)hH = 0. (5.10) The tensor hA is skew-Hermitian, so trghA = 0. Hence trghH = trgh−trghA = 0. To show that δghH = 0 we use the Weitzenböck formula (5.6) to deduce from (5.10) that ψ = hH◦J satisfies ∆ψ = 0. The weight of ψ satisfies −τ < −m just as hH , so by Lemma 5.3.5 ψ is d∗-closed. Equation (5.4) then tells us that δghH = 0. Hence hH ∈ ∞−τ (g). Let (X, J, g) be a complex Kähler manifold. Let Jt be a smooth curve of complex structures through J and let gt be a corresponding smooth curve of metrics through g such that each gt is Kähler with respect to Jt. Let h = ddtgt|t=0 and I = ddtJt|t=0. The (1, 1)- tensor I = I ij is identified with a (2, 0)-tensor g ◦ I . The components of the latter tensor 97 Chapter 5. ALE Ricci-flat deformations we denote Iij . According to [KOI83, section 9] the tensors h and Iij are related via 2ihαβ + (Iαβ + Iβα) = 0 (5.11) 2i(Dαhβγ¯ −Dβhαγ¯) = Dγ¯(Iαβ − Iβα), (5.12) where we use Greek indices as explained in the preliminaries. We say that a symmetric (2, 0)-tensor h and an infinitesimal complex deformation I are Kähler related when they satisfy the relations (5.11) and (5.12). The space ∞−τ (g) can be written as  ∞ −τ (g) =  ∞ −τ (g)H ⊕ ∞−τ (g)A, where ∞−τ (g)H and ∞−τ (g)A denote the subspaces of  ∞ −τ (g) of Hermitian and skew-Hermitian tensor fields respectively. In the following two subsections the subspaces ∞−τ (g)H and  ∞ −τ (g)A will be studied separately. Hermitian symmetric 2-tensors Let (X, J, g) be an ALEτ Ricci-flat Kähler manifold asymptotic to Cm/G of order τ > m. As explained above, any Hermitian symmetric (2, 0)-tensor h can be identified with the (2, 0)-tensor h ◦ J which is a real differential 2-form of type (1, 1). Let h ∈ ∞−τ (g)H , then (∇∗∇− 2 ◦R)h = 0 and the Weitzenböck formula, (5.6), gives us ∆(h ◦ J) = (∇∗∇− 2 ◦R)h ◦ J = 0. Proposition 5.3.1 guarantees that ∆(h ◦ J) = 0 implies h ◦ J ∈ H1,1−τ,R(X). This way we have identified elements of ∞−τ (g)H with real harmonic (1, 1)-forms with weight −τ , i.e with element of the space H1,1−τ,R(X). Denote the Kähler form of g by ω. As outlined in (5.5) the trace of h with respect to g becomes trg(h) = (h ◦ J, ω)g. The tensor h is trace-free, so the form h ◦ J is orthogonal to ω. Assume that τ = 2m − 1. Then, by Theorem 5.3.2, we know that H1,1−2m+1,R(X) ' H1,1R (X) The map ∞−2m+1(g) → H1,1R (X) : h 7→ h ◦ J is linear. Locally the relation between h and the harmonic form ψ = h ◦ J is ψαβ¯ = −ihαβ¯ , from which we see that the map is both injective and surjective. The map is therefore an isomorphism of vector spaces and we have ∞−2m+1(g)H ' H1,1R (X)/R · ω. 98 5.5. ALE Ricci-flat deformations We have shown Proposition 5.5.6. Let (X, J, g) be an ALE2m−1 Ricci-flat Kähler manifold asymptotic to Cm/G of order 2m− 1. Then ∞−2m+1(g)H ' H1,1R (X)/R · ω is an isomorphism and dim ∞−2m+1(g)H = dimH 1,1 R (X)− 1. Remark 5.5.7. [KOI83, Lemma 9.8] implies that the infinitesimal Ricci-flat deformation h ∈ ∞−2m+1(g) Kähler related to 0 are exactly the elements h ∈ ∞−2m+1(g)H . Skew-Hermitian symmetric 2-tensors Let (X, J, g) be an ALE2m−1 Ricci-flat Kähler manifold asymptotic to Cm/G of order 2m − 1. From the above remarks about anti-Hermitian tensors hA ∈ ∞−2m+1(g)A we know that the endomorphisms I satisfying hA ◦ J = g ◦ I defines a ∂¯-closed element I ∈ A0,1−2m+1(T 1,0X). The calculation in (3.19) is valid also on non-compact manifolds and it shows that this I satisfies g ◦ I(X, Y ) = g ◦ I(Y,X) so I is symmetric. Lemma 5.5.8. Let (X, J, g) be an ALE2m−1 Ricci-flat Kähler manifold asymptotic to Cm/G of order 2m− 1. The map ∞−2m+1(g)A → ker(∆∂¯) ∩ ICD∞−2m+1(J)S : h 7→ I, (5.13) where I is given by the equation h ◦ J = g ◦ I , is bijective. Proof. For h ∈ ∞−2m+1(g)A the corresponding I ∈ A0,1−2m+1(T 1,0X) satisfies (5.8) and (5.9), which imply ∂¯∗I = 0 and ∆∂¯I = 0 respectively. I has weight −2m + 1, so by integration by parts, 0 = (0, I)g = (∆∂¯I, I)g = (∂¯ ∗∂¯I, I)g = ||∂¯I||2g. Hence I is ∂¯-closed and satisfies I ∈ ker(∆∂¯) ∩ ICD∞−2m+1(J)S . The maps h 7→ h ◦ J and Iij 7→ I ij are injective, so h 7→ I is injective. To show surjectivity take an arbitrary I ∈ ker(∆∂¯)∩ICD∞−2m+1(J)S . Observe that h(X, Y ) = −h◦J(X, JY ) = −g ◦I(X, JY ) = g ◦ (JI)(X, Y ), so define a h ∈ C∞−2m+1(Sym2(T ∗X)) by h(X, Y ) = g ◦ (JI)(X, Y ). 99 Chapter 5. ALE Ricci-flat deformations This h satisfies h ◦ J = g ◦ I , i.e. h is mapped to I . By (5.9) the tensor h satisfies (∇∗∇ − 2 ◦R)h = 0. The same calculation as in (3.20) shows that the tensor h is anti- Hermitian, i.e. h(JX, JY ) = −h(X, Y ). This implies that trgh = 0. The tensor h also satisfies (5.8), so δgh = 0. Hence (2δg + d trg)h = 0 and so h ∈ ∞−2m+1(g)A. Remark 5.5.9. The anti-symmetry of I ∈ A0,1−2m+1(T 1,0X)A means that g ◦ I is a differ- ential 2-form. The form g ◦ I is anti-Hermitian, so it is not of type (1, 1). I is a (0, 1)-form with values in T 1,0X , so the form g ◦ I is naturally identified with a (0, 2)-form. The same argument as in the proof of Corollary 3.7.9 shows that I is harmonic exactly if g ◦I is har- monic. Hence elements of ker(∆∂¯)∩ ICD∞−2m+1(J)A are in a one-to-one correspondence withH0,2−2m+1(X) ' H0,2(X). Proposition 5.5.10. Let (X, J, g) be an ALE2m−1 Ricci-flat Kähler manifold asymptotic to Cm/G of order 2m− 1. Then dim ∞−2m+1(g)A = 2 dimCH 1(X, TX)− 2 dimCH0,2(X, J). Proof. Take hA ∈ ∞−2m+1(g)A. From Lemma 5.5.8 it follows that the image of hA in (5.13) is a ∂¯-harmonic and symmetric element in A0,1−2m+1(T 1,0X) which defines a class [I] ∈ H1−2m+1(X, TX). Harmonic anti-symmetric infinitesimal complex deformations cor- respond bijectively to harmonic (0, 2)-forms on X via Remark 5.5.9. This completes the proof. Remark 5.5.11. The relation h 7→ I from Lemma 5.5.8 can alternatively be written like hαβ = iIαβ , which is equivalent to saying that h is Kähler related to I . See [KOI83, Lemma 9.3] for details. 5.6 Deformations of a pair (X¯,D) In this section, we introduce the deformation theory by Kawamata for a pair (X¯,D), where X¯ is a compact complex manifold and D is a smooth divisor, and link it to the deformations of complex structures on X = X¯\D. The section is divided into two parts. In the first part we introduce Kawamata’s defor- mation theory and in the second part we link infinitesimal deformations of the pair (X¯,D) to infinitesimal complex deformations of X = X¯\D. 100 5.6. Deformations of a pair (X¯,D) We are interested in studying complex deformations of X via the deformation theory for pairs (X¯,D) because we do not have a non-compact version of Kodaira’s results about the stability of the Kähler property for small deformations of a complex structure over non-compact manifolds. In the recent preprint [DER-KRO17] Deruelle-Kröncke proved an ALE version of the stability result of the Kähler property for complex deformations (more about this in Section 5.10). This could potentially have been used to simplify our work, but we decided not to follow their approach because it would have made a potential generalization of our results to asymptotically conical manifolds less natural (more about this in Section 5.11). We have for the same reason also formulated Kawamata’s deformation theory in slightly more general terms than what would have been strictly needed for the context of ALE manifolds. Deformations of the pair (X¯,D) Let X¯ be a compact complex manifold and let D be a smooth divisor on X¯ . Following [KAW78] we introduce the deformation theory of the pair (X¯,D). Definition 5.6.1. Let X¯ be a compact complex manifold with a smooth divisor D. We define the sheaf of logarithmic p-forms Ωp X¯ (logD) to be the subsheaf of the sheaf of mero- morphic p-forms consisting of those forms with a simple pole along D. The complex (Ω•X(logD), d) is called the holomorphic log complex. A family of deformations of the pair (X¯,D) is defined as follows. Definition 5.6.2. Let X¯ be a compact complex manifold and let D be a smooth divisor on X¯ . A deformation of the pair (X¯,D) is a tuple (X¯ ,X ,D, p¯i, S, s0, ψ), where X¯ and S are complex spaces and p¯i : X¯ → S is a proper smooth morphism. Fix s0 ∈ S and assume that the tuple satisfies i) ψ : X¯ → p¯i−1(s0) is an isomorphism of complex manifolds satisfying ψ(X) = p¯i−1(s0) ∩ X , where X = X¯\D. ii) D is a closed analytic subset of X¯ and X = X¯ − D. iii) p¯i is locally a projection of a product space, that is, for each p ∈ X¯ there exists an open neighbourhood Up of p in X¯ and an isomorphism φp : Up ' p¯i(Up)×W , where 101 Chapter 5. ALE Ricci-flat deformations W = Up ∩ p¯i−1(p¯i(p)) such that the diagram Up φp // p¯i "" p¯i(Up)×W pr1xx p¯i(Up) commutes. iv) p¯i is the restriction of a product space to D, i.e. for p ∈ X¯ we have φp(Up ∩ D) = p¯i(Up)× (W ∩ D). For each t ∈ S in Definition 5.6.2 the pre-image p¯i−1(t) is a compact complex subman- ifold of X¯ diffeomorphic to X¯ and p¯i−1(t) ∩ D is a smooth divisor on Xt diffeomorphic to D. In this case the family of deformations of the pair (X¯,D) produces in particular a family of complex deformations X¯t of X and a family complex deformation Dt of D for t ∈ S. Definition 5.6.3. For a pair (X¯,D) define the logarithmic tangent sheaf TX¯(logD) to be the subsheaf of the sheaf of holomorphic sections TX¯ consisting of derivations of OX¯ which sends the ideal sheaf of D, OD, into itself. Definition 5.6.4. For a pair (X¯,D), TX¯(logD) is the sheaf of infinitesimal automorphisms of X¯ which send D into itself. For a pair (X¯,D) let S be the parameter space from Definition 5.6.2. By [KAW78, p. 249] there is a Kodaira-Spencer map ρs0 : Ts0S → H1(X¯, TX¯(logD)), and H1(X¯, TX¯(logD)) is the space of infinitesimal deformations of the pair (X¯,D). It follows from [KAW78, Theorem 1] that deformations of a pair (X¯,D) always exists and it follows from [KAW78, Corollary 4] that when H2(X¯, TX¯(logD)) = 0, then the parameter space S is regular at s0 and each infinitesimal deformation integrates into a smooth curve of deformations (X¯t, Dt). Linking ∞−2m+1(g)A to infinitesimal deformations of (X¯,D) Let X¯ be a compact complex manifold and let D be a smooth divisor. The Riemann ex- tension theorem tells us when a holomorphic function on X¯\D extends across the divisor 102 5.7. Stability results for deformations of a pair (X¯,D) D. Theorem 5.6.5. Let f1 be a non-zero holomorphic function on a polydisc ∆ ⊂ Cm and let f2 be a bounded and holomorphic function on ∆¯\{f1 = 0}. Then f2 extends uniquely to a holomorphic function f¯2 on all of the polydisc ∆. ([GUN-ROS65, Theorem C.3]). For the pair (X¯,D) we can always extend a bounded holomorphic p-form η on X to X¯ , by extending the coefficients of ηU in each chart (U, φ) from U∩X to U using Theorem 5.6.5. Lemma 5.6.6. Let (X, J, g) be a complex ALEτ manifold asymptotic to Cm/G of order τ > 0 and assume that X arises as the complement of a smooth divisor D in a compact complex manifold (X¯, J¯). For I ∈ A0,1−τ (T 1,0X) there exists I¯ ∈ A0,1(T 1,0X¯) satisfying I¯|X = I . If ∂¯I = 0, then ∂¯I¯ = 0. Also, any I¯ ∈ A0,1−τ (T 1,0X) is in particular an element of A0,1(TX¯(logD)). Proof. Outside a compact set the form I ∈ A0,1−τ (T 1,0X) decays to zero, so it is bounded. In particular, all coefficients of I in the asymptotic coordinates pi are holomorphic and bounded. By Theorem 5.6.5 all coefficients of I extend uniquely across D to holomorphic functions on all of X¯ . Hence there exists an I¯ ∈ A0,1(T 1,0X¯) satisfying I¯|X = I . If I ∈ A0,1(T 1,0X) satisfies ∂¯I = 0, then the extended form I¯ satisfies ∂¯I¯ = 0 on X and the same argument as before shows that ∂¯I¯ = 0 on all of X¯ . Lemma 5.6.7. Assume the hypotheses of Lemma 5.6.6 and Lemma 5.5.8. Then each h ∈ ∞−2m+1(g)A corresponds to an infinitesimal deformation [I¯] ∈ H1(X¯, TX¯(logD)) of the pair (X¯,D). 5.7 Stability results for deformations of a pair (X¯,D) In this section, we prove stability results for deformations of the pair (X¯,D) for a compact Kähler manifold X¯ and a smooth divisor D. Lemma 5.7.1. Let X¯ be a compact Kähler manifold and let D be a divisor biholomorphic to CPm−1. Assume that the pair (X¯,D) admits a family of deformations (X¯t, Dt). Then for small t the divisor Dt is biholomorphic to CPm−1. 103 Chapter 5. ALE Ricci-flat deformations Proof. A deformation (X¯t, Dt) of the pair (X¯,D) is in particular a complex deformation of D, so for small t, Dt is Kähler. For small t, Dt is diffeomorphic to D ' CPm−1. It follows from [BES87, Theorem 11.30] that any Kähler manifold homeomorphic toCPm−1 is biholomorphic to CPm−1. In particular, Dt is biholomorphic to CPm−1. Lemma 5.7.2. Let X¯ be a compact Kähler manifold and letD be a smooth divisor. Assume that (X¯,D) admits a smooth family of deformation (X¯t, Dt) and assume that the line bundle LD is ample. Then for small t, LDt is ample. Proof. This is a direct consequence of the Kodaira Embedding Theorem. It says that a line bundle on a compact Kähler manifold (X¯, J) is positive if and only if it is ample. By definition a line bundle is positive if the first Chern class can be represented by a posi- tive real (1, 1)-form, which is equivalent to the positive definiteness of the corresponding symmetric (2, 0)-tensor, which is an open condition. Proposition 5.7.3. Let X¯ be a compact complex manifold with a smooth divisor D satis- fying KX¯ = −βLD, i.e. c1(KX¯) = −βc1(LD), for some β ≥ 1. Let (X¯t, Dt) be a smooth family of deformations of the pair (X¯,D). Then for small t, KX¯t = −βLDt . Proof. The first Chern class of the line bundles KX¯t and LDt take values in H 2(X¯,Z) and are therefore stable for small deformations of the pair (X¯,D), i.e. c1(KX¯t) = c1(KX¯) and c1(LDt) = c1(LD) for small t. In particular, c1(KX¯) = −βc1(LD) implies that c1(KX¯t) = −βc1(LDt) for small t. Proposition 5.7.4. Let X¯ be a compact Kähler manifold with a smooth ample divisor D satisfying KX¯ = −βLD for some β ≥ 1 and assume that X = X¯\D admits an ALE2m Ricci-flat Kähler metric g. Also assume that the pair (X¯,D) admits a smooth family of deformations (X¯t, Dt). Then there exists a smooth family of complete Kähler metrics gt for the family Xt = X¯t\Dt through g. Proof. For the smooth family of deformations (X¯t, Dt) of the pair (X¯,D) denote by J¯t the corresponding family of complex structures on X¯t and by Jt the restriction of J¯t to Xt. It satisfies J¯0 = J¯ , where J¯ is the complex structure on X¯ . From Proposition 5.7.3 it follows that c1(K−1X¯t ⊗ βL−1Dt ) = 0. For β = 1 [TIA-YAU90, Theorem 4.1] ensures that for every ρt ∈ c1(K−1X¯t ⊗L−1Dt ) = 0 there exists a complete Kähler metric gt on Xt compatible with Jt and with Ricci-form ρt. 104 5.8. Proof of Theorem 5.1.5 For β > 1 [TIA-YAU91, Theorem 1.1] ensures that for every ρt ∈ c1(K−1X¯t ⊗L−1Dt ) = 0 there exists a complete Kähler metric gt on Xt compatible with Jt and with Ricci-form ρt. By the explicit construction of the metric gt both in [TIA-YAU90, Theorem 4.1] and [TIA-YAU91, Theorem 1.1] and since g can be taken to arise as a result of these construc- tions, then there exists a smooth curve of complete Kähler metrics gt with respect to Jt and with Ricci-forms ρt passing through g. We feel that an ALE version of the Kähler stability result [KOD-SPE60, Theorem 15] which does not make use of a compactification would have been more natural than Proposition 5.7.4. This was recently achieved by Deruelle and Kröncke in the preprint [DER-KRO17]. To keep our work independent of [DER-KRO17] we have decided not to make use of their findings. See Section 5.10 for a discussion of [DER-KRO17]. 5.8 Proof of Theorem 5.1.5 In this section, we prove Theorem 5.1.5. proof of Theorem 5.1.5. Let (X, J, g) be an ALE2m Ricci-flat Kähler manifold asymptotic to Cm/G. Assume that X = X¯\D for a compact complex Kähler manifold (X¯, J¯ , g¯) with a smooth ample divisor D satisfying KX¯ = −βLD for some β ≥ 1. Also assume that all infinitesimal deformations of the pair (X¯,D) are integrable. Denote by J the parameter space for the family of deformations of the pair (X¯,D). The assumption that all infinitesimal complex deformations are integrable implies that the space J is smooth in a neighbourhood around J . Denote by V the vector bundle over J with fibres Vt = H1,1−2m+1,R(X, Jt) of harmonic real 2-forms with weight −2m+ 1 and of type (1, 1) with respect to Jt. Take h ∈ ∞−2m+1(g). It is Kähler related to some I ∈ A0,1−2m+1(T 1,0X) which by Lemma 5.6.7 gives rise to an infinitesimal deformation [I¯] ∈ H1(X¯, TX¯(logD)) of the pair (X¯,D). Infinitesimal deformations of the pair (X¯,D) are assumed to be integrable. For a smooth curve of deformations (X¯t, Dt) let Xt = X¯t\Dt and denote the corresponding family of complex structures on X¯t by J¯t and the restriction to Xt by Jt. Write Jt = J+It. Choose a smooth curve of deformations (X¯t, Dt) of the pair (X¯,D) which satisfies d dt Jt|t=0 = ddtIt|t=0 = I . For small t Proposition 5.7.3 says that KX¯t = −βLDt and Lemma 5.7.2 says that LDt is ample, so by Proposition 5.7.4 there exists a smooth curve 105 Chapter 5. ALE Ricci-flat deformations of complete Kähler metrics gt on Xt passing through g0 = g. Write gt = g + ht. Each metric gt is Kähler with respect to Jt so ddtgt|t=0 = g′0 = h′0 and ddtJt|t=0 = I ′0 = I are Kähler related (as explained in Section 5.5). The tensor I has weight −2m + 1, so g′0 also has weight −2m + 1, and then so does each ht. The Kähler metrics gt are therefore ALE2m−1. Define the form κ = ( ddtgt|t=0 − h) ◦ J . It is a real (1, 1)-form and by (5.11) and (5.12) it is closed and Hermitian. Take a curve κt in V of Hermitian forms such that κ0 = κ. Denote by ωt the Kähler form gt ◦ Jt. The form ω˜t = ωt − tκt is closed and the tensor g˜t = (ωt − tκt) ◦ Jt is positive definite, so g˜t is a Kähler metric with respect to Jt. The symmetric tensor tκt ◦ Jt has weight −2m+ 1 and the metric g˜t is ALE2m−1. By Theorem 5.3.6 the real (1, 1)-forms cohomologous to the Kähler form ˜omegat = ωt− tκt in H2−2m+1(X,R) are parametrized by u ∈ C∞−2m+3(X) via ωt − tκt + i∂∂¯u. Solutions u ∈ Ck+2,α−2m+3(X) to the complex Monge-Ampère equation (ωt − tκt + i∂∂¯u)m + ef (ωt − tκt)m = 0 (5.14) produces Kähler forms ωˆt = ωt − tκt + i∂∂¯u. Note that for t = 0 the function u = 0 produces the already known ALE2m Ricci-flat Kähler metric g on (X, J). For each t the Ricci-form ρ˜t of gt belongs to 2pic1(Jt) = 0, so there exists a smooth curve of functions ft ∈ C∞(V ) satisfying ρ˜t = i∂∂¯ft. Similarly to what was done in Section 3.5 we can isolate f in (5.14) to produces a function F : V × C5,α−2m+3(X) → C3,α−2m+1(X) which in holomorphic coordinates {z1, . . . , zm} is given by F ((t, κt), u) = log det ( g˜αβ¯ + ∂2u ∂zα∂z¯β¯ ) − log det(g˜αβ¯)− ft. A similar calculation to (3.3) shows that the linearization of F in u is d ds F ((t, κt), u+ sv)|s=0 = ∆gˆv, where gˆ is the ALE2m−1 Kähler metric with Kähler form ωˆ = ω˜ + i∂∂¯u. For each v′ ∈ C3,α−2m+1(X) we can by [JOY00, Theorem 8.3.6] find a unique solution v ∈ C5,α−2m+3(X) such that ∆gˆv = v′. The linearization of F is therefore surjective and F is a submersion. From the implicit function theorem for Banach spaces (Theorem 2.1.10) we conclude that there exists a smooth function ψ : U ⊂ V → C5,α−2m+3(X) from a neighbourhood U 106 5.8. Proof of Theorem 5.1.5 of (0, 0) in V to a neighbourhood of 0 in C5,α−2m+3(X) such that for each (Jt, κt) ∈ U , ψ produces an ALE2m−1 Ricci-flat Kähler metric gˆt = ψ(Jt, κt), where ωˆt = ωt−tκt+i∂∂¯u and gˆt is the Ricci-flat Kähler metric satisfying gˆt ◦Jt = ωˆt. It follows from [DeT-KAZ81, Theorem 6.1] that the metric gˆt is actually smooth. Restrict ψ to a subset U˜ of U such that ψ|U˜ only takes values in the slice S2m−1. The image ψ(U˜) is contained in the premoduli space Pg and dψ(T(0,0)U˜) is contained in ∞−2m+1(g). The rest of the proof consists in showing that ψ surjects onto a a neighbourhood of g in the slice S2m−1. This part of the proof is similar to the latter part of the proof of Theorem 3.1.3. We repeat it for the convenience of the reader but with fewer details. Take the h ∈ ∞−2m+1(g) Kähler related to I = ddtJt|t=0 from before. For the Kähler form ω˜t = ωt − tκt the form ω˜′0 − h ◦ J satisfies ω˜′0 − h ◦ J = d dt (gt − tφt)|t=0 ◦ J + g˜ ◦ I − h ◦ J = g˜ ◦ Ia + g˜ ◦ Is, where φt denotes the tensor satisfying κt = φt ◦ Jt. The tensor g˜ ◦ Is is symmetric and the tensor g˜ ◦ Ia satisfies J(g˜ ◦ Ia)(X, Y ) = −g˜ ◦ Ia(X, Y ), so the 2-form g˜ ◦ Ia is not of type (1, 1). Hence [ω˜′0 − h ◦ J ] = [g˜ ◦ I] = 0 in H1,1−2m+1(X, J). Now, ωˆ′0 = d dt (ω˜t + i∂∂¯ut)|t=0 = ω˜′0 + i∂∂¯u′0 so [ωˆ′0] = [ω˜ ′ 0]. Hence [ωˆ ′ 0 − h ◦ J ] = 0. The symmetric tensors gˆ′0 and h are both Kähler related to I = J ′ 0, so gˆ ′ 0 − h is Kähler related to 0 and it is therefore Hermitian by Remark 5.5.7. For the Hermitian tensor gˆ′0− h ∈ ∞−2m+1(g) the form (gˆ′0− h) ◦ J is harmonic by the Weitzenböck formula for Hermitian tensors (5.6), since ∆((gˆ′0 − h) ◦ J) = (∇∗∇− 2 ◦ R)(gˆ ′ 0 − h) ◦ J = 0. Now, 0 = [ωˆ′0 − h ◦ J ] = [(gˆ′0 − h) ◦ J + gˆ ◦ I] = [(gˆ′0 − h) ◦ J ] + [gˆ ◦ I] = [(gˆ′0 − h) ◦ J ]. A harmonic form cohomologous to zero is itself zero, so (gˆ′0 − h) ◦ J = 0, which implies 107 Chapter 5. ALE Ricci-flat deformations that gˆ′0 = h. Hence the differential of the map ψ is surjective and ψ surjects onto a neighbourhood of g in S2m−1. This concludes the proof. 5.9 Moduli space of ALE Ricci-flat structures In this section, we introduce the moduli space of ALE Ricci-flat structures and prove that in a neighbourhood of an ALE Ricci-flat Kähler structure it is, up to an action of a finite group, a finite dimensional manifold and we find an expression for its dimension using cohomology groups. This is the content of Theorem 5.1.6, which is an ALE version of Theorem 2.7.2. The moduli space constructed in this section is different from the one used in Section 2.7 and Chapter 3, reflecting the fact that we no longer work over a compact base. Let (X, J, g) be an ALEτ Ricci-flat Kähler manifold asymptotic to Cm/G of order τ > 0. In Section 5.4 we denoted byM the space of complete Riemannian metrics and byMALEτ the subspace ofM consisting of ALEτ metrics. In Section 5.4 we showed that the group of diffeomorphismDτ+1 acts onMALEτ . We introduce the equivalence relation ∼ onM. Two metrics g, g′ ∈ MALEτ are equivalent if there exists φ ∈ Dτ+1 such that g′ = φ∗g. The equivalence class of an ALEτ metric is called an ALEτ structure. The quotient space M˜ALEτ =MALEτ/ ∼ is equipped with the quotient topology. Denote by R ⊂ M the complete Ricci-flat metrics and by RALEτ ⊂ MALEτ the ALEτ Ricci-flat metrics. Denote by R˜ALEτ the subspace of M˜ALEτ consisting of ALEτ structures containing a Ricci-flat metric. We call such structures for ALEτ Ricci-flat struc- tures and call R˜ALEτ for the moduli space of ALEτ Ricci-flat structures. The pull-back of a Ricci-flat metric by a diffeomorphism is Ricci-flat, so any metric in a Ricci-flat structure is Ricci-flat. We can therefore equivalently express R˜ALEτ asRALEτ/ ∼ orRALEτ/Dτ+1. Lemma 5.9.1. Let (X, g) be an ALEτ manifold asymptotic to Rn/G of order τ > 0 and assume that g is not flat. Then the isometry group Iso(X, g) is a finite dimensional compact Lie group. Proof. By [BES87, Theorem 1.77] the group Iso(X, g) is a Lie group. A metric space is compact if every sequence has a convergent subsequence with a limit point in the space. From [KOV06, proof of Lemma 3.6] we know that a sequence {ηi}i∈Z≥0 in Iso(X, g) has a convergent subsequence if for some p ∈ X the sequence {ηi(p)}i∈Z≥0 is convergent. The 108 5.9. Moduli space of ALE Ricci-flat structures metric g is not flat so we can take some p0 ∈ X with positive curvature, say |Rp0|g >  > 0. On ALE manifolds the Riemann curvature tensor decays to zero, so |Rp|g <  for ρ(p) > C for some C > 0, where ρ is the radius function on X . Hence η(p0) is contained in the compact ball of radius C for any isometry η. Hence the sequence {ηi(p0)}i∈Z≥0 has a limit point in X and the sequence {ηi}i∈Z≥0 has a convergent subsequence. Iso(X, g) is therefore compact. Denote by I0g the connected component of the identity element in the Lie group of isometries Iso(X, g). It is called the identity component and it is a normal subgroup. When the Lie group Iso(X, g) is compact the quotient Iso(X, g)/I0g is a finite group. Lemma 5.9.2. Let (X, g) be an ALEn−1 Ricci-flat manifold asymptotic to Rn/G. The identity component I0g acts trivially on Pg. Proof. Let g′ ∈ Sn−1. Then η ∈ Iso(X, g′) fixes g′ in Sn−1 so Theorem 5.4.7(2) implies that Iso(X, g′) ⊂ Iso(X, g). In particular I0g′ ⊂ I0g . Let g′ ∈ RALEn−1 . On (X, g′) let α ∈ C∞−n+1(∧1T ∗X) be a harmonic 1-form. As g′ is Ricci-flat the Weitzenböck formula ensures that ∆g′α = ∇∗g′∇g′α and integration by parts then gives us ∇g′α = 0. For 1- forms δ∗g′α = ∇α − 12dα = 0 + 0 = 0. Hence Lα]g = δ∗g′α = 0, so α] is a Killing vector field. On the other hand, Let X be a Killing vector field, then δ∗g′X [ = LXg = 0, and from Proposition 5.4.5 we get ∆g′X[ = (2δg′ + d trg)δ∗g′X [ = 0, so X[ is harmonic. Killing fields are therefore in bijection with harmonic 1-forms. The Laplace operator is Fredholm, so the space of harmonic 1-forms is finite dimensional. For any g′ ∈ Pg the connected Lie subgroup I0g′ therefore coincides with I 0 g . Proof of Theorem 5.1.6. Let (X, J, g) be an ALE2m Ricci-flat Kähler manifold asymptotic to Cm/G satisfying the assumptions of Theorem 5.1.5. From Section 5.4 we know that there exists a slice S2m−1 ⊂ MALE2m−1 for the action of D2m. From Theorem 5.5.3 we know that there exists a finite dimensional manifold Z ⊂ S2m−1 with tangent space TgZ =  ∞ −2m+1(g) at g and with the premoduli space of Ricci-flat metrics Pg as a real analytic subset. From Theorem 5.1.5 we know that all infinitesimal Ricci-flat deformations integrate into Ricci-flat deformations, so Pg is itself a manifold with TgPg = ∞−2m+1(g). As a result of the slice construction we know that a neighbourhood U ⊂ R˜ALE2m−1 of [g] is homeomorphic to Pg/Iso(X, g). By Lemma 5.9.1 and Lemma 5.9.2 it is homeomorphic to the orbifold Pg/(Iso(X, g)/I0g ). The dimension of R˜ALE2m−1 at [g] is therefore the 109 Chapter 5. ALE Ricci-flat deformations dimension of Pg at g. We can give an expression for this dimension via Proposition 5.5.6 and Proposition 5.5.10. dim TgPg = dim ∞−2m+1(g) = dim ∞−2m+1(g)H + dim  ∞ −2m+1(g)A = dimH1,1R (X, J)− 1 + 2 dimCH1(X, TX)− 2 dimCH0,2(X, J). Since H0,2(X, J) = 0 by [JOY00, Theorem 8.4.3], this concludes the proof. 5.10 ALE Ricci-flat deformations revisited In this section, we discuss a recent preprint ([DER-KRO17]), where Deruelle and Kröncke prove a similar result to Theorem 5.1.5. Their work and the work in this thesis have been done independently. In this section we briefly summarize their findings and highlight a few differences between their approach and ours. Their result ([DER-KRO17, Theorem 2.17]) is Theorem 5.10.1. Let (X, J, g) be an ALE Calabi-Yau manifold asymptotic to Cm/G and let δ ∈ (−2m+ 1,−2m+ 2) be not exceptional. Then for any h ∈ kerL2(∆L), there exists a smooth family gt of Ricci-flat metrics in some slice S ⊂ M2k,δ through g for the action of D2k+1,δ+1, with g0 = g and ddtgt|t=0 = h. Each metric gt is ALE and Kähler with respect to some complex structure Jt which is L2k,δ-close to J . Here D2k+1,δ+1 denotes diffeomorphisms from X to itself which are L2k+1,δ+1-close to the identity. The space kerL2(∆L) is defined as ker(∆L) ∩ L2(Sym2(T ∗X)), where ∆L is the Lichnerowicz Laplacian ∆L : C∞(Sym2(T ∗X)) → C∞(Sym2(T ∗X)). Any h ∈ kerL2(∆L) can be shown to satisfy h = O(ρ−2m−1) for a radius function ρ on the ALE manifold (X, g) of complex dimension m.M2k,δ is the space of metrics L2k,δ-close to g. In the proof of Theorem 5.1.5 we made use of a compactification X = X¯\D and deformation theory of the pair (X¯,D) to deform the complex structure in a way that pre- served the Kähler property. Deruelle-Kröncke use a different approach. Instead of using a 110 5.11. Asymptotically conical manifolds compactification they adapt the original proof ([KOD-SPE60, Theorem 15]) by Kodaira- Spencer to the ALE setting and show that the Kähler property is preserved under small deformations of the complex structure. This is the content of [DER-KRO17, Theorem 2.15] which we state next. Denote by J 2k,δ the space L2k,δ(∧0,1X ⊗ T 1,0X) ∩ ker(∆). Theorem 5.10.2. Let (X, J, g) be an ALE Ricci-flat Kähler manifold asymptotic to Cm/G and let δ < −2m + 2 be not exceptional and J 2k,δ defined as above. Then there exists a L2k,δ-neighbourhood U of J in J 2k,δ and a smooth map Ψ : U →M2k,δ which associates to each Jt ∈ U ⊂ J 2k,δ sufficiently close to J0 = J a metric gt which is L2k,δ-close to g0 and Kähler with respect to Jt. While we made use of the auxiliary hypothesis that all infinitesimal deformations of the pair (X¯,D) are integrable, then Kröncke-Deruelle show that infinitesimal complex deformations of ALE Calabi-Yau manifolds always are integrable. This is the content of [DER-KRO17, Theorem 2.14]. Theorem 5.10.3. Let (X, J, g) be an ALE Kähler manifold asymptotic to Cm/G with a holomorphic volume form. Let k > m + 1, δ < −m not exceptional and I ∈ L2k,δ(∧0,1 ⊗ T 1,0X) such that ∂¯I = 0 and ∂¯∗I = 0. Then there exists a smooth family of complex structures Jt with J0 = J such that Jt − J ∈ L2k,δ(T ∗X ⊗ TX) and ddtJt|t=0 = I . There are a number of similarities between both statement and proof of [DER-KRO17, Theorem 2.17] and Theorem 5.1.5, but there are also differencies. Our approach is closer in spirit to the constructions in [CAL79], [BAN-KOB88], [BAN-KOB90], [TIA-YAU90] and [TIA-YAU91], where a defining section of the divisor is used to construct a complete metric on the complement of the divisor. An advantage of the approach by Deruelle- Kröncke is that they do not make use of a compactification. This would have been an attractive simplification of the hypothesis of Theorem 5.1.5 as the compactification does not appear in the dimension of the moduli space of Ricci-flat structures in Theorem 5.1.6. An advantage of our approach, however, is that we expect that it should generalize more easily to asymptotically conical manifolds. 5.11 Asymptotically conical manifolds In this section, we discuss asymptotically conical manifolds. ALE manifolds are a special class of asymptotically conical (AC) manifolds. One could therefore ask if the proofs of 111 Chapter 5. ALE Ricci-flat deformations Theorem 5.1.5 and Theorem 5.1.6 extend to AC manifolds. In this section we address this question. While not giving an affirmative answer in either direction, we provide com- ments on the role of the special cone structure R+ × Sn−1/G of the ALE manifold in the proofs of Theorem 5.1.5 and Theorem 5.1.6. AC manifolds admit a more general cone structure R+ × L, for a compact connected manifold L, then ALE manifolds. The cone structure is therefore what sets ALE manifolds apart from other AC manifolds. For an introduction to AC manifolds see for instance [BOY-GAL08, Chapter 9], [CON-HEI13] and [CON-HEI15]. We start by introducing AC manifolds and continue with a discussion of the possibility of extending various components of Chapter 5 to the AC setting. Definition 5.11.1 (Tangent cone). Let (L, g) be a compact connected Riemannian mani- fold. A tangent cone (C, h0) with link L is defined to be the manifold C = R+ × L with the product metric h0 = dr2 ⊕ r2g and where r is the distance from the apex in the metric completion of the cone. Definition 5.11.2 (Asymptotically conical manifold). Let (X, g) be a complete Rieman- nian manifold and let (C, h0) be a tangent cone. We say that (X, g) is asymptotically con- ical (AC) with tangent cone (C, h0) if there exists a diffeomorphism pi : X\K ′ → C\K with K ′, K compact, such that∇k(pi∗g − h0) = O(r−τ−k) for some τ > 0 and all k ≥ 0. The flat metric h0 on Rn\{0} ' R+ × Sn−1 can be written as h0 = dr2 ⊕ r2g0, for metric g0 on Sn−1 and distance function r from 0 in Rn with respect to h0. An ALE manifolds asymptotic to Rn/G is therefore a special case of an AC manifolds with tangent cone Rn\{0}/G and link Sn−1/G. Remark 5.11.3 (Tools from analysis). Weighted Sobolev and Hölder spaces can be intro- duced for a general tangent cone R+ × S as was done for ALE manifolds in Section 5.2 (see [CON-HEI13, Section 2.2]). The basic set-up for the analysis on an AC manifold is therefore the same as it was on a ALE manifold. Theorem 5.2.8 is actually borrowed from an identical result about AC manifolds. Replacing Theorem 5.2.7 with [CON-HEI13, The- orem 2.11] will suffice for the application in Section 5.8. Note that the set of exceptional weights may change as it depends on the set of eigenvalues of the Laplacian on the link of the cone (see [CON-HEI13, Theorem 2.9]). This difference will affect the expression for the dimension of the moduli space of AC Ricci-flat structures as well. 112 5.11. Asymptotically conical manifolds Remark 5.11.4 (AC differential geometry). ALE manifolds are crepant resolutions of Cm/G, while AC manifolds are in general not. This means that we cannot apply [JOY00, Theorem 8.4.3] to conclude that H2,0(X) = H0,2(X) = 0. General AC manifolds may therefore admit skew-symmetric infinitesimal complex deformations. Remark 5.11.5 (Slice construction). We expect that the slice equation (2δg + d trg)h = 0 would also be a good choice on AC manifolds. Whether all details of the slice construction generalize to the AC context remains to be checked. Remark 5.11.6 (Ricci-flat deformations). The defining equations of infinitesimal Ricci-flat deformations do not make use of the special ALE cone structure, so the space of infinites- imal ALE Ricci-flat deformations could be defined in the same way as in Definition 5.5.1 on AC manifolds. The key component of Theorem 5.5.3 is Theorem 2.1.11. The argument why we can apply it is based on the Fredholm properties of the involved operators, and carries over to AC manifolds as explained in Remark 5.11.3 with the appropriate changes. The same is true for the analysis involved in proving Lemma 5.5.2. The equations used for the splitting of the space of infinitesimal Ricci-flat deformations in Proposition 5.5.5 are not affected by the more general cone structure, so the proposition should hold if we replace the ALE manifold with an AC manifold. The identifications of the Hermitian and skew-Hermitian infinitesimal Ricci-flat deformations with Harmonic real (1, 1)-forms and symmetric infinitesimal complex deformations respectively might require some more careful considerations in the AC context, especially for the skew-Hermitian infinitesimal Ricci-flat deformations. Remark 5.11.7 (Deformations of the pair (X¯,D)). Section 5.6 about deformations of the pair (X¯,D) makes no reference to the special ALE cone structure and is readily applica- ble for AC manifolds that arise as the complement of a smooth divisor inside a compact manifold. Remark 5.11.8 (Stability results). The stability results in Section 5.7 are based on the Kodaira Embedding theorem, general theory of Chern classes, deformation theory of pairs (X¯,D) and on [TIA-YAU90] and [TIA-YAU91]. All of these ingredients apply to AC manifolds just as well as to ALE manifolds. Lemma 5.7.2 and Proposition 5.7.3 generalize directly to AC manifolds. Proposition 5.7.4 would probably require some more work. Also, we do not know if the ALE version by Deruelle and Kröncke (Theorem 5.10.2) of Kodaira 113 Chapter 5. ALE Ricci-flat deformations and Spencer’s stability result for Kähler metrics for complex deformations [KOD-SPE60, Theorem 15] can be extended to AC manifolds. Remark 5.11.9. In Section 5.8 and Section 5.9 we use the material from the previous sections to prove Theorem 5.1.5 and Theorem 5.1.6 respectively. We do not know if the isometry group would act as a finite group on the premoduli space of AC Ricci-flat metrics. We make no additional use of the ALE cone structure in these two sections apart from that. Assuming a suitable AC version of the material leading up to Section 5.8 and Section 5.9 had been achieved and that the isometry group acts as a finite group on the premoduli space of AC Ricci-flat metrics, then the content of Section 5.8 and Section 5.9 should generalize from ALE manifolds to AC manifolds. 114 Chapter 6 Examples: ALE manifolds In this chapter, we consider examples of ALE Ricci-flat Kähler manifolds. The first ex- amples were produced by Eguchi and Hanson in [EGU-HAN78], who constructed explicit ALE hyperkähler manifolds of complex dimension 2. Gibbon and Hawking generalized this construction in [GIB-HAW78]. In [CAL79, p.285] Calabi gave an explicit construc- tion of ALE Ricci-flat Kähler manifolds of arbitrary dimension, which in dimension 2 recovers the Eguchi-Hanson metrics. Hitchin introduced in [HIT79] a hyperkähler quo- tient construction of ALE Ricci-flat Kähler manifolds. Kronheimer used it in [KRO89-1] to produce many more examples and in [KRO89-2] he classified all ALE hyperkähler manifolds. We start with the following example from [EGU-HAN78]. Example 6.0.10. Consider R4\{0} ' R+×S3 with spherical coordinates (r, x, y, z) and metric g = f 21 (r)dr 2 + r2f 22 (r)((dx) 2 + (dy)2) + r2(dz)2, (6.1) with coefficients f1(r) = 1 2 ( 1 + 1√ 1 + (a r )4 ) and f2(r) = √√√√1 2 ( 1 + √ 1− (a r )4) , where a is some positive constant. Because of spherical symmetry in (6.1) we can quotient out by the group G = {γ1, γ2} where γ1 = id and γ2 is the identification of antipodal points (r, x, y, z) 7→ (r,−x,−y,−z). The function f1 and f2 tends to 1 for r → ∞ so 115 Chapter 6. Examples: ALE manifolds (R4\{0}/G, g) is asymptotic to R4\{0}/G = R+ × S3/G with the ALE metric dr2 + r2g˜ where g˜ = dx2+dy2+dz2 is metric on the link S3. In [JOY00, Example 7.2.2] Joyce views this example from a complex perspective. Equip C2 with the complex coordinates (z1, z2). Let G = {id, γ} where γ is the involution (z1, z2) 7→ (−z1,−z2). Let (X, pi) be the blow- up of C2/G at 0. X is then a crepant resolution of C2/G. Define f : X\pi−1(0) → R by f = √ r4 + 1 + 2 log(r)− log( √ r4 + 1 + 1). Here r is the radius function r(z1, z2) = √|z1|2 + |z2|2 onX . Let ω = i∂∂¯f . It is a closed real (1, 1)-form on X\pi−1(0) and it extends uniquely across the exceptional divisor. ω is the Eguchi-Hanson metric on X and it is Ricci-flat Kähler. Furthermore, for large r the function f satisfies f(z1, z2) = r2 + O(r−2), so ω = i∂∂¯(r2) + O(r−4). The standard Kähler metric h0 on C2 has Kähler form i∂∂¯(r2), so the Kähler metric g of ω satisfies pi∗g = h0 +O(r−4). The metric g is therefore an ALE4 Ricci-flat Kähler metric. In [CON-HEI13, Section 4.1] Conlon and Hein explain the following example of an ALE manifold due to Calabi([CAL79]). Example 6.0.11. Consider the complex projective space CPm. Let L be the total space of the U(1)-bundle in the tautological bundle O(−1) → CPm and let Lk = L/Zk. Denote by C(Lm) the Kähler cone of Lm. As L ' S2m−1 the cone C(Lm) is Cm/Zm. Denote the metric completion C(Lm) ∪ {0} by Vm. The total space X of the canonical bundle KCPm is Ricci-flat Kähler with a global holomorphic volume form and it is a crepant resolution of the Calabi-Yau cone Vm with link S2m−1/Zm. It is therefore an ALE2m manifold. Also for this example has Joyce given an enlightening presentation ([JOY00, Example 8.2.5]). The m’th root of unity η = e 2pii m acts on Cm = (z1, . . . , zm) by η : (z1, . . . , zm) 7→ (ηz1, . . . , ηzm). The group G =< η > is a subgroup of SU(m) and it acts freely on Cm/G\{0}. Let (X, pi) be the blow-up of Cm/G and define f : Cm/G\{0} → R by f(x) = m √ r2m + 1 + 1 m ∑m−1 j=0 η j log( m √ r2m + 1 − ηj), where log( m√r2m + 1 − ηj) is defined by slicing C along the negative real axis and setting log(Reiθ) = log(R) + iθ for R > 0 and θ ∈ (−pi, pi). Define a (1, 1)-form ω on X\pi−1(0) by ω = ddcpi∗(f). The form ω extends to a smooth closed positive (1, 1)-form on all of X . Let g be the Kähler metric of ω. The metric g is in fact complete and Ricci-flat and has holonomy hol(g) = SU(m). 116 On Cm/G\{0} we can for large r rewrite f as f = r2 − 1 m(m− 1)r 2−2m +O(r−2m). Note that the Kähler form of the Euclidean metric on Cm/G is ω0 = ddc(r2). The push- forward of ω is pi∗ω = ω0 − 1m(m−1)ddc(r−2m+2) + ddcη on Cm/G\{0} where η = f − r2 + 1 m(m−1)r −2m+2. For large r we further have ∇kη = O(r−2m−k) on Cm/G\{0}. The metric g is therefore an ALE2m Ricci-flat Kähler metric. In [JOY00, p.178] Joyce Conjectures that for m ≥ 3 this is the only example of an ALE metric with holonomy SU(m) that can be written down explicitly in coordinates. In the next example we consider a pair (X¯,D) which satisfies the hypothesis of Theo- rem 5.1.5 but does not admit any ALE Ricci-flat deformations. Example 6.0.12. Consider the pair (CPm,CPm−1). The hyperplane divisor CPm−1 is a smooth ample divisor. The line bundles KCPm and LCPm−1 are related via KCPm = O(−m− 1) = −(m+ 1)O(1) = −(m+ 1)LCPm−1 . We see that TCPm ' Ωm−1CPm ⊗ det(TCPm) ' Ωm−1CPm ⊗OCPm(m+ 1). TCPm(−CPm−1) ' TCPm ⊗−LCPm−1 ' TCPm ⊗OCPm(−1) ' Ωm−1CPm ⊗OCPm(m). TCPm−1 ' Ωm−2CPm−1 ⊗ det(TCPm−1) ' Ωm−2CPm−1 ⊗OCPm−1(m). By Kodaira vanishing theorem, H2(CPm, TCPm(−CPm−1)) = H2(CPm,Ωm−1CPm ⊗OCPm(m)) = 0 H2(CPm−1, TCPm−1) = H2(CPm−1,Ωm−2CPm−1 ⊗OCPm−1(m)) = 0 as 2+(m−1) > m = dimCPm in the first case and 2+(m−2) > m−1 = dimCPm−1 in the second case. From [KAW78, Proposition 1] we have the following short exact sequence 0 −→ TX¯(−D) −→ TX¯(logD) −→ TD −→ 0. The second level of the corresponding long exact sequence in sheaf cohomology reduces 117 Chapter 6. Examples: ALE manifolds to 0 −→ H2(CPm, TCPm(logCPm−1)) −→ 0, Hence H2(CPm, TCPm(logCPm−1)) = 0. All infinitesimal complex deformations of the pair (CPm,CPm−1) are therefore integrable by [KAW78, Corollary 4]. The complement of the divisor CPm−1 is Cm and it trivially admits an ALE2m Ricci-flat Kähler metric. The space Cm is naturally a crepant resolution of Cm, so H0,2(Cm) = 0 by [JOY00, Theorem 8.4.3]. The dimension of the premoduli space of ALE2m−1 Ricci-flat Kähler metrics is now given by dimH1,1R (Cm, J)− 1 + 2 dimCH1(Cm, TCm). On Euclidean space the Betti numbers bi vanish for all i ≥ 1, so H1,1R (Cm, J) = 0. There are therefore no Hermitian infinitesimal Ricci-flat deformations. Cartan’s theorem B says that for a Stein manifold X and a coherent sheaf F , H i(X,F) = 0 for all i ≥ 0. The manifold Cm is Stein and the sheaf of sections of a holomorphic vector bundle is locally free, hence coherent, so by Cartan’s theorem B, H1(Cm, TCm) = 0. All cohomology groups involved in the expression for the dimension of the space of infinitesimal ALE2m−1 Ricci-flat deformations therefore vanish. The standard Euclidean metric on Cm therefore does not admit any ALE2m−1 Ricci-flat deformations. We have been unable to find more interesting applications of Theorem 5.1.5 and Theo- rem 5.1.6, but as we have discussed in Section 5.11, Theorem 5.1.5 and Theorem 5.1.6 can also be seen as a step on the way towards a version of Koiso’s results for asymptotically conical manifolds. 118 Concluding remarks An ideal candidate for a distinguished metric should preferably always exist, be unique and have interesting properties. While Koiso’s results and the Calabi conjecture do not provide actual existence and uniqueness of Einstein metrics on general compact manifolds, they do tell us that there are in some sense not too many Einstein structures on a compact Kähler manifold and that Einstein metrics always exist on compact Kähler manifold with vanishing first Chern class. This is part of the justification for promoting Einstein metrics as candidates for the role of distinguished metrics on compact manifolds given in [BES87]. In this thesis we have studied two ways of generalizing results by Koiso from [KOI83]. In Chapter 3 we studied the effect of introducing quotient singularities on Koiso’s results. It turned out that this effect was minor and we managed to prove a satisfying generalization. For a compact Ricci-flat Kähler orbifold we showed that orbifold Ricci- flat deformations of orbifold Ricci-flat Kähler metrics are Kähler possibly with respect to a perturbed complex structure. We also found an expression for the dimension of the moduli space of orbifold Ricci-flat structures in a neighbourhood of an orbifold Ricci-flat Kähler structure. In addition to this, we showed that the proof of the Calabi conjecture goes through on compact orbifolds. Another way to generalize Koiso’s results is to relax the compactness assumption to allow complete metrics on non-compact manifolds. In Chapter 5 we studied a particu- lar class of complete non-compact manifolds known as asymptotically locally Euclidean manifolds (ALE). For an ALE Ricci-flat Kähler manifold X that arises as X = X¯\D for a compact Kähler manifold X¯ and a smooth ample divisor D satisfying KX¯ = −βLD for some β ≥ 1 we showed that if all infinitesimal deformations of the pair (X¯,D) are integrable, then ALE Ricci-flat deformations of an ALE Ricci-flat Kähler metric is Kähler possibly with respect to a perturbed complex structure. We also found an expression for the moduli space of ALE Ricci-flat deformations in a neighbourhood of a ALE Ricci-flat 119 Chapter 6. Examples: ALE manifolds Kähler structure. Based on the work we did in Chapter 3 and Chapter 5 it seems reasonable to view Einstein metrics as a potential candidate for the role of a distinguished metric also on orbifolds and ALE manifolds. ALE manifolds are a special class of asymptotically conical manifolds. Conlon and Hein proved in [CON-HEI15] an asymptotically conical version of the Calabi conjecture. In Section 5.11 we discussed the possibility of extending Theorem 5.1.5 and Theorem 5.1.6 from ALE manifolds to asymptotically conical manifolds. It would be interesting to pursue such a generalization. 120 Bibliography [BAG-ZHU07] Bagaev, A. V. and Zhukova, N. I., The Isometry Groups of Riemannian Orbifolds, Siberian Mathematical Journal, Vol. 48, No. 4, pp. 579-592, 2007. [BAI56] Baily, Walter L. The decomposition theorem for V-Manifolds, Amer. J. Math., Vol. 78, No. 4, pp. 862-888, 1956. [BAN-KAS-NAK89] Bando, Shigetoshi and Kasue, Atsushi and Nakajima, Hiraku, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math., Vol. 97, pp. 313-349, 1989. [BAN-KOB88] Bando, Shigetoshi and Kohayashi, Ryoichi, Ricci-flat Kähler metrics on affine algebraic manifolds, Geometry and analysis on manifolds (Katata/Kyoto, 1987), Lecture Notes in Math., Vol. 1339, pp. 20-31, Springer-Verlag, 1988. [BAN-KOB90] Bando, Shigetoshi and Kohayashi, Ryoichi, Ricci-flat Kähler metrics on affine algebraic manifolds. II, Math. Ann., Vol. 287, No. 1, pp. 175-180, 1990. [BER-EBI69] Berger, M. and Ebin, D, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Diff. Geom., Vol. 3, pp. 379-392, 1969. [BES87] Besse, Arthur L., Einstein Manifolds, Springer-Verlag, New York, 1987. [BIQ00] Biquard, Olivier, Asymptotically Symmetric Einstein Metrics. Translation from French. SMF/AMF Texts and Monographs. Vol. 13, No. 265, 2006 (original 2000). [BOY-GAL08] Boyer, Charles and Galicki, Krzysztof, Sasakian Geometry, Oxford math- ematical monographs, Oxford university press, 2008. 121 Bibliography [CAL54] Calabi, Eugenio, The space of Kähler metrics, Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, vol. 2, pp. 206-207. North-Holland, Amsterdam, 1956. [CAL57] Calabi, Eugenio, On Kähler manifolds with vanishing canonical class, Alge- braic geometry and topology, A symposium in honor of S. Lefschetz, pp. 78-89, Princeton University Press, Princeton, N. J., 1957. [CAL79] Calabi, Eugenio, Métriques kählériennes et fibrés holomorphes. Ann. Sci. École Norm. Sup., Vol. 12, No. 2, pp. 269-294, 1979. [CAN-LYN90] Candelas, P. and Lynker, M, Calabi-Yau manifolds in weighted P4, Nu- clear Physics, Vol. B341, pp. 383-402, 1990. [CAN81] Cantor, Murray, Elliptic operators and the decomposition of tensor fields, Bull. Amer. Math. Soc., Vol. 5, No. 3, pp. 235-262, 1981. [CON-HEI13] Conlon, Ronan J. and Hein, Hans-Joachim, Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J., Vol. 162, No. 15, pp. 2855-2902, 2013. [CON-HEI15] Conlon, Ronan J. and Hein, Hans-Joachim, Asymptotically conical Calabi-Yau metrics on quasi-projective varieties, Geom. Funct. Anal., Vol. 25, pp. 517-552, 2015. [DER-KRO17] Deruelle, Alix and Kröncke, Klaus, Stability of ALE Ricci-flat manifolds under Ricci flow, arXiv:1707.09919v1, 31. Jul. 2017. [DeT-KAZ81] DeTurck, Dennis M. and Kazdan, Jerry L., Some regularity theorems in Riemannian geometry, Annals Scientifiques de l’É.N.S. 4e série, tome 14, No. 3, pp. 249-260, 1981. [DOL82] Dolgachev, Igor, Weighted Projective Varieties. Group actions and vector fields, Lecture notes in math., Vol. 956, pp. 34-71, Springer-Verlag, 1982. [EBI70] Ebin, David. G., The Manifold of Riemannian Metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), pp. 11-40, Amer. Math. Soc., Providence, R.I., 1970. 122 Bibliography [EGU-HAN78] Eguchi, Tohru and Hanson, Andrew J., Asymptotically flat self-dual solu- tions to Euclidean gravity, Phys. Lett. Vol. 74B, pp. 249-251, 1978. [GAF54] Gaffney, Matthew P., A Special Stoke’s Theorem for Complete Riemannian Manifolds, Ann. Math. 2nd series, Vol. 60, No. 1, pp. 140-145, 1954. [GIB-HAW78] Gibbons, G.W. and Hawking, S.W., Gravitational Multi-Instantons, Physics Letters, Vol. 78B, pp. 430-432, 1978. [GRI-HAR94] Griffiths, Phillip and Harris, Joseph, Principles of Algebraic Geometry, Wiley and Sons, 1994. [GUN-ROS65] Gunning, R. and Rossi, H., Analytic functions of several complex vari- ables, Prentice Hall, 1965. [HAM82] Hamilton, Richard, Three-manifolds with positive Ricci-curvature. J. Diff. Geom., Vol. 17, pp. 255-306, 1982. [HIT79] Hitchin, Nigel J., Polygons and gravitons, Math. Proc. Camb. Phil. Soc., No. 85, pp. 465-476, 1979. [HUY05] Huybrechts, Daniel, Complex Geometry, an introduction, Springer-Verlag, 2005. [JOY00] Joyce, Dominic D., Compact Manifolds with Special Holonomy, Oxford Gradu- ate texts in mathematics, 2007. [KAC88] El Kacimi-Alaoui, Aziz, Stabilité des V -variétés kahlérienne, Holomorphic dynamics (Mexico 1986), Lecture Notes in Mathematics, Vol. 1345, pp. 111-123, Springer-Verlag, 1988. [KAW78] Kawamata, Yujiro, On Deformations of Compactifiable Complex Manifolds, Math. Ann., Vol. 235, pp. 247-265, 1978. [KOD86] Kodaira, K., Complex Manifolds and Deformation of Complex Structures, Grundlehren der mathematicschen Wissenschaften, Vol. 283, Springer-Verlag, 1986. 123 Bibliography [KOD-NIR-SPE58] Kodaira, K. Nirenberg, L. Spencer, D. C., On the Existence of Defor- mations of Complex Analytic Structures, Ann. Math. 2nd series, Vol. 68, No. 2, pp. 450-459, 1958. [KOD-SPE60] Kodaira, K. and Spencer, D. C., On deformations of complex analytic structures, III. Stability theorems for complex structures, Ann. Math. 2nd series, Vol. 71, No. 1, pp. 43-76, 1960. [KOI83] Koiso, Norihito, Einstein Metrics and Complex Structures, Invent. Math., Vol. 73, pp. 71-106, 1983. [KOV-NOR10] Kovalev, Alexei and Nordström, Johannes, Asymptotically cylindrical 7- manifolds of holonomy G2 with applications to compact irreducible G2-manifolds, Ann. Glob. Anal. Geom., Vol. 38, pp. 221-257, 2010. [KOV06] Kovalev, Alexei, Ricci-flat deformations of asymptotically cylindrical Calabi- Yau manifolds, Proceedings of Gökova Geometry-Topology Conference 2005, pp. 140-156, 2006. [KRO89-1] Kronheimer, Peter B., The construction of ALE spaces as hyperkähler quo- tients, J. Diff. Geom., Vol. 29, pp. 665-683, 1989. [KRO89-2] Kronheimer, Peter B., A Torelli-type theorem for gravitational instantons, J. Diff. Geom., Vol. 29, pp. 685-697, 1989. [LAN62] Lang, Serge, Introduction to differential manifolds, Interscience, New York, 1962. [LEE-PAR87] Lee, J. M. and Parker, T. H., The Yamabe problem, Bull. Amer. Math. Soc., Vol. 17, No. 1, pp. 37-91, 1987. [LOC-McO85] Lockhart, Robert B. and Mc Owen, Robert C., Elliptic differential opera- tors on noncompact manifolds, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 12, No. 3, pp. 409-447, 1985. [MOE-PRO97] MOERDIJK, I. and PRONK, D. A., Orbifolds, Sheaves and Groupoids, K-Theory, Vol. 12, pp. 3-21, 1997. 124 Bibliography [MYE-STE39] Myers, S. B. and Steenrod, N. E., The Group of Isometries of a Rieman- nian Manifold, Ann. Math. 2nd series, Vol. 40, No. 2, pp. 400-416, 1939. [NOR08] Nordström, Johannes, Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy, PhD thesis, Cambridge, 2008. [SAT56] Satake, Ichiro, On a generalization of the notion of manifold, Proceeding of the national academy of Science, Vol. 42, pp. 359-363, 1956. [SAT57] Satake, Ichiro, The Gauss-Bonnet Theorem for V-manifolds, Journal of the Math- ematical Society of Japan, Vol. 9, No. 4, pp. 464-492, 1957. [THU78] Thurston, William, The geometry and topology of 3-manifolds, Lecture notes, Princeton, 1978. [TIA-YAU90] Tian, Gang and Yau, Shing-Tung, Complete Kähler manifolds with zero Ricci curvature I, J. Amer. Math. Soc., Vol. 3, No. 3, pp. 579-609, 1990. [TIA-YAU91] Tian, Gang and Yau, Shing-Tung, Complete Kähler manifolds with zero Ricci curvature II, Invent. math., Vol. 106, pp. 27-60, 1991. [YAU78] Yau, Shing-Tung, On the Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampère Equation, I, Communications on Pure and Applied Math- ematics, Vol. XXXI, pp. 339-411, 1978. 125 Notation Index Banach manifold, page 7 Fréchet manifold, page 7 Hilbert manifold, page 7 (Hk,αβ )r(X) d and d∗-closed forms in Ck,αβ (∧rT ∗X), ALE, page 85 δg formal adjoint of δ∗g , page 24 δ∗g symmetric part of∇, page 14 (g) infinitesimal Ricci-flat deformations, compact manifolds, page 25 (g)orb infinitesimal Ricci-flat deformations, orbifolds, page 56 (g)orbA skew-Hermitian part of (g) orb, page 61 (g)orbH Hermitian part of (g) orb, page 61 ∞−τ (g)A skew-Hermitian part of  ∞ −τ (g), ALE, page 98 ∞−τ (g)H Hermitian part of  ∞ −τ (g), ALE, page 98 ∞−τ (g) infinitesimal Ricci-flat deformations, ALE, page 93 Γ orbifold group, orbifolds, page 32 Γx isotropy subgroup of Γ, orbifolds, page 33 D diffeomorphism group, page 6 Dorbk L2k(F )orb ∩ C1Dorb-diffeomorphisms, orbifolds, page 51 126 Notation Index Dτ diffeomorphisms generated by vector fields with weight −τ , ALE, page 89 FV local uniformizing systems of V , orbifolds, page 33 LV injections of V , orbifolds, page 33 M space of Riemannian metrics, page 12 Morb space of orbifold Riemannian metrics, orbifolds, page 50 Mk L2k-symmetric (2, 0)-tensors which are C0-metrics, compact manifolds, page 12 Morbk Lk2-symmetric (2, 0)-tensors which are C0-metrics, orbifolds, page 50 MALEτ space of ALEτ metrics, ALE, page 87 P exceptional weights, ALE, page 84 ∇ covariant derivative, page 6 Ωk(U)Γ Γ-invariant k-forms on U ⊆ Rn, orbifolds, page 35 Ωk(V )orb orbifold differential k-forms on V , orbifolds, page 35 φΓ orbifold chart homeomorphism, orbifolds, page 32 pi asymptotic coordinate system, ALE, page 76 ρ Ricci-form ρ(x, y) = Ric(Jx, y), page 20 Ric Ricci curvature tensor, page 13 ◦ R action of the Riemann curvature tensor on (2, 0)-tensors, page 25 J , page 6 ALEτ ALE of order τ , ALE, page 76 vol(g) total volume of metric g, page 6 vol(g)orb volume of metric g, orbifolds, page 36 M˜orb quotient space of Riemannian structures, orbifolds, page 67 127 Notation Index R˜ moduli space of Ricci-flat structures, compact manifolds, page 24 R˜orb moduli space of orbifold Ricci-flat structures, orbifolds, page 68 R˜ALEτ moduli space of ALEτ Ricci-flat structures, ALE, page 108 {U,Γ, φ} local uniformizing system, orbifolds, page 32 Ck(E)orb Ck-sections of orbifold vector bundle E, orbifolds, page 37 CkD(V )orb Ck-diffeomorphisms, orbifolds, page 51 CkM space of Ck-metrics, compact manifold, page 12 CkMorb space of Ck-metrics, orbifolds, page 50 C∞(E)orb sections of orbifold vector bundle E, orbifolds, page 34 C∞c (M) space of test functions, page 7 Ck,α(E)orb Ck,α-sections of orbifold vector bundle E, orbifolds, page 37 c1(V ) orb orbifold first Chern class, orbifolds, page 40 d exterior differential operator, page 6 d∗ formal adjoint of d, page 6 dV orbg volume form of metric g, orbifolds, page 35 dVg volume element of metric g, page 6 gij components of g−1ij , page 6 gαβ¯ components of g in holomorphic coordinates, page 6 gij components of the metric g, page 6 i, j, α, β tensor index following Joyce §4.2.1. Roman letter for real index, Greek letters for complex index, page 6 Ia anti-symmetric part of I , ALE, page 96 128 Notation Index Is symmetric part of I , ALE, page 96 Ig isometry group of metric g, page 52 Iij components of tensor g ◦ I , ALE, page 98 ICD(J) infinitesimal complex deformations, orbifolds, page 43 ICD∞−τ (J) space of infinitesimal complex deformations, ALE, page 96 Lp(E)orb Lp-sections of orbifold vector bundle E, orbifolds, page 37 Lpk(E) orb Lpk-sections of orbifold vector bundle E, orbifolds, page 37 NΓ order of Γ, i.e. number of elements in the isotropy group Γp, orbifolds, page 36 O(rk) big-O notation, ALE, page 75 Pg premoduli space of Ricci-flat metrics, ALE, page 92 Pg premoduli space of Ricci-flat metrics, compact manifolds, page 24 Pg premoduli space of Ricci-flat metrics, orbifolds, page 56 R Riemann curvature tensor, page 13 S slice inMorb for the action of Dorb, orbifolds, page 54 sg scalar curvature, page 13 Tg total scalar curvature, page 13 Vsing singular locus, orbifolds, page 33 AC asymptotically conical, ALE, page 112 ALE asymptotically locally Euclidean, ALE, page 76 129