On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

Lie Fu; Grégoire Menet

doi:10.1007/s00209-020-02682-7

On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

Fu, Lie; Menet, Grégoire 2021-10-01 00:00:00 We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon’s relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds. Keywords Hyper-Kähler varieties · Orbifolds · Betti numbers · singularities · Riemann–Roch theorem Mathematics Subject Classiﬁcation 57R18 · 14J42 · 14J28 · 53C26 · 19L10 · 14B05 1 Introduction A compact Kähler manifold is called holomorphic symplectic if it admits a holomorphic 2-form that is nowhere degenerate. In particular, it is even-dimensional and has trivial canon- ical bundle. Such a manifold is called irreducible if moreover it is simply connected and the holomorphic symplectic form is unique up to scalar. Irreducible holomorphic symplectic (IHS) manifolds (also known as compact hyper-Kähler manifolds) admit a Ricci-ﬂat Rie- mannian metric [59], and are characterized by the condition that the holonomy group is the compact symplectic group. The importance of IHS manifolds is manifested in the Beauville– Bogomolov decomposition theorem [2,8]: any compact Kähler manifold with vanishing ﬁrst Chern class has a ﬁnite étale cover which can be written as a product of a complex torus, Lie Fu fu@math.univ-lyon1.fr Grégoire Menet gregoire.menet@univ-grenoble-alpes.fr Institut Camille Jordan, Université Claude Bernard Lyon 1, Lyon, France IMAPP, Radboud Universiteit, Nijmegen, The Netherlands Institut Fourier, 100 rue des Mathématiques, Gières, France 123 204 L. Fu, G. Menet Calabi–Yau varieties and IHS manifolds. We refer to [2,27]and [20, Part III] for the basic theory on such manifolds. Irreducible holomorphic symplectic surfaces are nothing but K3 surfaces. The construction problem for IHS manifolds in higher dimensions seems quite hard: up to deformation, in each even dimension (≥ 4), we so far only have two examples constructed by Beauville [2] (Hilbert schemes of points on K3 surfaces and generalized Kummer varieties) together with two sporadic examples constructed by O’Grady [43,44] in dimensions 6 and 10. The limitedness of available examples suggests the possibility to bound or even classify IHS manifolds (see [28] for diffeomorphic types). As the second cohomology of an IHS manifold, together with the Beauville–Bogomolov quadratic form [2] and the weight-2 Hodge structure, controls most of its geometry [4,5,29,36,38,57,58], it is natural to ask the following question. Question 1.1 In a given dimension, what values can the second Betti number of an irreducible holomorphic symplectic manifold take? In dimension 4, Guan [21] proved the following result in the direction of Question 1.1. Theorem 1.2 (Guan) The second Betti number of a 4-dimensional irreducible holomorphic symplectic manifold is no more than 8, or equal to 23. Moreover, if b = 23, the Hodge diamond must be the same as that of the Hilbert square of a K3 surface. The fact that b ≤ 23 was attributed to Beauville. See [21,22] for extra constraints on each cases; see [35,51] for related results in dimension 6 and the more recent work [33]for a conjectural bound in arbitrary dimension based on [19]. When b = 23, let us mention the work [31,45], which aims at determining the deformation type of IHS fourfolds upon ﬁxing some extra topological data. In the point of view of birational geometry, or more precisely the minimal model program (cf.[34]), it is important to treat varieties with mild singularities. With recent intensive efforts [12,13,16,18,26], the Beauville–Bogomolov decomposition theorem is now extended to projective varieties with klt singularities and numerically trivial canonical class. Naturally, boundedness results for possibly singular irreducible holomorphic symplectic varieties [18, Deﬁnition 8.16] are desired. In particular, Question 1.1 can be posed in this broader setting. This article is our ﬁrst experimental attempt towards the boundedness problem, where we will focus on the classical approach of enlarging the category of IHS manifolds to the so-called primitively symplectic orbifolds, pioneered by Fujiki [15]. Roughly speaking, a primitively symplectic orbifold is a compact Kähler space with quotient singularities in codimension ≥ 4, such that the smooth locus carries a holomorphic symplectic form which is unique up to scalar. See Deﬁnition 3.1. Our ﬁrst main result extends Guan’s Theorem 1.2. Theorem 1.3 Let X be a primitively symplectic orbifold of dimension 4. Then b ( X ) ≤ 23. Moreover, the equality occurs only in the smooth case. Our second main result bounds the size of the singularities. Theorem 1.4 Let X be a primitively symplectic orbifold of dimension 4. Then (i) X has at most 91 singular points. (ii) For each singular point of X , the order of the local fundamental group is at most 1424. 123 Betti numbers of 4-dimensional symplectic orbifolds 205 The proof of Theorems 1.3 and 1.4 is given in the end of Sect. 3. The bound for b in Theorem 1.3 being the same as in the smooth case (note however that no numbers between 9 and 22 are excluded as in [21], despite of our effort in Sect. 4 where we generalize the Hitchin–Sawon formula), the construction methods in the orbifold setting are much richer. Indeed, staying in the smooth category of IHS fourfolds, the only available values for b are 23 and 7, achieved by Hilbert squares of K3 surfaces and generalized Kummer fourfolds respectively; while we are able to construct much more examples within the enlarged category of symplectic orbifolds, ﬁlling many “gaps” in the possibilities of the Betti number. More precisely, we have the following result. Theorem 1.5 There are 4-dimensional primitively symplectic orbifolds with second Betti number 3, 5, 6, 7, 8, 9, 10, 11, 14, 16 and 23. We refer to Sect. 5 for details of these examples. 2 Riemann–Roch theorem for orbifolds 2.1 Orbifolds and V-bundles We ﬁrst ﬁx the notion of orbifolds (“V-manifolds” in [49,50]). Deﬁnition 2.1 (Orbifolds) An n-dimensional complex orbifold is a connected paracompact Hausdorff complex analytic space X such that for any point x ∈ X, there exists an open neighborhood U of x and a triple (V , G,π) with V an open subset of C , G a ﬁnite subgroup of the biholomorphic automorphism group of V,and π : V → U the composition of the quotient map V → V /G and an isomorphism V /G U. Remark 2.2 Note that an orbifold is always normal (see for instance [11, Théorème 4]). In particular, the singular locus is of codimension at least 2. Deﬁnition-Proposition 2.3 [47, Proposition 6] Let X be an n-dimensional complex orbifold. Let x ∈ Sing X . Then there exist a ﬁnite subgroup G of GL (C) and an open neighbourhood x n n n V ⊂ C of the origin 0 ∈ C , stable under the action of G , with V /G isomorphic to an x x x x open neighbourhood U of x , and such that Codim Fix(g) ≥ 2 for all g ∈ G \{id}. Such a group G is unique up to conjugation. Let π : V → V /G U ,wecall x x x x x x (U , V , G ,π ) a local uniformizing system of x , and G the local fundamental group of x x x x x Xat x. Remark 2.4 In modern literature (see for example [1]), orbifold is a synonym for (analytic) Deligne–Mumford stack. In particular, the collection of charts (V , G,π) is part of the data of an orbifold and the group G is sometimes allowed to act non-effectively on V.Ifweuse the terminology orbifold in this generalized sense, then what is deﬁned in Deﬁnition 2.1 cor- responds to effective orbifolds [1, Deﬁnition 1.2] with all nontrivial elements of all stabilizer groups acting with a non-empty ﬁxed locus of codimension at least 2. By the above result of Prill, these are equivalent notions, as such an orbifold/stacky structure is determined by the underlying complex analytic space. The notion of vector bundles naturally generalizes to orbifolds. 123 206 L. Fu, G. Menet Deﬁnition 2.5 (V-bundles) Let X be an orbifold. • AV-bundle (or orbibundle)on X is a vector bundle F on X := X Sing X such that reg for any local uniformizing system (U , V , G,π), there exists a vector bundle F on V endowed with an equivariant action of G such that: F π ( F ), V |V Fix G |U reg where Fix G := Fix(g). g∈G,g =id • Let F be a coherent sheaf on X. The sheaf F is said to be locally V-free if for any x ∈ X, there exist a local uniformizing system (U , V , G,π), a free coherent sheaf F on V , ˆ ˆ and a G-action on F such that F π F .By[7, 4.2], the local V-freeness of V |U ∗ a coherent sheaf F is equivalent to the condition that F is reﬂexive and the reﬂexive [∗] ∗ ∨∨ pull-back π (F ) := π (F ) is locally free for any local uniformizing system. |U |U As in the smooth case, there is an equivalence of categories between the category of locally V-free sheaves and that of V-bundles. Example 2.6 (Reﬂexive differentials) Givenanorbifold X of dimension n, the sheaf of reﬂex- ive differential forms ( [17,2.D], [46, Section 2.5]) [i ] i ∨∨ i := ( ) ι ( ) X X X reg is a locally V-free sheaf for any i ∈ N,where ι : X → X is the natural inclusion of the reg smooth part. The sheaf of reﬂexive forms of top degree is identiﬁed with the dualizing sheaf: [n] ω . Remark 2.7 (Hodge decomposition) Let X be a compact Kähler orbifold. For any integer k ≥ 0, the rational singular cohomology group H ( X , Q) carries a pure Hodge structure of weight k and in the Hodge decomposition k p,q H ( X , C) = H ( X ), p+q=k [ p] p,q q p,q we have H ( X ) = H ( X, ),see [46, Section 2.5]. We denote h ( X ) := p,q p,q q, p dim H ( X ).Wehave h = h . Notation 2.8 Let X be an orbifold, x ∈ X and F a locally V-free sheaf. Let (U , V , G,π) be a local uniformizing system of x and let F be a locally free sheaf on V endowed with an action of G as in Deﬁnition 2.5. Hence, the ﬁber of F at 0 is endowed with an action of G, which provides a representation of G.Wedenoteby ρ the representation of G x ,F associated with x and F . 2.2 Characteristic classes on orbifolds We recall the deﬁnition of Chern classes of V-bundles on orbifolds, by adapting the Chern– Weil approach. Deﬁnition 2.9 [7, Deﬁnition 2.9] Let F be a V-bundle of rank r on an orbifold X.A metric on F is a Hermitian metric h on F as bundle on X such that for all local uniformizing reg systems (U , V , G,π), the Hermitian metric π (h ) extends to a Hermitian metric on F . |U V reg 123 Betti numbers of 4-dimensional symplectic orbifolds 207 Deﬁnition 2.10 [7, Deﬁnition 1.5] Let X be an orbifold. A smooth differential k-form ϕ on X is a C differential k-form on X such that for all local uniformazing system (U , V , G,π), reg ∗ ∞ π (ϕ ) extends to a C -differential k-form on V . (We always mean C-valued forms.) |U reg Notation 2.11 We denote by A the sheaf of differential k-forms. As explained in [7, Deﬁnition 2.10], we can deﬁne the Chern classes of a V-bundle as follows. Let F be a V-bundle of rank r on an orbifold X. We can ﬁrst construct the Chern forms on X as in the smooth case. To a Hermitian metric h on F, we associate the Chern reg connection D on F,and to D, we associate the curvature D .Let be the corresponding i 2k r × r matrix of curvature 2-forms, then we set c (h) = P ( ) ∈ ( X , A ),where P k k reg k 2π r ×r is the k-th elementary invariant polynomial function C → C. The same process can be also done on all local uniformizing systems (U , V , G,π). ˆ ˆ The metric π (h ) extends to a Hermitian metric h on F which gives rise to the |U V reg ˆ ˆ ˆ Chern connection D on F and hence the curvature D . As previously, we can construct 2k ∗ ˆ ˆ c (h) ∈ (V , A ). By construction π (c (h) ) extends to c (h) on V . Hence, we obtain k k |U k reg 2k c (h) ∈ ( X , A ). As in the smooth case, we show that c (h) is a closed form, and that the k k 2k cohomology class c ( F ) := [c (h)] ∈ H ( X , C), called the k-th Chern class of F, does k k not depend on the choice of the metric h. Other characteristic classes, like Todd classes and Chern characters, are deﬁned in terms of Chern classes by the usual formulas. A characteristic class of an orbifold is that of its tangent V-bundle. 2.3 Riemann–Roch and Gauss–Bonnet theorems for orbifolds One key ingredient in the proof of Theorem 1.3 is the following orbifold version of the Hirzebruch–Riemann–Roch theorem due to Blache [7, Theorems 3.5 and 3.17]. Theorem 2.12 (Blache [7]) Let X be a compact complex orbifold with only isolated singu- larities and let F be a locally V-free sheaf. Then we have ⎡ ⎤ ⎢ ⎥ 1 tr(ρ (g)) x ,F ⎢ ⎥ χ( X , F ) = ch(F ) · td( X ) + , ⎣ ⎦ |G | det(id −ρ (g)) X x x ,T x ∈Sing X g∈G g =id where g is viewed as an endomorphism on T V with (U , V , G ,π) a local uniformizing 0 x system of x . Blache also established the following orbifold version of Gauss–Bonnet theorem. Theorem 2.13 [7, Theorem 2.14] Let X be an n-dimensional compact complex orbifold with only isolated singularities. Then we have the following formula for its topological Euler characteristic: χ ( X ) = c ( X ) + 1 − , top n |G | X x x ∈Sing X where G is the local fundamental group of X at x . 123 208 L. Fu, G. Menet 3 Bounding Betti numbers and singularities The aim of this section is to show Theorems 1.3 and 1.4. Let us ﬁrst make precise the class of (possibly singular) symplectic varieties that we consider. 3.1 Symplectic orbifolds Deﬁnition 3.1 (Fujiki [15]) A compact Kähler orbifold X is called primitively symplectic if (i) the smooth locus X := X Sing X is endowed with a non-degenerated holomorphic reg 2-form which is unique up to scalar; and (ii) the singular locus Sing X has codimension at least 4. If moreover X is simply connected, X is called an irreducible symplectic orbifold. reg Remark 3.2 As in the smooth case, a primitively symplectic orbifold X has even (complex) dimension and trivial dualizing sheaf ω O . Moreover, the symplectic form extends to X X a symplectic form on any local uniformizing system. In particular, the contraction with the [1] symplectic form induces an isomorphism T . By deﬁnition, if X has dimension 4, then X has isolated quotient singularities. Remark 3.3 As quotient singularities are rational singularities, the singularities appearing in Deﬁnition 3.1 are symplectic singularities in the sense of Beauville [3]. Moreover, an irreducible symplectic orbifold deﬁned above is an irreducible symplectic variety in the sense of [18]and [26, Deﬁnition 1.4]. Remark 3.4 (Hodge diamond) Let X be a 4-dimensional primitively symplectic orbifold. Fujiki [15, Proposition 6.7] showed that X has vanishing irregularity, hence b ( X ) = 0. Serre–Grothendieck duality gives that [1] [1] 3 1 ∨ 1 ∨ ∼ ∼ H ( X, ) = H ( X,ω ⊗ T ) = H ( X, ) . X X X X 3,1 1,1 3,0 1,0 In particular, h = h . Similarly, h = h = 0. In conclusion, the Hodge diamond of X takes the following form. 1,1 1 h 1 2,1 2,1 0 h h 0 1,1 2,2 1,1 1 h h h 1. 2,1 2,1 0 h h 0 1,1 1 h 1 3.2 Quotient symplectic singularities in dimension 4 For later use, we classify in this section all symplectic quotient singularities in dimension 4. As the germ of a quotient singularity is determined by the local fundamental group, one needs to classify all ﬁnite subgroups of the Lie group Sp(4, C). Since any ﬁnite subgroup must be contained in some compact maximal subgroup, we are to classify ﬁnite subgroups of the compact symplectic group Sp(2) := Sp(4, C) ∩ SU(4). 123 Betti numbers of 4-dimensional symplectic orbifolds 209 2i π Proposition 3.5 Let n > 0 be an integer, we denote ξ := e the primitive n-th root of unity. For integers 1 ≤ k ≤ n, we denote ⎛ ⎞ 00 1 0 ⎜ ⎟ 00 0 1 ⎜ ⎟ T := . n,k k ⎝ ⎠ ξ 00 0 −k 0 ξ 00 Let G be a ﬁnite subgroup of the compact symplectic group Sp(2). Then, up to conjugation, (i) there exists ﬁnite subgroups H ,H of SU(2),integersn > 0 and k ∈{1, ..., n}, and 1 2 a normal subgroup G of G of index at most 2, such that any element M of G has the form A 0 M = , 0 B with A ∈ H ,B ∈ H , and G/G = T if G = G. 1 2 n,k (ii) If moreover C /G has only the image of 0 as singularity, then there exists a ﬁnite subgroup Hof SU(2) and θ an automorphism of H such that any element M ∈ G has the form A 0 M = , 0 θ( A) for some A ∈ H. Proof Hanany and He classiﬁed in [23] the ﬁnite subgroups of SU(4). Hence it is enough to identify those groups in the list that preserve a symplectic form. In the sequel, we follow their notation. The ﬁrst category of groups are the so-called primitive simple groups described in [23, Section 3.1.1] and they are numbered from I to VI. However, none of them are symplectic. Indeed, the following matrices are considered: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 10 0 0 100 2 3 000 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 01 0 0 1 0 −1 20 1 0 −12 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ F := , F := √ and F := , 1 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 00 w 0 02 −12 0 21 0 3 02 2 −1 00 0 w 20 0 −1 2i π where w := e . The matrices F and F do not preserve any common symplectic form, 1 2 hence the groups I and III, which are partially generated by these two matrices, cannot be symplectic. Similarly, the group II cannot be symplectic because it is partially generated by the two matrices F and F which do not ﬁx the same symplectic form. 2i π 4 2 Let β := e . The matrices S := diag(1,β,β ,β ) and D := diag(w,w,w, 1) are not symplectic, hence the groups IV, V and VI, which are partially generated by one of these two matrices, are not symplectic. In [23, Section 3.1.2], Hanany and He consider the groups VII, VIII and IX which cannot be symplectic since they are partially generated by the groups I, II and III. In [23, Section 3.1.3], they consider group obtained from Kronecker products of matrices of SU(2).Let 1 1 −1 + i −1 + i 1 + i 0 S := and U := √ . SU(2) SU(2) 1 + i −1 − i 01 − i 123 210 L. Fu, G. Menet The following couples of matrices (S ⊗ S , U ⊗ U ) and (S ⊗ SU(2) SU(2) SU(2) SU(2) SU(2) 2 2 S , U ⊗ U ) both do not ﬁx the same symplectic form. Hence the groups from X SU(2) SU(2) SU(2) to XVI cannot be symplectic since they are all partially generated by one of theses couples of matrices. Also the groups from XVII to XXI cannot be symplectic because they are partially generated by the groups XI, X, XVI and XIV. The matrices A := diag(1, 1, −1, −1) and A := diag(1, −1, −1, 1) do not ﬁx the same 1 2 symplectic form. Hence all the groups from XXII to XXX are not symplectic since they are all partially generated by these two matrices. Finally, we consider the group: 2i π j k l − j −k−l := diag(w ,w ,w ,w ) w = e , j , k, l ∈{1, ..., n} , which is not symplectic, hence all the groups from XXXI to XXXIII which are partially generated by are not symplectic. It only remains to consider the group XXXIV and the intransitive groups. We will study the group XXXIV in the end. The intransitive groups are the groups coming from diagonal embedding of subgroups of SU(2) or SU(3) (see [23, Deﬁnition 2.1] for the precise deﬁni- tion). We will see that all the symplectic groups constructed from a diagonal embedding of a subgroup of SU(3) are actually constructed from a diagonal embedding of subgroups of SU(2).Let G be such a group and M an element in G.Let (e , e , e , e ) be the canonical 1 2 3 4 basis of C .Wehave: ξ 0 M = , 0 A where ξ is a root of the unity and A ∈ U (3). We can ﬁnd a basis (v ,v ,v ) of C in which 1 2 3 −1 −1 A is diagonalized: A = diag(ξ ,ζ,ζ ). In the basis (e ,v ,v ,v ), the symplectic form 1 1 2 3 has to be ⎛ ⎞ 01 00 ⎜ ⎟ −10 0 0 ⎜ ⎟ J := . ⎝ ⎠ 00 −10 We consider now another matrix of G expressed in the basis (e ,v ,v ,v ): 1 1 2 3 ⎛ ⎞ a 00 0 ⎜ ⎟ 0 bc d ⎜ ⎟ N := , ⎝ ⎠ 0 ef g 0 hj k where a, b, c, d, e, f , g, h, j , k are in C.Since N is symplectic. It follows: ab = 1 ac = 0 ad = 0 −hf + ej = 0 −hg + ek = 0 − jg + fk = 1. ej ek Hence c = d = 0. If h = 0, then f = and g = . This is impossible because, it h h contradicts − jg + fk = 1. So h = 0. For the same reason e = 0and G is actually a group 123 Betti numbers of 4-dimensional symplectic orbifolds 211 composed by matrices of the forms: ⎛ ⎞ a 00 0 ⎜ ⎟ 0 b 00 ⎜ ⎟ N := . ⎝ ⎠ 00 fg 00 jk It only remains to study the case of the group XXXIV. In this case, G = G , T , with G 0 n,k 0 composed of matrices of type A 0 , with A, B ∈ SU(2). 0 B We consider A 0 G := M = , ∈ G | A, B ∈ SU(2) . 0 B The group G is a normal subgroup of G and the class T ∈ G/G has order 2. n,k Now, we prove (ii ).Let G be a ﬁnite subgroup of Sp(2) such that C /G admits only 0 as singularities. Then necessarily, the unique element of G with the eigenvalue 1 is the identity. In particular, this is true for G . Therefore, the following projections are isomorphisms: A 0 A 0 P : G −→ H , → A, and P : G −→ H , → B. 1 1 2 2 0 B 0 B −1 So, setting θ := P ◦ P ﬁnishes the proof. 3.3 Orbifold Salamon relation Salamon [48, (0.1)] discovered a remarkable linear relation among the Betti/Hodge numbers of a compact hyper-Kähler manifold. By applying Blache’s orbifold Hirzebruch–Riemann– Roch Theorem 2.12, we will establish (Proposition 3.6 below) a Salamon-type relation between the Hodge numbers and information from the singularities of a 4-dimensional prim- itively symplectic orbifold. A more general result can be obtained by adapting Salamon’s method. However, we decide to give an elementary proof here and leave the general result in Appendix A. In what follows, X is a primitively symplectic orbifold of dimension 4. For any (necessarily isolated) singular point x ∈ X, G is the local fundamental group of X at x and ρ is the x x ,• representation of G deﬁned in Notation 2.8. • Deﬁne for any integer p ≥ 0, tr(ρ [ p] (g)) 1 x , S := . (1) |G | det(id −ρ (g)) x x ,T x ∈Sing X g =id • Applying Theorem 2.12 to F = O , we obtain: td ( X ) = 3 − S , (2) 4 0 [1] • Applying Theorem 2.12 to F = , we obtain, using Remark 3.4: 2,1 1,1 h ( X ) − 2h ( X ) = 4 td ( X ) − c ( X ) + S , (3) 4 4 1 X X 123 212 L. Fu, G. Menet [2] • Applying Theorem 2.12 to F = , we obtain, using Remark 3.4: 2,1 2,2 2 − 2h ( X ) + h ( X ) = 6 td ( X ) + c ( X ) + S , (4) 4 4 2 X X • Applying Theorem 2.13, we obtain: 1,1 2,1 2,2 8 + 4h − 4h + h = c ( X ) + η, (5) where η = 1 − . (6) |G | x ∈Sing X Combining (2), (3), and (5), we can eliminate c ( X ) and td ( X ) to obtain: 4 4 X X 2,1 2,2 1,1 2h + h − 8h = 64 + 6(S − 4S ) + η. (7) 1 0 Similarly, by combining (2), (4), and (5), it yields that 2,1 2,2 1,1 2h + h − 8h = 64 + 3(S − 6S ) − 2η. (8) 2 0 Then, (7)and (8) provide the following orbifold version (in dimension 4) of Salamon’s famous relation [48] for hyper-Kähler varieties. See Appendix A for a generalization. Proposition 3.6 (Orbifold Salamon relation) Let X be a primitively symplectic orbifold of dimension 4. We have: 2,1 2,2 1,1 2h + h − 8h = 64 + s, (9) or equivalently, b + b − 10b = 46 + s, 4 3 2 where s := 6(S − 4S ) + η = 3(S − 6S ) − 2η = 4S + S − 22S (10) 1 0 2 0 1 2 0 is a correction term determined by the singularities. In particular: η = S − 2S + 2S = (−1) S . 2 1 0 i i =0 1,1 2,1 Remark 3.7 Proposition 3.6 shows that the knowledge of h , h and the singularities is enough to compute the topological Euler characteristic and all the Betti numbers of a 4-dimensional primitively symplectic orbifold. 3.4 Estimate of the contribution of singularities We turn to a more careful study of the quantity s in the orbifold Salamon relation (9), which is the local contribution of singularities. Using (1), (6)and (10), we can write s = s x ∈Sing X with: ⎡ ⎛ ⎞ ⎤ tr(ρ [1] (g)) − 4 ⎢ ⎜ ⎟ ⎥ 1 x , ⎢ ⎜ ⎟ ⎥ s = 6 +|G |− 1 , (11) x x ⎣ ⎝ ⎠ ⎦ |G | det(ρ (g) − id) x x ,T g∈G g =id 123 Betti numbers of 4-dimensional symplectic orbifolds 213 with (U , V , G ,π) a local uniformizing system around x. In particular, the action of g ∈ G x x −1 −1 on T is symplectic. We can therefore write that g = diag(ξ ,ξ ,ξ ,ξ ), with ξ = V ,0 1 2 j 1 2 2i π k e , k , n ∈ N for all j ∈ {1, 2}. Hence: j j 2π k 2π k 1 2 2 cos + cos − 4 tr(ρ 1 (g)) − 4 n n x , 1 2 2π k 2π k det(g − id) 1 2 4 1 − cos 1 − cos n n 1 2 1 1 =− − . 2π k 2π k 1 2 2 1 − cos 2 1 − cos n n 1 2 So: tr(ρ (g)) − 4 x , 1 ≤− (12) det(g − id) 2 Hence for any x ∈ Sing X,wehave |G |− 1 s ≤−2 . (13) |G | In particular, s ≤−1 and the quantity s, which is an integer by (9), is at most −| Sing X |. Using Proposition 3.5, we can be more precise. The local fundamental group G is a ﬁnite subgroup of Sp(2). Hence, there exists a normal subgroup G of G of index at most 2, H a ﬁnite subgroup of SU(2) and an automorphism θ of H such that any element M ∈ G has the form A 0 M = , 0 θ( A) with A ∈ H. As we have noticed previously, if A is a matrix of SU(2) of ﬁnite order, we have det( A − id) =− tr( A) + 2. Therefore tr(ρ (g)) − 4 [1] x , 1 1 =− + , det(ρ (g) − id) 2 − tr( A) 2 − tr(θ ( A)) x ,T g∈G g∈G g =id g =id A 0 where on the right-hand side, we write a non-trivial element g of G as for 0 θ( A) A ∈ H. Reordering the sum of the second term, we obtain the following equation: tr(ρ [1] (g)) − 4 x , 1 =−2 . (14) det(ρ (g) − id) 2 − tr( A) x ,T g∈G A∈ H A =id g =id Example 3.8 We compute explicitly s in the following cases. • G = C is a cyclic group of order n. x n 2i π 2i π n n In this case, G = G and H = g , with g = diag(e , e ).By(14), we have: x n n n−1 tr(ρ [1] (g)) − 4 2 x , 1 n − 1 =− =− , (15) 2kπ det(ρ (g) − id) 6 x ,T 1 − cos( ) g∈C k=1 g =id 123 214 L. Fu, G. Menet where we used the identity n−1 1 n − 1 = . kπ sin ( ) k=1 n As a result, s (C ) =−(n − 1). (16) x n • G = D is a binary dihedral group of order 4n. x n ˜ ˜ In this case, G = G and H = D . The binary dihedral group D is generated by x n n ξ 0 01 B := and P = , with ξ a2n-root of the unity. The group D can −1 0 ξ −10 be partitioned into the disjoint union of the following two sets: 2 2n 2 2n B, B , ..., B and BP, B P, ..., B P . Hence by (14) and the fact that tr( B P) = 0for all k ∈{1, ..., 2n},wehavethat ⎛ ⎞ tr(ρ [1] (g)) − 4 tr(ρ [1] (g)) − 4 ⎜ ⎟ x , x , X X ⎜ ⎟ = − 2n. ⎝ ⎠ det(ρ (g) − id) det(ρ (g) − id) x ,T x ,T X X ˜ g∈C 2n g∈ D g =id g =id Using (15), we obtain tr(ρ (g)) − 4 2 [1] x , 4n + 12n − 1 =− . (17) det(ρ (g) − id) 6 x ,T g∈ D g =id Therefore ( D ) =−(n + 2). (18) x n 3.5 Orbifold Guan inequality In the smooth case, Guan [21, Section 2] has proved Theorem 1.3 using two ingredients: the Hirzebruch–Riemann–Roch formula and the Verbitsky theorem [56, Theorem 1.5]. The orbifold extension of the former being explained in Sect. 2.3, we give the generalization of the latter here. Proposition 3.9 [5, Proposition 5.16] Let X be a primitively symplectic orbifold of dimension 2n. Then the following map induced by the cup-product is injective for any k ≤ n: k 2 2k Sym H ( X , C) → H ( X , C). Remark 3.10 When n = 2, we can also prove the previous proposition with an elementary method using the Fujiki relation and the fact that the Beauville–Bogomolov form is non- degenerate (see [38, Section 3.4]). Corollary 3.11 Let X be a primitively symplectic orbifold of dimension 4. Then: 2,1 1,1 2 1,1 4h ≤−(h ) + 15h + 126 + 2s, where s is (the non-positive integer) deﬁned in (11). In particular: 0 ≥ s ≥−91 (19) 123 Betti numbers of 4-dimensional symplectic orbifolds 215 Proof Proposition 3.9 provides the following inequality: (b + 1)b 2 2 b ≥ , which can be rewritten as 1,1 2,2 1,1 1,1 4 + 4h + 2h ≥ (3 + h )(2 + h ). Combining this inequality with Proposition 3.6, we obtain our result. 3.6 Proof of the main results Proof of Theorem 1.3 Thanks to Corollary 3.11,wehave: 1,1 2 1,1 0 ≤−(h ) + 15h + 126 + 2s. Or equivalently, 1,1 1,1 −2s ≤ (21 − h )(h + 6). Because of (13), s ≤ 0and when X is singular s < 0. Then Theorem 1.3 follows. Proof of Theorem 1.4 Statement (i) follows from (13)and (19). Let us prove (ii). By Proposition 3.5, there exist a normal subgroup G of G of index at most 2, H a ﬁnite subgroup of SU(2) and an automorphism θ of H such that any element M ∈ G has the form A 0 M = , 0 θ( A) for some A ∈ H. We only need the well-known classiﬁcation of the ﬁnite subgroups of SU(2) to have a full description of all possible G . The ﬁnite subgroups of SU(2) are the so-called Kleinian groups corresponding to the A-D-E Dynkin diagrams: the cyclic groups C , the binary dihedral groups D and the three sporadic groups E , E and E . The biggest n n 6 7 8 sporadic group E has order 120. Let us check the maximal size of the groups of A-D types. When G = G , we already have computed in Example 3.8 that s (C ) =−(n − 1) and x x n s ( D ) =−(n + 2). x n Now, we assume that G /G has order 2. By (12), we have: ⎡ ⎛ ⎞ ⎤ tr(ρ [1] (g)) − 4 tr(ρ [1] (T g)) − 4 n,k ⎢ ⎜ ⎟ ⎥ x , x , X X ⎢ ⎜ ⎟ ⎥ s (G ) = 6 + +|G |− 1 x x x ⎣ ⎝ ⎠ ⎦ |G | det(ρ (g) − id) det(ρ (T g) − id) x x ,T x ,T n,k X X g∈G g∈G g =id ⎡ ⎛ ⎞ ⎤ tr(ρ [1] (g)) − 4 ⎢ ⎜ ⎟ ⎥ 1 x , |G | ⎢ ⎜ ⎟ ⎥ ≤ 6 − +|G |− 1 . ⎣ ⎝ ⎠ ⎦ |G | det(ρ (g) − id) 2 x x ,T g∈G g =id [2] [2] ˜ ˜ Denote by C (resp. D ) a ﬁnite subgroup of Sp(2) such that C (resp. D ) is a normal n n n n subgroup of index 2. We have then by (15)and (17): n + 1 n + 4 [2] [2] s (C ) ≤− and s D ≤− . (20) x x n n 2 2 123 216 L. Fu, G. Menet However by (19), we know that s (G) ≥−91. Hence, the biggest possible groups are the groups which have a binary dihedral group D as normal subgroup of index 2. These groups have order 8 × 178 = 1424. Remark 3.12 Using (19), we can be more precise about the maximal cardinality of each kind of groups. • If G = C is a cyclic group of order n,thenby(16), n ≤ 92. x n • If G = D is a binary dihedral group of order 4n,thenby(18), n ≤ 89. x n [2] • If G = C is a group with a cyclic group of order n as normal subgroup of index 2, x n then by (20), n ≤ 181. [2] • If G = D is a group with a binary dihedral group of order 4n as normal subgroup of x n index 2, then by (20), n ≤ 178. Remark 3.13 Using Corollary 3.11, we can get sharper constraints on singularities for each ﬁxed second Betti number. For example, if b = 22 (resp. 21, 20, etc.), then there are at most 13 (resp. 25, 36, etc.) singular points. 4 Hitchin–Sawon formula We can try to improve Theorem 1.3 using the same method as in [21, Section 3]. The method of Guan is based on an equation of Hitchin–Sawon [25]. This section is just an attempt and is not needed in the rest of the paper. First, we recall the following generalized Fujiki formula. Lemma 4.1 [38, Lemma 4.6] Let X be a primitively symplectic orbifold of dimension 2n. 4 p If β ∈ H ( X , C) is of type (2 p, 2 p) on all small deformations of X , then there exists a 2 2(n− p) constant N (β) depending on β such that for all α ∈ H ( X , C), one has β · α = n− p ! " 2n N (β) α . Proposition 4.2 Let X be a primitively symplectic orbifold of dimension 2n. Then: n−1 n ((2n)!) N (c ) = A, (24n(2n − 2)!) a /2 2n i ˆ ˆ where Ais the A-genus deﬁned by ,where a ’s are the Chern roots of the i =1 sinh a /2 tangent bundle of X . Proof We adapt Hitchin–Sawon’s proof [25]. We will note that quotient singularities do not have any effect on Hitchin–Sawon formula. We can consider N (c ) as deﬁned in Lemma 4.1. In the smooth case, the equation of Hitchin–Sawon provides an expression of N (c ) in terms of Pontryagin classes. The main tool used by Hitchin and Sawon are Rozansky–Witten invariants (see [25, Section 2]). These invariants are constructed from the curvature of the manifold and a trivalent graph with 2n vertices. The tangent sheaf T on X can be deﬁned as the unique reﬂexive sheaf such that T X X | X reg is the usual holomorphic tangent sheaf on X . It is a locally V-free sheaf. We consider g reg a Kähler metric on T . As explained in Deﬁnition 2.9, this provides a metric g on T X X reg such that for all local uniformizing system (U , V , G,π), π (g ) extends to a metric g |U V reg on T . Then the Riemannian curvature K of ( X , g) is obtained on X by the Riemannian V reg curvature of ( X , g) and on all local uniformizing systems by the Riemann curvature of reg 123 Betti numbers of 4-dimensional symplectic orbifolds 217 (V , g ). For the same reason as in the smooth case, we can associated to the curvature a class 1 3 1 []∈ H ( X , Sym ) (see [25, Section 2] or [54]). From this class [], the deﬁnition of the Rozansky–Witten invariants being purely algebraic, it can be generalized, word by word, to the case of primitively symplectic orbifold. We denote these invariants b ( X ) for a trivalent graph with 2n vertices. In the smooth case, it is well known that: % & tr K 2c − c = , (21) (2π) where K is the curvature of X. Because of our deﬁnition of Chern classes in Sect. 2,(21)is also true in the orbifold case. In the symplectic case (21)gives: % & tr K 2c = . (2π) Then, using this expression for c exactly as Hitchin and Sawon did in [25, Section 3], we can provide an expression of N (c ) ([25, equations (7) and (8)]): n n 2 n−1 n−1 c ω · ω = 16π n c · ω · ω , (22) X X 2,0 where ω generated H ( X ) and c can be express by: n! b n ( X ) = c vol( X ), (23) 2 n (2π ) 2n (ω+ω) with vol( X ) = and the trivalent graph with two vertices. Equation (22) can be 2n 2 (2n)! rewritten: 2n 2 n−1 c (ω + ω) 16π n c · (ω + ω) X X = . ! " ! " 2n 2(n−1) n n−1 That is: 32π (2n − 1) N (c ) c = . ' ( 1/n 2n (ω + ω) Then, with (23), we obtain: n n n n!4 (2n − 1) N (c ) b ( X ) = . (24) (2n)! In general, we can write: % & 2m tr(K ) s = , 2m 2m (2π i ) where 2m ch(T ) = . (2m)! Using these expressions and important results on graphs (see [25, Section 5]), Hitchin and Sawon provide an expression of b in terms of the Pontryagin classes. The results on graph 123 218 L. Fu, G. Menet are not affected by having singularities on X, hence, the same expression can be obtained in the symplectic orbifold case: b n ( X ) = 48 n! A. Combined with (24) this equation provides our proposition. Lemma 4.3 Let X be a primitively symplectic orbifold of dimension 4, then: 2 2 3b N (c ) ≤ (b + 2)c . 2 2 2 Proof The proof of [21, Lemma 3] can be adapted in the case of primitively symplectic orbifolds. Indeed, it is a consequence of Lemma 4.1 and the Hodge–Riemann bilinear relation which have been generalized in [38, Proposition 2.14]. Corollary 4.4 Let X be a primitively symplectic orbifold of dimension 4, then: (b + 1)b ≤ 4b + 2(S + 20S − 62)b + 736 + 2(S − 124S ), 2 3 1 0 2 1 0 where S and S are deﬁned in (1), introduced in Sect. 3.3. 0 1 Proof In our case, Proposition 4.2 provides: 2 2 4! N (c ) 1 = td − . 2 2 (24 × 2 × 2) 2 8 × 12 That is: 2 2 N (c ) = 192 td − . 2 4 So, using Lemma 4.3: 576b td ≤ 2(b + 1)c . 2 4 2 In Sect. 3.3, we found expressions for c and td , so it is more convenient to replace c by 4 4 720 td +c 4 4 720 td +c 4 4 576b td ≤ 2(b + 1) . 2 4 2 Then (3) provides: 576b td ≤ 2(b + 1)(248 td +4b − b − 8 + 2S ). 2 4 2 4 2 3 1 This is: (b + 1)b ≤ 4b + (2S − 40 td −4)b + 248 td −8 + 2S . 2 3 1 4 2 4 1 Using (2) to replace td by 3 − S , we obtain our result. 4 0 Example 4.5 We can apply Corollary 4.4 to orbifolds with singularities C / ±id. It provides: (b + 1)b ≤ 4b + ( N − 124)b + 736 − 8 N , 2 3 2 where N is the number of singular points. • If N = 28: (b + 1)b ≤ 4(16 − b )(8 − b ). 2 3 2 2 123 Betti numbers of 4-dimensional symplectic orbifolds 219 • If N = 36: (b + 1)b ≤ 4(14 − b )(8 − b ). 2 3 2 2 This corresponds exactly to the second Betti numbers of examples in [15, Section 13] (see also Sect. 5.13). Remark 4.6 Unfortunately, Corollary 4.4 is not restrictive enough to exclude more second Betti numbers. For instance, a primitively symplectic orbifold with second Betti number 22 2i π 2i π 2i π 3 3 3 and 3 isolated singularities of analytic type (C /g , 0) with g = diag(e , e , e , 3 3 2i π e ) is not in contradiction with Corollary 4.4 and Proposition 3.11. To improve Theo- rem 1.3, we need some techniques to exclude some conﬁgurations of singularities. 5 Examples of primitively symplectic orbifolds of dimension 4 For each Betti number between 3 and 23, we provide an example of primitively symplectic orbifold when we know one. See [15, Section 13] for more examples; many additional exam- ples could also be obtained by considering partial resolution in codimension 2 of quotients [2] of K 3 -type and Kum -type manifolds. We summarize all the numerical results in a table in Sect. 5.13. In this section, an isolated cyclic quotient singularity of order n refers to the germ (C /C , 0),where C is an order-n cyclic subgroup of Sp(k, C) such that the origin is n n the only ﬁxed point. 5.1 A construction of Fujiki Most of the know examples of primitively symplectic orbifolds of dimension 4 was con- structed by Fujiki in [15, Section 13]. We recall his construction. Let H be a ﬁnite group of symplectic automorphisms on a K3 or an abelian surface S. First, we assume that H is abelian. Let θ be an involution on H.The action of H on S × S is given by h ·(s, t ) = (hs,θ(h)t ) for all (h, s, t ) ∈ H × S × S. Moreover, we deﬁne G to be the group of automorphisms of S × S generated by H and the involution (s, t ) → (t , s). The quotient (S × S)/G has isolated singularities as well as singularities in codimension 2. The singularities in codimension 2 can be resolved crepantly (see [6]) and we denote by Y ( H ) (resp. Y ( H )) the primitively symplectic orbifold obtained when S is K 3 T aK3surface (resp.when S is a complex torus of dimension 2). As explained in [15, Section 13], the deformation class of Y ( H ) (resp. Y ( H )) only depends on H. K 3 T When the group H is non abelian, the situation is more complicated (the deformation class does not only depends on H) and Fujiki only provides 5 additional examples. 5.2 b = 3 Mongardi [41, Example 4.5.1 and Example 4.5.2] constructed two manifolds ( X ,σ ) and 1 1 [2] ( X ,σ ) of K 3 -type endowed with symplectic automorphisms of order 11 such that 2 2 ⎛ ⎞ ⎛ ⎞ 62 2 21 0 2 σ 2 σ 1 2 ⎝ ⎠ ⎝ ⎠ H ( X , Z) 28 −3 and H ( X , Z) 16 0 . 1 2 2 −38 0022 123 220 L. Fu, G. Menet We denote the quotients M = X /σ with i ∈ {1, 2}. The primitively symplectic orbifolds i i 1 2 M and M both have second Betti number equal to 3 and have 5 isolated cyclic quotient 11 11 singularities of order 11. Moreover their Beauville–Bogomolov forms were computed in [40, Theorem 1.2]. The pair ( X ,σ ) was independently discovered in [14]. 1 1 [2] In general, the fourth Betti number of the quotient of a manifold of K 3 -type by an automorphism of prime order p = 2, 5 can be computed using the Boissière–Nieper- Wisskirchen–Sarti invariants and the fact that: H ( X , Z) = (Z /2 Z) ⊕ (Z /5 Z). Sym H ( X , Z) See [9, Section 2, Proposition 5.1, Proposition 6.6 and Lemma 6.14] for more details. We obtain b ( M ) = 26 for all i ∈ {1, 2}. 5.3 b = 5 [2] Let X be a manifold of K 3 -type endowed with a symplectic automorphism σ of order 7. 2 σ As explained in [41, Section 7.3], we always have rk H ( X , Z) = 5and ( X,σ ) is standard [2] [2] [2] (i.e. deformation equivalent to a natural pair (S ,σ ),where S is a K3 surface and σ induced by an automorphism σ on S). We denote M := X /σ , which is a primitively sym- plectic orbifold with second Betti number 5, and has 9 isolated cyclic quotient singularities of order 7. Moreover, its Beauville–Bogomolov form have been computed in [39, Theorem 1.3]. [m] In general, the Betti numbers of the quotient of a Hilbert scheme S of m points on a K3 surface S by a natural automorphism of prime order can be computed using the Boissière– ∗ [m] Nieper-Wisskirchen–Sarti invariants and the Qin–Wang integral basis of H (S , Z) ([53, Theorem 1.1 and Remark 5.6]). See [39, Proof of Corollary 5.2] for more details. We can compute b ( M ) = 42. 4 7 5.4 b = 6 We consider the complex torus T = C /,where = (1, 0), (i , 0), (0, 1), (0, i ).The torus is given by the product of elliptic curves T = E × E, with E := C/ 1, i .Let σ be the symplectic automorphism of order 4 on T deﬁned by the action on C givenbythe matrix: 0 −1 σ = . We remark that 2 ν ⊕2 H (T , Z) = U ⊕−2 . (25) [2] Since σ is a linear automorphism on T , it extends to an automorphism σ on K (T ).We 4 2 [2] want to consider a crepent resolution in codimension 2 of K (T )/σ . It can be obtain as follows. We have σ =− id. As explained in [55, Section 1.2.1], the induced automorphism [2] (− id) on K (T ) has 36 isolated ﬁxed points and a ﬁxed surface = { ξ ∈ K (T )| Supp ξ = {0, x , −x } , x ∈ T {0}}. [2] We can consider r : K (T ) → K (T ) the blow-up of K (T ) in .Then σ extends to an 2 2 2 [2] [2] automorphism ) σ on K (T ) and K := K (T )/) σ is a primitively symplectic orbifold, 2 2 4 4 4 [2] which is a crepent resolution in codimension 2 of K (T )/σ .From(25), we deduce that: 123 Betti numbers of 4-dimensional symplectic orbifolds 221 b (K ) = 4 + 1 + 1 = 6. (26) Moreover, we can also describe K as a quotient of K (constructed in Sect. 5.6). The orbifold [2] [2] K is given by K (T )/(− id) .Then ) σ induced an involution on K that we denote by [2] [2] σ and K = K /σ .By[30, Proposition 8.23] b (K ) = 0, it follows that: 4 4 4 b (K ) = 0. (27) Now, we are going to determine the singularities of K , which are all isolated cyclic quotient singularities of order 2 or 4. Let us denote the number of such singular points by a and a respectively. It turns out to be quite technical to determine a directly. However, order-4 4 2 [2] cyclic quotient singular points correspond to the singularities of K which are ﬁxed by σ , [2] hence they correspond to the ﬁxed points of σ outside of . After determining a , we will deduce a with a numerical method. We have: { | } Fix σ = (a, a) a ∈ E [2] . 1 1 i i 1+i 1+i [2] That is, we have 4 isolated ﬁxed points (0, 0), ( , ), ( , ) and ( , ). Hence σ 2 2 2 2 2 2 4 ﬁxes only one point outside of the diagonal of K (T ) which is the scheme supported on [2] 1 1 i i 1+i 1+i ( , ), ( , ), ( , ) . This point is also outside of . Furthermore, σ acts on 2 2 2 2 2 2 4 Z := { ξ ∈ K (T )| Supp ξ = {(0, 0)}} . 0 2 [2] 2 [2] As explained in [55, Section 1.2.1], the ﬁxed locus of (σ ) = (− id) on Z is given by the vertex x and a line which is contained in ;the vertex x is out of .Necessarily,the [2] [2] vertex x will also be ﬁxed by σ . We conclude that σ only ﬁxes 2 points outside of 4 4 and so: a = 2. (28) Now, we are going to deduce the number a by considering the double cover: [2] π : K → K /σ = K . Since π has only isolated ramiﬁcation points, the Hurwitz formula can be used in this frame- work and provides: χ(K ) = 2χ(K ) − R, (29) where R is the number of ramiﬁcation points. An order-2 cyclic quotient singular point in K can arise in two different ways. First, it can be the image by π of a non-ramiﬁed singular point of K , or second, it can be the image by π of a ramiﬁed smooth point of K . Since the π-ramiﬁed singular points of K provide cyclic quotient singularities of order 4 in K ,we obtain the following formula: #Sing(K ) − a a = + R − a . 2 4 We have seen that K has 36 isolated singularities, hence by (28): a = 15 + R. (30) By [30, Proposition 8.23], χ(K ) = 108. Hence (29), (27) and Proposition 3.6 provide: 108 = 2(48 + 12b (K ) + s) − R. Applying (30), (26)and (16), we obtain: 108 = 2(48 + 12 × 6 − 3 × 2 − a ) − a + 15. 2 2 123 222 L. Fu, G. Menet One deduces that a = 45. 5.5 b = 7 The generalized Kummer fourfold [2]. For the sake of completeness, we also provide a singular example with b = 7. We consider the same complex torus as previously T = C /,where = (1, 0), (i , 0), (0, 1), (0, i ). The torus is given by the product of elliptic curves T = E × E, with E := C/ 1, i .Let ν be the symplectic automorphism of order 3 on T deﬁned by the action on C given by the matrix: 0 −1 ν = . 1 −1 We remark that: 2 ν H (T , Z) = U ⊕ A . (31) Since ν is a linear automorphism on T , it extends to an automorphism on K (T ). Furthermore, ν veriﬁes the following relation: id +ν + ν = 0. (32) [2] [2] We denote ν the automorphism induced by ν on K (T ). The relation (32) shows that ν [2] admits one ﬁxed surface which induces a surface of singularities in the quotient K (T )/ν . This surface is isomorphic to the K3 surface obtained after resolving the singularities of T /ν. Because of (32), the ﬁxed points of ν are in T [3], there are 9 points of the form (a, 2a),where 9×8×1 [2] a ∈ E [3].Let x , ..., x be these 9 ﬁxed points. It induces = 12 ﬁxed points of ν 1 9 of the form x , x , −x − x , with x = x .Let i j i j i j Z := { ξ ∈ K (T )| Supp ξ = {τ }} , τ 2 for all τ = (a, 2a), with a ∈ E [3]. As explained in [24, Section 4]: Z P(1, 1, 3). [2] Hence, the action of ν on Z ﬁxes 2 lines which intersect in the vertex of Z .These two τ τ lines are necessarily included in the surface . Hence there is no additional isolated ﬁxed [2] point in Z ;thatis ν has 12 isolated ﬁxed points. [2] [2] Let be the image of in K (T )/ν .Let x ∈ . The singularity (K (T )/ν , x ) 2 2 2i π −1 2 2 is analytically equivalent to (C × (C /g ), 0), with g = diag(ξ ,ξ ) and ξ = e . 3 3 3 3 The singularity (C /g , 0) is of type A and can be resolved crepently by a blow-up and 3 2 this resolution has two exceptional lines which intersect in one point. This shows that the singular surface can be resolved crepently and the exceptional locus is the union of two [2] irreducible divisors. Let K → K (T )/ν be such a resolution. The orbifold K is primi- 3 3 tively symplectic with 12 isolated cyclic quotient singularities of order 3. Moreover by (31), we have: b (K ) = 4 + 1 + 2 = 7. Furthermore, because of the action of ν on and [30, Corollary 6.3], the third Betti [2] number of K (T )/ν is trivial. Since is a simply connected surface, b (K ) = 0. 2 3 123 Betti numbers of 4-dimensional symplectic orbifolds 223 5.6 b = 8 This example has already been introduced in Sect. 5.4 as an intermediate consequence of computation. In full generality, it can be constructed as follows. We consider a symplectic involution ι on X a manifold of Kum -type. As it is explained in [30, Theorem 7.5], X /ι has a surface of singularities and 36 isolated ﬁxed points. By resolving the surface of singularities, we obtain a primitively symplectic orbifold that we denote K and which has b = 8. Moreover, its Beauville–Bogomolov form has been computed in [30, Theorem 1.1] and its Betti numbers in [30, Proposition 8.23]. It has been proved that K is irreducible symplectic in [38, Proposition 3.8]. 5.7 b = 9 We describe Fujiki’s example with second Betti number 9. Let T be a complex torus which admits a symplectic binary dihedral linear automorphism group D of order 12. For instance, i π we consider T = E × E ,where E = C/ 1,ξ and ξ = e .Then D is generated ξ ξ ξ 6 6 3 6 6 6 by the linear automorphisms: ξ 0 01 and . −1 0 ξ −10 Let N be the center of D which is generated by − id. We consider the K3 surface S obtained as a resolution of T / N. The group H = D / N is isomorphic to the dihedral group of order 6, denoted by D . There is a natural lifting of a symplectic action of H on S. Then, as in the abelian case, we form the automorphisms group G on S × S generated by H acting diagonally and the involution (s, t ) → (t , s) with θ = id. Fujiki resolves the singularities in codimension 2 of (S × S)/G and shows in [15, Section 13] that we obtain a primitively symplectic orbifold Y ( D ) with second Betti number 9. K 3 3 Since θ = id, the resolution in codimension 2 considered by Fujiki in [15,Section7] [2] [2] corresponds to S / H → (S × S)/G,where S is the Hilbert scheme of 2 points on S and [2] H the induced automorphisms group. That is Y ( D ) = S / H. K 3 3 To determine the singularities of Y ( D ), we start by computing the singularities of K 3 3 S/ H. As there is no ﬁxed point of S by the entire group H, by classiﬁcation of the ﬁnite subgroups of SL(2, C), we know that the singularities of S/ H can only be of type A or A . 1 2 We denote by N (resp. N ) the number of singularities of S/ H of type A (resp. A ). Then 2 1 2 1 the integers N and N can be computed by Riemann–Roch (a direct computation is also 2 1 possible, but is more technical). We apply Theorem 2.12 to S/ H and the V-bundles O , S/ H [1] . We obtain respectively: S/ H c 2 N N 2 2 1 2 = + + , (33) 12 9 8 and 5c 2 N N 2 2 1 − 6 =− − − . (34) 6 9 4 Finally, Theorem 2.13 provides: 2 N N 2 1 10 = c + + . (35) 3 2 123 224 L. Fu, G. Menet Combining (33), (34)and (35), we obtain N = 0and N = 7. Then, we can deduce 1 2 7×6 [2] the singularities of S / H.There are = 21 singular points of the form (a, b) with a = b ∈ Sing S/ H and 2 × 7 = 14 singular points on the diagonal. So Y ( D ) has 35 K 3 3 isolated cyclic quotient singularities of order 3. 5.8 b = 10 We have examples of Fujiki, for instance for H = Z /4 Z and S a K3 surface. 5.9 b = 11 [2] Let X be a manifold of K 3 -type endowed with a symplectic automorphism σ of order 3. As explained in [41, Section 7.3], there are two possibilities: σ has 27 isolated ﬁxed points or σ 2 σ { } has an abelian ﬁxed surface. When Fix σ = 27 points ,wealwayshaverk H ( X , Z) = 11. We denote M := X /σ which is a primitively symplectic orbifold with second Betti number 11 and 27 isolated cyclic quotient singularities of order 3. Moreover its Beauville–Bogomolov form has been computed in [40, Theorem 1.3]. 5.10 b = 14 We have examples of Fujiki, for instance for H = (Z /2 Z) and S a K3 surface. 5.11 b = 16 [2] We consider a symplectic involution ι on X amanifoldof K 3 -type. As it is explained in [42], X /ι has a surface of singularities and 28 isolated ﬁxed points. If we resolve the surface of singularities we obtain a primitively symplectic orbifold that we denote M and which has b = 16. Moreover, its Beauville–Bogomolov form has been computed in [37, Theorem 2.5] and its fourth Betti number in [37, Proposition 2.40]. It has been proved that M is symplectic irreducible in [38, Proposition 3.8]. 5.12 b = 23 The Hilbert square of K3 surface [2]. 123 Betti numbers of 4-dimensional symplectic orbifolds 225 5.13 Summary b X b b Singularities B–B form Irreducible 2 3 4 ⎛ ⎞ ⎛ ⎞ 2 −13 21 0 1 2 ⎝ ⎠ ⎝ ⎠ 3 M ; M 0 26 a = 5 −18 −1 ; 160 No 11 11 3 −16 002 4 ? ? ? ? ? ? 4 −3 5 M 0 42 a = 9 U ⊕ ⊕ (−14) No 7 7 −34 6 K 0 55 a = 45, a = 2 ? No 2 4 7 K ( A) 8 108 Smooth U ⊕ (−6) Yes 7 K 0 92 a = 12 ? ? −5 −4 8 K 0 90 a = 36 U (3) ⊕ Yes −4 −5 9 Y ( D ) 0 66 a = 35 ? No K 3 3 3 10 Y (Z /4 Z) 0 118 a = 10, a = 6 ? ? K 3 2 4 2 2 11 M 0 102 a = 27 U (3) ⊕ U ⊕ A ⊕ (−6) No 3 3 12-13 ? ? ? ? ? ? ! " 14 Y (Z /2 Z) 0 150 a = 36 ? ? K 3 2 15 ? ? ? ? ? ? 3 2 16 M 0 178 a = 28 U (2) ⊕ E (−1) ⊕ (−2) Yes 2 8 17-22 ? ? ? ? ? ? [2] 3 2 23 K 3 0 276 Smooth U ⊕ E (−1) ⊕ (−2) Yes Here,wedenoteby a the number of isolated cyclic quotient singularities of order k. The singularities of the Fujiki examples are described in [15, Section 13, table 1]. We can then compute their fourth Betti numbers using Proposition 3.6 and Example 3.8.Samething for all the orbifolds with known singularities and b . Acknowledgements We would like to thank Roland Bacher, Arnaud Beauville, Daniel Huybrechts and Radu Laza for useful discussions. We also want to thank the referee for his or her careful reading and helpful comments. We are very grateful to the Second Japanese-European Symposium on Symplectic Varieties and Moduli Spaces where our collaboration was initiated. Funding L. Fu is supported by the Radboud Excellence Initiative programme. G. Menet was ﬁnanced by the Fapesp grant 2014/05733-9, the Marco Brunella grant of Burgundy University and the ERC-ALKAGE, grant No. 670846. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A: Salamon’s relation for orbifolds In this Appendix, we extend some of Salamon’s results [48] to the setting of complex orbifolds. Let us ﬁrst recall some notation. Given a complex orbifold X,wedenoteby [ p] p p ∨∨ := ( ) ι ( ) the sheaf of reﬂexive p-forms, where ι : X → X is the ∗ reg X X X reg 123 226 L. Fu, G. Menet inclusion of the smooth part. The main result of “Appendix” is the following generalization of [48, Corollary 3.4]. Theorem A.1 Let X be a compact complex orbifold of dimension n. Then p 2 orb c c = (−1) 6 p − n(3n + 1) χ , (36) 1 n−1 p=0 [ p] orb where χ := ch · td( X ). X X [ p] [n− p] Proof Using the fact that = ⊗ ω , we easily see from the deﬁnition that X X orb n orb χ = (−1) χ . (37) p n− p We deﬁne the orbifold χ -genus of X to be the following polynomial in t: n n orb p n orb p χ (t ) := χ t = (−1) χ t . (38) orb p n− p p=0 p=0 Let c(T ) = (1 + x ) be the formal factorization of the Chern class of X,where X i i =1 x ,..., x are the Chern roots. Following [48], we deﬁne a new polynomial in t with coef- 1 n ﬁcients characteristic classes of X: K (t ) := x + t · . −x 1 − e i =1 ∗ k Denote by K ∈ H ( X , Q) the coefﬁcient of t . Clearly, K = c .Moregenerally,asshown k 0 n in [48, Proposition 3.2], K has the nice property that K − c is in the ideal generated by k k n−k c ,..., c for any k ≥ 1. For example, n−k+1 n K = c + nc ; (39) 1 n−1 n 1 1 K = c + (n − 1)c + (2c c + n(3n − 5)c ). (40) 2 n−2 n−1 1 n−1 n 2 24 It is easy to relate K (t ) and χ (t ): orb −x K (t ) = td( X ) · (t + 1 − e ) X X i =1 p n− p −x −···−x j j n− p = td( X ) · (t + 1) (−1) e p=0 1≤ j <···< j ≤n 1 n− p [n− p] p n− p = td( X ) · (t + 1) (−1) ch p=0 n p orb = (−1) (−1 − t ) χ n− p p=0 = χ (−1 − t ). orb 123 Betti numbers of 4-dimensional symplectic orbifolds 227 Therefore for any k ≥ 0, 1 (−1) (k) (k) K = K (0) = χ (−1). (41) orb k! k! X X Taking k = 0in (41), we obtain p orb c = (−1) χ . (42) p=0 (This is essentially Blache’s Gauss–Bonnet Theorem 2.13.) Taking k = 1in (41), combined with (39), we ﬁnd p orb c = (−1) pχ . (43) p=0 (This is equivalent to (42) by taking into account the symmetry (37).) Taking k = 2in (41), combined with (40), it yields that 1 n(3n − 5) (−1) orb c c + c = p( p − 1)χ . (44) 1 n−1 n 12 24 2 X X p=0 We deduce (36) by combining (42), (43), and (44). [ p] p,q q Remark A.2 We are mainly interested in the Hodge numbers h := dim H ( X, ) and their alternating sum [ p] q p,q χ := χ( X, ) := (−1) h , q=0 orb rather than χ . Therefore, in practice, Theorem A.1 is often combined with orbifold Riemann–Roch formula relating these two quantities: when X has only isolated singularities, orb Blache’s Riemann–Roch theorem 2.12 implies that for any integer p ≥ 0, χ = χ − S p p with tr(ρ [ p] (g)) x , S := (45) |G | det(id −ρ (g)) x x ,T x ∈Sing X g =id being the correction term determined by the singularities of X, where for any singular point x ∈ X, G is the local fundamental group and for any locally V-free bundle F, ρ is the x x , F associated representation of G ; see Sect. 2. In the general case where the singularities are not necessarily isolated, we can use Kawasaki’s Riemann–Roch formula [32], by replacing in the above deﬁnition of S by + , [ p] 1 tr S := ch · td(T ), (46) p X • ∨ m tr ∧ N μ X μ μ where μ runs over all connected components of the inertia orbifold IX except the component X. We refer to [32,52] for the explanation of the notation as well as more details. Remark A.3 Givenaninteger p, from (37) and the fact χ = (−1) χ ,wesee that p n− p S = (−1) S . p n− p 123 228 L. Fu, G. Menet In fact, if X has only isolated singularities, it is a simple exercise in linear algebra to see more precisely that for any x ∈ Sing X, S = (−1) S , p,x n− p,x where tr(ρ [ p] (g)) 1 x , S := . p,x |G | det(id −ρ (g)) x x ,T g =id When specializing Theorem A.1 to the symplectic case, we get the following orbifold analogue of [48, Theorem 4.1]. Theorem A.2 Let X be a compact Kähler orbifold of even complex dimension n = 2msuch p,q n− p,q that the Hodge numbers satisfy the “mirror symmetry” h = h for any p, q ≥ 0. Then 2n n 1 1 k 2 p 2 c c = (−1) 3k − n(6n + 1) b − (−1) 6 p − n(3n + 1) S , 1 n−1 k p 2 2 p=0 k=0 (47) where S is deﬁned in (45) when X has only isolated singularities and in (46) in general. The most important case we have in mind is when X is a primitively symplectic orbifold (Deﬁnition 3.1), hence the left-hand side of (47) vanishes and we get a linear relation among the Betti numbers and contributions of singularities. orb Proof We keep the same notation as above. Since χ = χ − S , the equations (36)and p p (42) imply that n n 1 1 p+q 2 p,q p 2 p 2 orb (−1) p h − (−1) p S = (−1) p χ = c c + n(3n + 1) c . p 1 n−1 n 6 12 X X p,q=0 p=0 (48) p,q q, p Using the Hodge symmetry h = h ,wehave 1 1 p+q 2 p,q p 2 (−1) q h − (−1) p S = c c + n(3n + 1) c . (49) p 1 n−1 n 6 12 X X p,q=0 p p,q p,n−q The mirror symmetry relation h = h implies that (remember that n is even) n n p+q p,q p+q p,q (−1) pq h = (−1) p(n − q)h . p,q=0 p,q=0 Hence n n n p+q p,q p+q p,q p 2 (−1) pq h = (−1) nph = n (−1) pχ . p,q=0 p,q=0 p=0 orb Writing χ = χ + S , together with (43), we obtain p p n n n n p+q p,q p orb p p 2 (−1) pq h = n (−1) pχ + n (−1) pS = c + n (−1) pS . p n p p,q=0 p=0 p=0 p=0 (50) 123 Betti numbers of 4-dimensional symplectic orbifolds 229 Combining (48), (49), and (50), we ﬁnd n n 1 1 p+q 2 p,q p 2 (−1) ( p + q) h − (−1) p(2 p + n)S = c c + n + n c , p 1 n−1 n 3 6 X X p,q=0 p=0 which is nothing else but 2n n 1 1 k 2 p 2 (−1) k b − (−1) p(2 p + n)S = c c + n + n c , (51) k p 1 n−1 n 3 6 X X k=0 p=0 On the other hand, (42)saysthat 2n n k p c = (−1) b − (−1) S . (52) n k p k=0 p=0 Putting (51)and (52) together, we obtain the following formula, 2n n 1 1 k 2 p 2 2 c c = (−1) 3k − n(6n + 1) b − (−1) 6 p + 3np − 3n − n S . 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Abstract

We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon’s relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds. Keywords Hyper-Kähler varieties · Orbifolds · Betti numbers · singularities · Riemann–Roch theorem Mathematics Subject Classiﬁcation 57R18 · 14J42 · 14J28 · 53C26 · 19L10 · 14B05 1 Introduction A compact Kähler manifold is called holomorphic symplectic if it admits a holomorphic 2-form that is nowhere degenerate. In particular, it is even-dimensional and has trivial canon- ical bundle. Such a manifold is called irreducible if moreover it is simply connected and the holomorphic symplectic form is unique up to scalar. Irreducible holomorphic symplectic (IHS) manifolds (also known as compact hyper-Kähler manifolds) admit a Ricci-ﬂat Rie- mannian metric [59], and are characterized by the condition that the holonomy group is the compact symplectic group. The importance of IHS manifolds is manifested in the Beauville– Bogomolov decomposition theorem [2,8]: any compact Kähler manifold with vanishing ﬁrst Chern class has a ﬁnite étale cover which can be written as a product of a complex torus, Lie Fu fu@math.univ-lyon1.fr Grégoire Menet gregoire.menet@univ-grenoble-alpes.fr Institut Camille Jordan, Université Claude Bernard Lyon 1, Lyon, France IMAPP, Radboud Universiteit, Nijmegen, The Netherlands Institut Fourier, 100 rue des Mathématiques, Gières, France 123 204 L. Fu, G. Menet Calabi–Yau varieties and IHS manifolds. We refer to [2,27]and [20, Part III] for the basic theory on such manifolds. Irreducible holomorphic symplectic surfaces are nothing but K3 surfaces. The construction problem for IHS manifolds in higher dimensions seems quite hard: up to deformation, in each even dimension (≥ 4), we so far only have two examples constructed by Beauville [2] (Hilbert schemes of points on K3 surfaces and generalized Kummer varieties) together with two sporadic examples constructed by O’Grady [43,44] in dimensions 6 and 10. The limitedness of available examples suggests the possibility to bound or even classify IHS manifolds (see [28] for diffeomorphic types). As the second cohomology of an IHS manifold, together with the Beauville–Bogomolov quadratic form [2] and the weight-2 Hodge structure, controls most of its geometry [4,5,29,36,38,57,58], it is natural to ask the following question. Question 1.1 In a given dimension, what values can the second Betti number of an irreducible holomorphic symplectic manifold take? In dimension 4, Guan [21] proved the following result in the direction of Question 1.1. Theorem 1.2 (Guan) The second Betti number of a 4-dimensional irreducible holomorphic symplectic manifold is no more than 8, or equal to 23. Moreover, if b = 23, the Hodge diamond must be the same as that of the Hilbert square of a K3 surface. The fact that b ≤ 23 was attributed to Beauville. See [21,22] for extra constraints on each cases; see [35,51] for related results in dimension 6 and the more recent work [33]for a conjectural bound in arbitrary dimension based on [19]. When b = 23, let us mention the work [31,45], which aims at determining the deformation type of IHS fourfolds upon ﬁxing some extra topological data. In the point of view of birational geometry, or more precisely the minimal model program (cf.[34]), it is important to treat varieties with mild singularities. With recent intensive efforts [12,13,16,18,26], the Beauville–Bogomolov decomposition theorem is now extended to projective varieties with klt singularities and numerically trivial canonical class. Naturally, boundedness results for possibly singular irreducible holomorphic symplectic varieties [18, Deﬁnition 8.16] are desired. In particular, Question 1.1 can be posed in this broader setting. This article is our ﬁrst experimental attempt towards the boundedness problem, where we will focus on the classical approach of enlarging the category of IHS manifolds to the so-called primitively symplectic orbifolds, pioneered by Fujiki [15]. Roughly speaking, a primitively symplectic orbifold is a compact Kähler space with quotient singularities in codimension ≥ 4, such that the smooth locus carries a holomorphic symplectic form which is unique up to scalar. See Deﬁnition 3.1. Our ﬁrst main result extends Guan’s Theorem 1.2. Theorem 1.3 Let X be a primitively symplectic orbifold of dimension 4. Then b ( X ) ≤ 23. Moreover, the equality occurs only in the smooth case. Our second main result bounds the size of the singularities. Theorem 1.4 Let X be a primitively symplectic orbifold of dimension 4. Then (i) X has at most 91 singular points. (ii) For each singular point of X , the order of the local fundamental group is at most 1424. 123 Betti numbers of 4-dimensional symplectic orbifolds 205 The proof of Theorems 1.3 and 1.4 is given in the end of Sect. 3. The bound for b in Theorem 1.3 being the same as in the smooth case (note however that no numbers between 9 and 22 are excluded as in [21], despite of our effort in Sect. 4 where we generalize the Hitchin–Sawon formula), the construction methods in the orbifold setting are much richer. Indeed, staying in the smooth category of IHS fourfolds, the only available values for b are 23 and 7, achieved by Hilbert squares of K3 surfaces and generalized Kummer fourfolds respectively; while we are able to construct much more examples within the enlarged category of symplectic orbifolds, ﬁlling many “gaps” in the possibilities of the Betti number. More precisely, we have the following result. Theorem 1.5 There are 4-dimensional primitively symplectic orbifolds with second Betti number 3, 5, 6, 7, 8, 9, 10, 11, 14, 16 and 23. We refer to Sect. 5 for details of these examples. 2 Riemann–Roch theorem for orbifolds 2.1 Orbifolds and V-bundles We ﬁrst ﬁx the notion of orbifolds (“V-manifolds” in [49,50]). Deﬁnition 2.1 (Orbifolds) An n-dimensional complex orbifold is a connected paracompact Hausdorff complex analytic space X such that for any point x ∈ X, there exists an open neighborhood U of x and a triple (V , G,π) with V an open subset of C , G a ﬁnite subgroup of the biholomorphic automorphism group of V,and π : V → U the composition of the quotient map V → V /G and an isomorphism V /G U. Remark 2.2 Note that an orbifold is always normal (see for instance [11, Théorème 4]). In particular, the singular locus is of codimension at least 2. Deﬁnition-Proposition 2.3 [47, Proposition 6] Let X be an n-dimensional complex orbifold. Let x ∈ Sing X . Then there exist a ﬁnite subgroup G of GL (C) and an open neighbourhood x n n n V ⊂ C of the origin 0 ∈ C , stable under the action of G , with V /G isomorphic to an x x x x open neighbourhood U of x , and such that Codim Fix(g) ≥ 2 for all g ∈ G \{id}. Such a group G is unique up to conjugation. Let π : V → V /G U ,wecall x x x x x x (U , V , G ,π ) a local uniformizing system of x , and G the local fundamental group of x x x x x Xat x. Remark 2.4 In modern literature (see for example [1]), orbifold is a synonym for (analytic) Deligne–Mumford stack. In particular, the collection of charts (V , G,π) is part of the data of an orbifold and the group G is sometimes allowed to act non-effectively on V.Ifweuse the terminology orbifold in this generalized sense, then what is deﬁned in Deﬁnition 2.1 cor- responds to effective orbifolds [1, Deﬁnition 1.2] with all nontrivial elements of all stabilizer groups acting with a non-empty ﬁxed locus of codimension at least 2. By the above result of Prill, these are equivalent notions, as such an orbifold/stacky structure is determined by the underlying complex analytic space. The notion of vector bundles naturally generalizes to orbifolds. 123 206 L. Fu, G. Menet Deﬁnition 2.5 (V-bundles) Let X be an orbifold. • AV-bundle (or orbibundle)on X is a vector bundle F on X := X Sing X such that reg for any local uniformizing system (U , V , G,π), there exists a vector bundle F on V endowed with an equivariant action of G such that: F π ( F ), V |V Fix G |U reg where Fix G := Fix(g). g∈G,g =id • Let F be a coherent sheaf on X. The sheaf F is said to be locally V-free if for any x ∈ X, there exist a local uniformizing system (U , V , G,π), a free coherent sheaf F on V , ˆ ˆ and a G-action on F such that F π F .By[7, 4.2], the local V-freeness of V |U ∗ a coherent sheaf F is equivalent to the condition that F is reﬂexive and the reﬂexive [∗] ∗ ∨∨ pull-back π (F ) := π (F ) is locally free for any local uniformizing system. |U |U As in the smooth case, there is an equivalence of categories between the category of locally V-free sheaves and that of V-bundles. Example 2.6 (Reﬂexive differentials) Givenanorbifold X of dimension n, the sheaf of reﬂex- ive differential forms ( [17,2.D], [46, Section 2.5]) [i ] i ∨∨ i := ( ) ι ( ) X X X reg is a locally V-free sheaf for any i ∈ N,where ι : X → X is the natural inclusion of the reg smooth part. The sheaf of reﬂexive forms of top degree is identiﬁed with the dualizing sheaf: [n] ω . Remark 2.7 (Hodge decomposition) Let X be a compact Kähler orbifold. For any integer k ≥ 0, the rational singular cohomology group H ( X , Q) carries a pure Hodge structure of weight k and in the Hodge decomposition k p,q H ( X , C) = H ( X ), p+q=k [ p] p,q q p,q we have H ( X ) = H ( X, ),see [46, Section 2.5]. We denote h ( X ) := p,q p,q q, p dim H ( X ).Wehave h = h . Notation 2.8 Let X be an orbifold, x ∈ X and F a locally V-free sheaf. Let (U , V , G,π) be a local uniformizing system of x and let F be a locally free sheaf on V endowed with an action of G as in Deﬁnition 2.5. Hence, the ﬁber of F at 0 is endowed with an action of G, which provides a representation of G.Wedenoteby ρ the representation of G x ,F associated with x and F . 2.2 Characteristic classes on orbifolds We recall the deﬁnition of Chern classes of V-bundles on orbifolds, by adapting the Chern– Weil approach. Deﬁnition 2.9 [7, Deﬁnition 2.9] Let F be a V-bundle of rank r on an orbifold X.A metric on F is a Hermitian metric h on F as bundle on X such that for all local uniformizing reg systems (U , V , G,π), the Hermitian metric π (h ) extends to a Hermitian metric on F . |U V reg 123 Betti numbers of 4-dimensional symplectic orbifolds 207 Deﬁnition 2.10 [7, Deﬁnition 1.5] Let X be an orbifold. A smooth differential k-form ϕ on X is a C differential k-form on X such that for all local uniformazing system (U , V , G,π), reg ∗ ∞ π (ϕ ) extends to a C -differential k-form on V . (We always mean C-valued forms.) |U reg Notation 2.11 We denote by A the sheaf of differential k-forms. As explained in [7, Deﬁnition 2.10], we can deﬁne the Chern classes of a V-bundle as follows. Let F be a V-bundle of rank r on an orbifold X. We can ﬁrst construct the Chern forms on X as in the smooth case. To a Hermitian metric h on F, we associate the Chern reg connection D on F,and to D, we associate the curvature D .Let be the corresponding i 2k r × r matrix of curvature 2-forms, then we set c (h) = P ( ) ∈ ( X , A ),where P k k reg k 2π r ×r is the k-th elementary invariant polynomial function C → C. The same process can be also done on all local uniformizing systems (U , V , G,π). ˆ ˆ The metric π (h ) extends to a Hermitian metric h on F which gives rise to the |U V reg ˆ ˆ ˆ Chern connection D on F and hence the curvature D . As previously, we can construct 2k ∗ ˆ ˆ c (h) ∈ (V , A ). By construction π (c (h) ) extends to c (h) on V . Hence, we obtain k k |U k reg 2k c (h) ∈ ( X , A ). As in the smooth case, we show that c (h) is a closed form, and that the k k 2k cohomology class c ( F ) := [c (h)] ∈ H ( X , C), called the k-th Chern class of F, does k k not depend on the choice of the metric h. Other characteristic classes, like Todd classes and Chern characters, are deﬁned in terms of Chern classes by the usual formulas. A characteristic class of an orbifold is that of its tangent V-bundle. 2.3 Riemann–Roch and Gauss–Bonnet theorems for orbifolds One key ingredient in the proof of Theorem 1.3 is the following orbifold version of the Hirzebruch–Riemann–Roch theorem due to Blache [7, Theorems 3.5 and 3.17]. Theorem 2.12 (Blache [7]) Let X be a compact complex orbifold with only isolated singu- larities and let F be a locally V-free sheaf. Then we have ⎡ ⎤ ⎢ ⎥ 1 tr(ρ (g)) x ,F ⎢ ⎥ χ( X , F ) = ch(F ) · td( X ) + , ⎣ ⎦ |G | det(id −ρ (g)) X x x ,T x ∈Sing X g∈G g =id where g is viewed as an endomorphism on T V with (U , V , G ,π) a local uniformizing 0 x system of x . Blache also established the following orbifold version of Gauss–Bonnet theorem. Theorem 2.13 [7, Theorem 2.14] Let X be an n-dimensional compact complex orbifold with only isolated singularities. Then we have the following formula for its topological Euler characteristic: χ ( X ) = c ( X ) + 1 − , top n |G | X x x ∈Sing X where G is the local fundamental group of X at x . 123 208 L. Fu, G. Menet 3 Bounding Betti numbers and singularities The aim of this section is to show Theorems 1.3 and 1.4. Let us ﬁrst make precise the class of (possibly singular) symplectic varieties that we consider. 3.1 Symplectic orbifolds Deﬁnition 3.1 (Fujiki [15]) A compact Kähler orbifold X is called primitively symplectic if (i) the smooth locus X := X Sing X is endowed with a non-degenerated holomorphic reg 2-form which is unique up to scalar; and (ii) the singular locus Sing X has codimension at least 4. If moreover X is simply connected, X is called an irreducible symplectic orbifold. reg Remark 3.2 As in the smooth case, a primitively symplectic orbifold X has even (complex) dimension and trivial dualizing sheaf ω O . Moreover, the symplectic form extends to X X a symplectic form on any local uniformizing system. In particular, the contraction with the [1] symplectic form induces an isomorphism T . By deﬁnition, if X has dimension 4, then X has isolated quotient singularities. Remark 3.3 As quotient singularities are rational singularities, the singularities appearing in Deﬁnition 3.1 are symplectic singularities in the sense of Beauville [3]. Moreover, an irreducible symplectic orbifold deﬁned above is an irreducible symplectic variety in the sense of [18]and [26, Deﬁnition 1.4]. Remark 3.4 (Hodge diamond) Let X be a 4-dimensional primitively symplectic orbifold. Fujiki [15, Proposition 6.7] showed that X has vanishing irregularity, hence b ( X ) = 0. Serre–Grothendieck duality gives that [1] [1] 3 1 ∨ 1 ∨ ∼ ∼ H ( X, ) = H ( X,ω ⊗ T ) = H ( X, ) . X X X X 3,1 1,1 3,0 1,0 In particular, h = h . Similarly, h = h = 0. In conclusion, the Hodge diamond of X takes the following form. 1,1 1 h 1 2,1 2,1 0 h h 0 1,1 2,2 1,1 1 h h h 1. 2,1 2,1 0 h h 0 1,1 1 h 1 3.2 Quotient symplectic singularities in dimension 4 For later use, we classify in this section all symplectic quotient singularities in dimension 4. As the germ of a quotient singularity is determined by the local fundamental group, one needs to classify all ﬁnite subgroups of the Lie group Sp(4, C). Since any ﬁnite subgroup must be contained in some compact maximal subgroup, we are to classify ﬁnite subgroups of the compact symplectic group Sp(2) := Sp(4, C) ∩ SU(4). 123 Betti numbers of 4-dimensional symplectic orbifolds 209 2i π Proposition 3.5 Let n > 0 be an integer, we denote ξ := e the primitive n-th root of unity. For integers 1 ≤ k ≤ n, we denote ⎛ ⎞ 00 1 0 ⎜ ⎟ 00 0 1 ⎜ ⎟ T := . n,k k ⎝ ⎠ ξ 00 0 −k 0 ξ 00 Let G be a ﬁnite subgroup of the compact symplectic group Sp(2). Then, up to conjugation, (i) there exists ﬁnite subgroups H ,H of SU(2),integersn > 0 and k ∈{1, ..., n}, and 1 2 a normal subgroup G of G of index at most 2, such that any element M of G has the form A 0 M = , 0 B with A ∈ H ,B ∈ H , and G/G = T if G = G. 1 2 n,k (ii) If moreover C /G has only the image of 0 as singularity, then there exists a ﬁnite subgroup Hof SU(2) and θ an automorphism of H such that any element M ∈ G has the form A 0 M = , 0 θ( A) for some A ∈ H. Proof Hanany and He classiﬁed in [23] the ﬁnite subgroups of SU(4). Hence it is enough to identify those groups in the list that preserve a symplectic form. In the sequel, we follow their notation. The ﬁrst category of groups are the so-called primitive simple groups described in [23, Section 3.1.1] and they are numbered from I to VI. However, none of them are symplectic. Indeed, the following matrices are considered: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 10 0 0 100 2 3 000 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 01 0 0 1 0 −1 20 1 0 −12 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ F := , F := √ and F := , 1 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 00 w 0 02 −12 0 21 0 3 02 2 −1 00 0 w 20 0 −1 2i π where w := e . The matrices F and F do not preserve any common symplectic form, 1 2 hence the groups I and III, which are partially generated by these two matrices, cannot be symplectic. Similarly, the group II cannot be symplectic because it is partially generated by the two matrices F and F which do not ﬁx the same symplectic form. 2i π 4 2 Let β := e . The matrices S := diag(1,β,β ,β ) and D := diag(w,w,w, 1) are not symplectic, hence the groups IV, V and VI, which are partially generated by one of these two matrices, are not symplectic. In [23, Section 3.1.2], Hanany and He consider the groups VII, VIII and IX which cannot be symplectic since they are partially generated by the groups I, II and III. In [23, Section 3.1.3], they consider group obtained from Kronecker products of matrices of SU(2).Let 1 1 −1 + i −1 + i 1 + i 0 S := and U := √ . SU(2) SU(2) 1 + i −1 − i 01 − i 123 210 L. Fu, G. Menet The following couples of matrices (S ⊗ S , U ⊗ U ) and (S ⊗ SU(2) SU(2) SU(2) SU(2) SU(2) 2 2 S , U ⊗ U ) both do not ﬁx the same symplectic form. Hence the groups from X SU(2) SU(2) SU(2) to XVI cannot be symplectic since they are all partially generated by one of theses couples of matrices. Also the groups from XVII to XXI cannot be symplectic because they are partially generated by the groups XI, X, XVI and XIV. The matrices A := diag(1, 1, −1, −1) and A := diag(1, −1, −1, 1) do not ﬁx the same 1 2 symplectic form. Hence all the groups from XXII to XXX are not symplectic since they are all partially generated by these two matrices. Finally, we consider the group: 2i π j k l − j −k−l := diag(w ,w ,w ,w ) w = e , j , k, l ∈{1, ..., n} , which is not symplectic, hence all the groups from XXXI to XXXIII which are partially generated by are not symplectic. It only remains to consider the group XXXIV and the intransitive groups. We will study the group XXXIV in the end. The intransitive groups are the groups coming from diagonal embedding of subgroups of SU(2) or SU(3) (see [23, Deﬁnition 2.1] for the precise deﬁni- tion). We will see that all the symplectic groups constructed from a diagonal embedding of a subgroup of SU(3) are actually constructed from a diagonal embedding of subgroups of SU(2).Let G be such a group and M an element in G.Let (e , e , e , e ) be the canonical 1 2 3 4 basis of C .Wehave: ξ 0 M = , 0 A where ξ is a root of the unity and A ∈ U (3). We can ﬁnd a basis (v ,v ,v ) of C in which 1 2 3 −1 −1 A is diagonalized: A = diag(ξ ,ζ,ζ ). In the basis (e ,v ,v ,v ), the symplectic form 1 1 2 3 has to be ⎛ ⎞ 01 00 ⎜ ⎟ −10 0 0 ⎜ ⎟ J := . ⎝ ⎠ 00 −10 We consider now another matrix of G expressed in the basis (e ,v ,v ,v ): 1 1 2 3 ⎛ ⎞ a 00 0 ⎜ ⎟ 0 bc d ⎜ ⎟ N := , ⎝ ⎠ 0 ef g 0 hj k where a, b, c, d, e, f , g, h, j , k are in C.Since N is symplectic. It follows: ab = 1 ac = 0 ad = 0 −hf + ej = 0 −hg + ek = 0 − jg + fk = 1. ej ek Hence c = d = 0. If h = 0, then f = and g = . This is impossible because, it h h contradicts − jg + fk = 1. So h = 0. For the same reason e = 0and G is actually a group 123 Betti numbers of 4-dimensional symplectic orbifolds 211 composed by matrices of the forms: ⎛ ⎞ a 00 0 ⎜ ⎟ 0 b 00 ⎜ ⎟ N := . ⎝ ⎠ 00 fg 00 jk It only remains to study the case of the group XXXIV. In this case, G = G , T , with G 0 n,k 0 composed of matrices of type A 0 , with A, B ∈ SU(2). 0 B We consider A 0 G := M = , ∈ G | A, B ∈ SU(2) . 0 B The group G is a normal subgroup of G and the class T ∈ G/G has order 2. n,k Now, we prove (ii ).Let G be a ﬁnite subgroup of Sp(2) such that C /G admits only 0 as singularities. Then necessarily, the unique element of G with the eigenvalue 1 is the identity. In particular, this is true for G . Therefore, the following projections are isomorphisms: A 0 A 0 P : G −→ H , → A, and P : G −→ H , → B. 1 1 2 2 0 B 0 B −1 So, setting θ := P ◦ P ﬁnishes the proof. 3.3 Orbifold Salamon relation Salamon [48, (0.1)] discovered a remarkable linear relation among the Betti/Hodge numbers of a compact hyper-Kähler manifold. By applying Blache’s orbifold Hirzebruch–Riemann– Roch Theorem 2.12, we will establish (Proposition 3.6 below) a Salamon-type relation between the Hodge numbers and information from the singularities of a 4-dimensional prim- itively symplectic orbifold. A more general result can be obtained by adapting Salamon’s method. However, we decide to give an elementary proof here and leave the general result in Appendix A. In what follows, X is a primitively symplectic orbifold of dimension 4. For any (necessarily isolated) singular point x ∈ X, G is the local fundamental group of X at x and ρ is the x x ,• representation of G deﬁned in Notation 2.8. • Deﬁne for any integer p ≥ 0, tr(ρ [ p] (g)) 1 x , S := . (1) |G | det(id −ρ (g)) x x ,T x ∈Sing X g =id • Applying Theorem 2.12 to F = O , we obtain: td ( X ) = 3 − S , (2) 4 0 [1] • Applying Theorem 2.12 to F = , we obtain, using Remark 3.4: 2,1 1,1 h ( X ) − 2h ( X ) = 4 td ( X ) − c ( X ) + S , (3) 4 4 1 X X 123 212 L. Fu, G. Menet [2] • Applying Theorem 2.12 to F = , we obtain, using Remark 3.4: 2,1 2,2 2 − 2h ( X ) + h ( X ) = 6 td ( X ) + c ( X ) + S , (4) 4 4 2 X X • Applying Theorem 2.13, we obtain: 1,1 2,1 2,2 8 + 4h − 4h + h = c ( X ) + η, (5) where η = 1 − . (6) |G | x ∈Sing X Combining (2), (3), and (5), we can eliminate c ( X ) and td ( X ) to obtain: 4 4 X X 2,1 2,2 1,1 2h + h − 8h = 64 + 6(S − 4S ) + η. (7) 1 0 Similarly, by combining (2), (4), and (5), it yields that 2,1 2,2 1,1 2h + h − 8h = 64 + 3(S − 6S ) − 2η. (8) 2 0 Then, (7)and (8) provide the following orbifold version (in dimension 4) of Salamon’s famous relation [48] for hyper-Kähler varieties. See Appendix A for a generalization. Proposition 3.6 (Orbifold Salamon relation) Let X be a primitively symplectic orbifold of dimension 4. We have: 2,1 2,2 1,1 2h + h − 8h = 64 + s, (9) or equivalently, b + b − 10b = 46 + s, 4 3 2 where s := 6(S − 4S ) + η = 3(S − 6S ) − 2η = 4S + S − 22S (10) 1 0 2 0 1 2 0 is a correction term determined by the singularities. In particular: η = S − 2S + 2S = (−1) S . 2 1 0 i i =0 1,1 2,1 Remark 3.7 Proposition 3.6 shows that the knowledge of h , h and the singularities is enough to compute the topological Euler characteristic and all the Betti numbers of a 4-dimensional primitively symplectic orbifold. 3.4 Estimate of the contribution of singularities We turn to a more careful study of the quantity s in the orbifold Salamon relation (9), which is the local contribution of singularities. Using (1), (6)and (10), we can write s = s x ∈Sing X with: ⎡ ⎛ ⎞ ⎤ tr(ρ [1] (g)) − 4 ⎢ ⎜ ⎟ ⎥ 1 x , ⎢ ⎜ ⎟ ⎥ s = 6 +|G |− 1 , (11) x x ⎣ ⎝ ⎠ ⎦ |G | det(ρ (g) − id) x x ,T g∈G g =id 123 Betti numbers of 4-dimensional symplectic orbifolds 213 with (U , V , G ,π) a local uniformizing system around x. In particular, the action of g ∈ G x x −1 −1 on T is symplectic. We can therefore write that g = diag(ξ ,ξ ,ξ ,ξ ), with ξ = V ,0 1 2 j 1 2 2i π k e , k , n ∈ N for all j ∈ {1, 2}. Hence: j j 2π k 2π k 1 2 2 cos + cos − 4 tr(ρ 1 (g)) − 4 n n x , 1 2 2π k 2π k det(g − id) 1 2 4 1 − cos 1 − cos n n 1 2 1 1 =− − . 2π k 2π k 1 2 2 1 − cos 2 1 − cos n n 1 2 So: tr(ρ (g)) − 4 x , 1 ≤− (12) det(g − id) 2 Hence for any x ∈ Sing X,wehave |G |− 1 s ≤−2 . (13) |G | In particular, s ≤−1 and the quantity s, which is an integer by (9), is at most −| Sing X |. Using Proposition 3.5, we can be more precise. The local fundamental group G is a ﬁnite subgroup of Sp(2). Hence, there exists a normal subgroup G of G of index at most 2, H a ﬁnite subgroup of SU(2) and an automorphism θ of H such that any element M ∈ G has the form A 0 M = , 0 θ( A) with A ∈ H. As we have noticed previously, if A is a matrix of SU(2) of ﬁnite order, we have det( A − id) =− tr( A) + 2. Therefore tr(ρ (g)) − 4 [1] x , 1 1 =− + , det(ρ (g) − id) 2 − tr( A) 2 − tr(θ ( A)) x ,T g∈G g∈G g =id g =id A 0 where on the right-hand side, we write a non-trivial element g of G as for 0 θ( A) A ∈ H. Reordering the sum of the second term, we obtain the following equation: tr(ρ [1] (g)) − 4 x , 1 =−2 . (14) det(ρ (g) − id) 2 − tr( A) x ,T g∈G A∈ H A =id g =id Example 3.8 We compute explicitly s in the following cases. • G = C is a cyclic group of order n. x n 2i π 2i π n n In this case, G = G and H = g , with g = diag(e , e ).By(14), we have: x n n n−1 tr(ρ [1] (g)) − 4 2 x , 1 n − 1 =− =− , (15) 2kπ det(ρ (g) − id) 6 x ,T 1 − cos( ) g∈C k=1 g =id 123 214 L. Fu, G. Menet where we used the identity n−1 1 n − 1 = . kπ sin ( ) k=1 n As a result, s (C ) =−(n − 1). (16) x n • G = D is a binary dihedral group of order 4n. x n ˜ ˜ In this case, G = G and H = D . The binary dihedral group D is generated by x n n ξ 0 01 B := and P = , with ξ a2n-root of the unity. The group D can −1 0 ξ −10 be partitioned into the disjoint union of the following two sets: 2 2n 2 2n B, B , ..., B and BP, B P, ..., B P . Hence by (14) and the fact that tr( B P) = 0for all k ∈{1, ..., 2n},wehavethat ⎛ ⎞ tr(ρ [1] (g)) − 4 tr(ρ [1] (g)) − 4 ⎜ ⎟ x , x , X X ⎜ ⎟ = − 2n. ⎝ ⎠ det(ρ (g) − id) det(ρ (g) − id) x ,T x ,T X X ˜ g∈C 2n g∈ D g =id g =id Using (15), we obtain tr(ρ (g)) − 4 2 [1] x , 4n + 12n − 1 =− . (17) det(ρ (g) − id) 6 x ,T g∈ D g =id Therefore ( D ) =−(n + 2). (18) x n 3.5 Orbifold Guan inequality In the smooth case, Guan [21, Section 2] has proved Theorem 1.3 using two ingredients: the Hirzebruch–Riemann–Roch formula and the Verbitsky theorem [56, Theorem 1.5]. The orbifold extension of the former being explained in Sect. 2.3, we give the generalization of the latter here. Proposition 3.9 [5, Proposition 5.16] Let X be a primitively symplectic orbifold of dimension 2n. Then the following map induced by the cup-product is injective for any k ≤ n: k 2 2k Sym H ( X , C) → H ( X , C). Remark 3.10 When n = 2, we can also prove the previous proposition with an elementary method using the Fujiki relation and the fact that the Beauville–Bogomolov form is non- degenerate (see [38, Section 3.4]). Corollary 3.11 Let X be a primitively symplectic orbifold of dimension 4. Then: 2,1 1,1 2 1,1 4h ≤−(h ) + 15h + 126 + 2s, where s is (the non-positive integer) deﬁned in (11). In particular: 0 ≥ s ≥−91 (19) 123 Betti numbers of 4-dimensional symplectic orbifolds 215 Proof Proposition 3.9 provides the following inequality: (b + 1)b 2 2 b ≥ , which can be rewritten as 1,1 2,2 1,1 1,1 4 + 4h + 2h ≥ (3 + h )(2 + h ). Combining this inequality with Proposition 3.6, we obtain our result. 3.6 Proof of the main results Proof of Theorem 1.3 Thanks to Corollary 3.11,wehave: 1,1 2 1,1 0 ≤−(h ) + 15h + 126 + 2s. Or equivalently, 1,1 1,1 −2s ≤ (21 − h )(h + 6). Because of (13), s ≤ 0and when X is singular s < 0. Then Theorem 1.3 follows. Proof of Theorem 1.4 Statement (i) follows from (13)and (19). Let us prove (ii). By Proposition 3.5, there exist a normal subgroup G of G of index at most 2, H a ﬁnite subgroup of SU(2) and an automorphism θ of H such that any element M ∈ G has the form A 0 M = , 0 θ( A) for some A ∈ H. We only need the well-known classiﬁcation of the ﬁnite subgroups of SU(2) to have a full description of all possible G . The ﬁnite subgroups of SU(2) are the so-called Kleinian groups corresponding to the A-D-E Dynkin diagrams: the cyclic groups C , the binary dihedral groups D and the three sporadic groups E , E and E . The biggest n n 6 7 8 sporadic group E has order 120. Let us check the maximal size of the groups of A-D types. When G = G , we already have computed in Example 3.8 that s (C ) =−(n − 1) and x x n s ( D ) =−(n + 2). x n Now, we assume that G /G has order 2. By (12), we have: ⎡ ⎛ ⎞ ⎤ tr(ρ [1] (g)) − 4 tr(ρ [1] (T g)) − 4 n,k ⎢ ⎜ ⎟ ⎥ x , x , X X ⎢ ⎜ ⎟ ⎥ s (G ) = 6 + +|G |− 1 x x x ⎣ ⎝ ⎠ ⎦ |G | det(ρ (g) − id) det(ρ (T g) − id) x x ,T x ,T n,k X X g∈G g∈G g =id ⎡ ⎛ ⎞ ⎤ tr(ρ [1] (g)) − 4 ⎢ ⎜ ⎟ ⎥ 1 x , |G | ⎢ ⎜ ⎟ ⎥ ≤ 6 − +|G |− 1 . ⎣ ⎝ ⎠ ⎦ |G | det(ρ (g) − id) 2 x x ,T g∈G g =id [2] [2] ˜ ˜ Denote by C (resp. D ) a ﬁnite subgroup of Sp(2) such that C (resp. D ) is a normal n n n n subgroup of index 2. We have then by (15)and (17): n + 1 n + 4 [2] [2] s (C ) ≤− and s D ≤− . (20) x x n n 2 2 123 216 L. Fu, G. Menet However by (19), we know that s (G) ≥−91. Hence, the biggest possible groups are the groups which have a binary dihedral group D as normal subgroup of index 2. These groups have order 8 × 178 = 1424. Remark 3.12 Using (19), we can be more precise about the maximal cardinality of each kind of groups. • If G = C is a cyclic group of order n,thenby(16), n ≤ 92. x n • If G = D is a binary dihedral group of order 4n,thenby(18), n ≤ 89. x n [2] • If G = C is a group with a cyclic group of order n as normal subgroup of index 2, x n then by (20), n ≤ 181. [2] • If G = D is a group with a binary dihedral group of order 4n as normal subgroup of x n index 2, then by (20), n ≤ 178. Remark 3.13 Using Corollary 3.11, we can get sharper constraints on singularities for each ﬁxed second Betti number. For example, if b = 22 (resp. 21, 20, etc.), then there are at most 13 (resp. 25, 36, etc.) singular points. 4 Hitchin–Sawon formula We can try to improve Theorem 1.3 using the same method as in [21, Section 3]. The method of Guan is based on an equation of Hitchin–Sawon [25]. This section is just an attempt and is not needed in the rest of the paper. First, we recall the following generalized Fujiki formula. Lemma 4.1 [38, Lemma 4.6] Let X be a primitively symplectic orbifold of dimension 2n. 4 p If β ∈ H ( X , C) is of type (2 p, 2 p) on all small deformations of X , then there exists a 2 2(n− p) constant N (β) depending on β such that for all α ∈ H ( X , C), one has β · α = n− p ! " 2n N (β) α . Proposition 4.2 Let X be a primitively symplectic orbifold of dimension 2n. Then: n−1 n ((2n)!) N (c ) = A, (24n(2n − 2)!) a /2 2n i ˆ ˆ where Ais the A-genus deﬁned by ,where a ’s are the Chern roots of the i =1 sinh a /2 tangent bundle of X . Proof We adapt Hitchin–Sawon’s proof [25]. We will note that quotient singularities do not have any effect on Hitchin–Sawon formula. We can consider N (c ) as deﬁned in Lemma 4.1. In the smooth case, the equation of Hitchin–Sawon provides an expression of N (c ) in terms of Pontryagin classes. The main tool used by Hitchin and Sawon are Rozansky–Witten invariants (see [25, Section 2]). These invariants are constructed from the curvature of the manifold and a trivalent graph with 2n vertices. The tangent sheaf T on X can be deﬁned as the unique reﬂexive sheaf such that T X X | X reg is the usual holomorphic tangent sheaf on X . It is a locally V-free sheaf. We consider g reg a Kähler metric on T . As explained in Deﬁnition 2.9, this provides a metric g on T X X reg such that for all local uniformizing system (U , V , G,π), π (g ) extends to a metric g |U V reg on T . Then the Riemannian curvature K of ( X , g) is obtained on X by the Riemannian V reg curvature of ( X , g) and on all local uniformizing systems by the Riemann curvature of reg 123 Betti numbers of 4-dimensional symplectic orbifolds 217 (V , g ). For the same reason as in the smooth case, we can associated to the curvature a class 1 3 1 []∈ H ( X , Sym ) (see [25, Section 2] or [54]). From this class [], the deﬁnition of the Rozansky–Witten invariants being purely algebraic, it can be generalized, word by word, to the case of primitively symplectic orbifold. We denote these invariants b ( X ) for a trivalent graph with 2n vertices. In the smooth case, it is well known that: % & tr K 2c − c = , (21) (2π) where K is the curvature of X. Because of our deﬁnition of Chern classes in Sect. 2,(21)is also true in the orbifold case. In the symplectic case (21)gives: % & tr K 2c = . (2π) Then, using this expression for c exactly as Hitchin and Sawon did in [25, Section 3], we can provide an expression of N (c ) ([25, equations (7) and (8)]): n n 2 n−1 n−1 c ω · ω = 16π n c · ω · ω , (22) X X 2,0 where ω generated H ( X ) and c can be express by: n! b n ( X ) = c vol( X ), (23) 2 n (2π ) 2n (ω+ω) with vol( X ) = and the trivalent graph with two vertices. Equation (22) can be 2n 2 (2n)! rewritten: 2n 2 n−1 c (ω + ω) 16π n c · (ω + ω) X X = . ! " ! " 2n 2(n−1) n n−1 That is: 32π (2n − 1) N (c ) c = . ' ( 1/n 2n (ω + ω) Then, with (23), we obtain: n n n n!4 (2n − 1) N (c ) b ( X ) = . (24) (2n)! In general, we can write: % & 2m tr(K ) s = , 2m 2m (2π i ) where 2m ch(T ) = . (2m)! Using these expressions and important results on graphs (see [25, Section 5]), Hitchin and Sawon provide an expression of b in terms of the Pontryagin classes. The results on graph 123 218 L. Fu, G. Menet are not affected by having singularities on X, hence, the same expression can be obtained in the symplectic orbifold case: b n ( X ) = 48 n! A. Combined with (24) this equation provides our proposition. Lemma 4.3 Let X be a primitively symplectic orbifold of dimension 4, then: 2 2 3b N (c ) ≤ (b + 2)c . 2 2 2 Proof The proof of [21, Lemma 3] can be adapted in the case of primitively symplectic orbifolds. Indeed, it is a consequence of Lemma 4.1 and the Hodge–Riemann bilinear relation which have been generalized in [38, Proposition 2.14]. Corollary 4.4 Let X be a primitively symplectic orbifold of dimension 4, then: (b + 1)b ≤ 4b + 2(S + 20S − 62)b + 736 + 2(S − 124S ), 2 3 1 0 2 1 0 where S and S are deﬁned in (1), introduced in Sect. 3.3. 0 1 Proof In our case, Proposition 4.2 provides: 2 2 4! N (c ) 1 = td − . 2 2 (24 × 2 × 2) 2 8 × 12 That is: 2 2 N (c ) = 192 td − . 2 4 So, using Lemma 4.3: 576b td ≤ 2(b + 1)c . 2 4 2 In Sect. 3.3, we found expressions for c and td , so it is more convenient to replace c by 4 4 720 td +c 4 4 720 td +c 4 4 576b td ≤ 2(b + 1) . 2 4 2 Then (3) provides: 576b td ≤ 2(b + 1)(248 td +4b − b − 8 + 2S ). 2 4 2 4 2 3 1 This is: (b + 1)b ≤ 4b + (2S − 40 td −4)b + 248 td −8 + 2S . 2 3 1 4 2 4 1 Using (2) to replace td by 3 − S , we obtain our result. 4 0 Example 4.5 We can apply Corollary 4.4 to orbifolds with singularities C / ±id. It provides: (b + 1)b ≤ 4b + ( N − 124)b + 736 − 8 N , 2 3 2 where N is the number of singular points. • If N = 28: (b + 1)b ≤ 4(16 − b )(8 − b ). 2 3 2 2 123 Betti numbers of 4-dimensional symplectic orbifolds 219 • If N = 36: (b + 1)b ≤ 4(14 − b )(8 − b ). 2 3 2 2 This corresponds exactly to the second Betti numbers of examples in [15, Section 13] (see also Sect. 5.13). Remark 4.6 Unfortunately, Corollary 4.4 is not restrictive enough to exclude more second Betti numbers. For instance, a primitively symplectic orbifold with second Betti number 22 2i π 2i π 2i π 3 3 3 and 3 isolated singularities of analytic type (C /g , 0) with g = diag(e , e , e , 3 3 2i π e ) is not in contradiction with Corollary 4.4 and Proposition 3.11. To improve Theo- rem 1.3, we need some techniques to exclude some conﬁgurations of singularities. 5 Examples of primitively symplectic orbifolds of dimension 4 For each Betti number between 3 and 23, we provide an example of primitively symplectic orbifold when we know one. See [15, Section 13] for more examples; many additional exam- ples could also be obtained by considering partial resolution in codimension 2 of quotients [2] of K 3 -type and Kum -type manifolds. We summarize all the numerical results in a table in Sect. 5.13. In this section, an isolated cyclic quotient singularity of order n refers to the germ (C /C , 0),where C is an order-n cyclic subgroup of Sp(k, C) such that the origin is n n the only ﬁxed point. 5.1 A construction of Fujiki Most of the know examples of primitively symplectic orbifolds of dimension 4 was con- structed by Fujiki in [15, Section 13]. We recall his construction. Let H be a ﬁnite group of symplectic automorphisms on a K3 or an abelian surface S. First, we assume that H is abelian. Let θ be an involution on H.The action of H on S × S is given by h ·(s, t ) = (hs,θ(h)t ) for all (h, s, t ) ∈ H × S × S. Moreover, we deﬁne G to be the group of automorphisms of S × S generated by H and the involution (s, t ) → (t , s). The quotient (S × S)/G has isolated singularities as well as singularities in codimension 2. The singularities in codimension 2 can be resolved crepantly (see [6]) and we denote by Y ( H ) (resp. Y ( H )) the primitively symplectic orbifold obtained when S is K 3 T aK3surface (resp.when S is a complex torus of dimension 2). As explained in [15, Section 13], the deformation class of Y ( H ) (resp. Y ( H )) only depends on H. K 3 T When the group H is non abelian, the situation is more complicated (the deformation class does not only depends on H) and Fujiki only provides 5 additional examples. 5.2 b = 3 Mongardi [41, Example 4.5.1 and Example 4.5.2] constructed two manifolds ( X ,σ ) and 1 1 [2] ( X ,σ ) of K 3 -type endowed with symplectic automorphisms of order 11 such that 2 2 ⎛ ⎞ ⎛ ⎞ 62 2 21 0 2 σ 2 σ 1 2 ⎝ ⎠ ⎝ ⎠ H ( X , Z) 28 −3 and H ( X , Z) 16 0 . 1 2 2 −38 0022 123 220 L. Fu, G. Menet We denote the quotients M = X /σ with i ∈ {1, 2}. The primitively symplectic orbifolds i i 1 2 M and M both have second Betti number equal to 3 and have 5 isolated cyclic quotient 11 11 singularities of order 11. Moreover their Beauville–Bogomolov forms were computed in [40, Theorem 1.2]. The pair ( X ,σ ) was independently discovered in [14]. 1 1 [2] In general, the fourth Betti number of the quotient of a manifold of K 3 -type by an automorphism of prime order p = 2, 5 can be computed using the Boissière–Nieper- Wisskirchen–Sarti invariants and the fact that: H ( X , Z) = (Z /2 Z) ⊕ (Z /5 Z). Sym H ( X , Z) See [9, Section 2, Proposition 5.1, Proposition 6.6 and Lemma 6.14] for more details. We obtain b ( M ) = 26 for all i ∈ {1, 2}. 5.3 b = 5 [2] Let X be a manifold of K 3 -type endowed with a symplectic automorphism σ of order 7. 2 σ As explained in [41, Section 7.3], we always have rk H ( X , Z) = 5and ( X,σ ) is standard [2] [2] [2] (i.e. deformation equivalent to a natural pair (S ,σ ),where S is a K3 surface and σ induced by an automorphism σ on S). We denote M := X /σ , which is a primitively sym- plectic orbifold with second Betti number 5, and has 9 isolated cyclic quotient singularities of order 7. Moreover, its Beauville–Bogomolov form have been computed in [39, Theorem 1.3]. [m] In general, the Betti numbers of the quotient of a Hilbert scheme S of m points on a K3 surface S by a natural automorphism of prime order can be computed using the Boissière– ∗ [m] Nieper-Wisskirchen–Sarti invariants and the Qin–Wang integral basis of H (S , Z) ([53, Theorem 1.1 and Remark 5.6]). See [39, Proof of Corollary 5.2] for more details. We can compute b ( M ) = 42. 4 7 5.4 b = 6 We consider the complex torus T = C /,where = (1, 0), (i , 0), (0, 1), (0, i ).The torus is given by the product of elliptic curves T = E × E, with E := C/ 1, i .Let σ be the symplectic automorphism of order 4 on T deﬁned by the action on C givenbythe matrix: 0 −1 σ = . We remark that 2 ν ⊕2 H (T , Z) = U ⊕−2 . (25) [2] Since σ is a linear automorphism on T , it extends to an automorphism σ on K (T ).We 4 2 [2] want to consider a crepent resolution in codimension 2 of K (T )/σ . It can be obtain as follows. We have σ =− id. As explained in [55, Section 1.2.1], the induced automorphism [2] (− id) on K (T ) has 36 isolated ﬁxed points and a ﬁxed surface = { ξ ∈ K (T )| Supp ξ = {0, x , −x } , x ∈ T {0}}. [2] We can consider r : K (T ) → K (T ) the blow-up of K (T ) in .Then σ extends to an 2 2 2 [2] [2] automorphism ) σ on K (T ) and K := K (T )/) σ is a primitively symplectic orbifold, 2 2 4 4 4 [2] which is a crepent resolution in codimension 2 of K (T )/σ .From(25), we deduce that: 123 Betti numbers of 4-dimensional symplectic orbifolds 221 b (K ) = 4 + 1 + 1 = 6. (26) Moreover, we can also describe K as a quotient of K (constructed in Sect. 5.6). The orbifold [2] [2] K is given by K (T )/(− id) .Then ) σ induced an involution on K that we denote by [2] [2] σ and K = K /σ .By[30, Proposition 8.23] b (K ) = 0, it follows that: 4 4 4 b (K ) = 0. (27) Now, we are going to determine the singularities of K , which are all isolated cyclic quotient singularities of order 2 or 4. Let us denote the number of such singular points by a and a respectively. It turns out to be quite technical to determine a directly. However, order-4 4 2 [2] cyclic quotient singular points correspond to the singularities of K which are ﬁxed by σ , [2] hence they correspond to the ﬁxed points of σ outside of . After determining a , we will deduce a with a numerical method. We have: { | } Fix σ = (a, a) a ∈ E [2] . 1 1 i i 1+i 1+i [2] That is, we have 4 isolated ﬁxed points (0, 0), ( , ), ( , ) and ( , ). Hence σ 2 2 2 2 2 2 4 ﬁxes only one point outside of the diagonal of K (T ) which is the scheme supported on [2] 1 1 i i 1+i 1+i ( , ), ( , ), ( , ) . This point is also outside of . Furthermore, σ acts on 2 2 2 2 2 2 4 Z := { ξ ∈ K (T )| Supp ξ = {(0, 0)}} . 0 2 [2] 2 [2] As explained in [55, Section 1.2.1], the ﬁxed locus of (σ ) = (− id) on Z is given by the vertex x and a line which is contained in ;the vertex x is out of .Necessarily,the [2] [2] vertex x will also be ﬁxed by σ . We conclude that σ only ﬁxes 2 points outside of 4 4 and so: a = 2. (28) Now, we are going to deduce the number a by considering the double cover: [2] π : K → K /σ = K . Since π has only isolated ramiﬁcation points, the Hurwitz formula can be used in this frame- work and provides: χ(K ) = 2χ(K ) − R, (29) where R is the number of ramiﬁcation points. An order-2 cyclic quotient singular point in K can arise in two different ways. First, it can be the image by π of a non-ramiﬁed singular point of K , or second, it can be the image by π of a ramiﬁed smooth point of K . Since the π-ramiﬁed singular points of K provide cyclic quotient singularities of order 4 in K ,we obtain the following formula: #Sing(K ) − a a = + R − a . 2 4 We have seen that K has 36 isolated singularities, hence by (28): a = 15 + R. (30) By [30, Proposition 8.23], χ(K ) = 108. Hence (29), (27) and Proposition 3.6 provide: 108 = 2(48 + 12b (K ) + s) − R. Applying (30), (26)and (16), we obtain: 108 = 2(48 + 12 × 6 − 3 × 2 − a ) − a + 15. 2 2 123 222 L. Fu, G. Menet One deduces that a = 45. 5.5 b = 7 The generalized Kummer fourfold [2]. For the sake of completeness, we also provide a singular example with b = 7. We consider the same complex torus as previously T = C /,where = (1, 0), (i , 0), (0, 1), (0, i ). The torus is given by the product of elliptic curves T = E × E, with E := C/ 1, i .Let ν be the symplectic automorphism of order 3 on T deﬁned by the action on C given by the matrix: 0 −1 ν = . 1 −1 We remark that: 2 ν H (T , Z) = U ⊕ A . (31) Since ν is a linear automorphism on T , it extends to an automorphism on K (T ). Furthermore, ν veriﬁes the following relation: id +ν + ν = 0. (32) [2] [2] We denote ν the automorphism induced by ν on K (T ). The relation (32) shows that ν [2] admits one ﬁxed surface which induces a surface of singularities in the quotient K (T )/ν . This surface is isomorphic to the K3 surface obtained after resolving the singularities of T /ν. Because of (32), the ﬁxed points of ν are in T [3], there are 9 points of the form (a, 2a),where 9×8×1 [2] a ∈ E [3].Let x , ..., x be these 9 ﬁxed points. It induces = 12 ﬁxed points of ν 1 9 of the form x , x , −x − x , with x = x .Let i j i j i j Z := { ξ ∈ K (T )| Supp ξ = {τ }} , τ 2 for all τ = (a, 2a), with a ∈ E [3]. As explained in [24, Section 4]: Z P(1, 1, 3). [2] Hence, the action of ν on Z ﬁxes 2 lines which intersect in the vertex of Z .These two τ τ lines are necessarily included in the surface . Hence there is no additional isolated ﬁxed [2] point in Z ;thatis ν has 12 isolated ﬁxed points. [2] [2] Let be the image of in K (T )/ν .Let x ∈ . The singularity (K (T )/ν , x ) 2 2 2i π −1 2 2 is analytically equivalent to (C × (C /g ), 0), with g = diag(ξ ,ξ ) and ξ = e . 3 3 3 3 The singularity (C /g , 0) is of type A and can be resolved crepently by a blow-up and 3 2 this resolution has two exceptional lines which intersect in one point. This shows that the singular surface can be resolved crepently and the exceptional locus is the union of two [2] irreducible divisors. Let K → K (T )/ν be such a resolution. The orbifold K is primi- 3 3 tively symplectic with 12 isolated cyclic quotient singularities of order 3. Moreover by (31), we have: b (K ) = 4 + 1 + 2 = 7. Furthermore, because of the action of ν on and [30, Corollary 6.3], the third Betti [2] number of K (T )/ν is trivial. Since is a simply connected surface, b (K ) = 0. 2 3 123 Betti numbers of 4-dimensional symplectic orbifolds 223 5.6 b = 8 This example has already been introduced in Sect. 5.4 as an intermediate consequence of computation. In full generality, it can be constructed as follows. We consider a symplectic involution ι on X a manifold of Kum -type. As it is explained in [30, Theorem 7.5], X /ι has a surface of singularities and 36 isolated ﬁxed points. By resolving the surface of singularities, we obtain a primitively symplectic orbifold that we denote K and which has b = 8. Moreover, its Beauville–Bogomolov form has been computed in [30, Theorem 1.1] and its Betti numbers in [30, Proposition 8.23]. It has been proved that K is irreducible symplectic in [38, Proposition 3.8]. 5.7 b = 9 We describe Fujiki’s example with second Betti number 9. Let T be a complex torus which admits a symplectic binary dihedral linear automorphism group D of order 12. For instance, i π we consider T = E × E ,where E = C/ 1,ξ and ξ = e .Then D is generated ξ ξ ξ 6 6 3 6 6 6 by the linear automorphisms: ξ 0 01 and . −1 0 ξ −10 Let N be the center of D which is generated by − id. We consider the K3 surface S obtained as a resolution of T / N. The group H = D / N is isomorphic to the dihedral group of order 6, denoted by D . There is a natural lifting of a symplectic action of H on S. Then, as in the abelian case, we form the automorphisms group G on S × S generated by H acting diagonally and the involution (s, t ) → (t , s) with θ = id. Fujiki resolves the singularities in codimension 2 of (S × S)/G and shows in [15, Section 13] that we obtain a primitively symplectic orbifold Y ( D ) with second Betti number 9. K 3 3 Since θ = id, the resolution in codimension 2 considered by Fujiki in [15,Section7] [2] [2] corresponds to S / H → (S × S)/G,where S is the Hilbert scheme of 2 points on S and [2] H the induced automorphisms group. That is Y ( D ) = S / H. K 3 3 To determine the singularities of Y ( D ), we start by computing the singularities of K 3 3 S/ H. As there is no ﬁxed point of S by the entire group H, by classiﬁcation of the ﬁnite subgroups of SL(2, C), we know that the singularities of S/ H can only be of type A or A . 1 2 We denote by N (resp. N ) the number of singularities of S/ H of type A (resp. A ). Then 2 1 2 1 the integers N and N can be computed by Riemann–Roch (a direct computation is also 2 1 possible, but is more technical). We apply Theorem 2.12 to S/ H and the V-bundles O , S/ H [1] . We obtain respectively: S/ H c 2 N N 2 2 1 2 = + + , (33) 12 9 8 and 5c 2 N N 2 2 1 − 6 =− − − . (34) 6 9 4 Finally, Theorem 2.13 provides: 2 N N 2 1 10 = c + + . (35) 3 2 123 224 L. Fu, G. Menet Combining (33), (34)and (35), we obtain N = 0and N = 7. Then, we can deduce 1 2 7×6 [2] the singularities of S / H.There are = 21 singular points of the form (a, b) with a = b ∈ Sing S/ H and 2 × 7 = 14 singular points on the diagonal. So Y ( D ) has 35 K 3 3 isolated cyclic quotient singularities of order 3. 5.8 b = 10 We have examples of Fujiki, for instance for H = Z /4 Z and S a K3 surface. 5.9 b = 11 [2] Let X be a manifold of K 3 -type endowed with a symplectic automorphism σ of order 3. As explained in [41, Section 7.3], there are two possibilities: σ has 27 isolated ﬁxed points or σ 2 σ { } has an abelian ﬁxed surface. When Fix σ = 27 points ,wealwayshaverk H ( X , Z) = 11. We denote M := X /σ which is a primitively symplectic orbifold with second Betti number 11 and 27 isolated cyclic quotient singularities of order 3. Moreover its Beauville–Bogomolov form has been computed in [40, Theorem 1.3]. 5.10 b = 14 We have examples of Fujiki, for instance for H = (Z /2 Z) and S a K3 surface. 5.11 b = 16 [2] We consider a symplectic involution ι on X amanifoldof K 3 -type. As it is explained in [42], X /ι has a surface of singularities and 28 isolated ﬁxed points. If we resolve the surface of singularities we obtain a primitively symplectic orbifold that we denote M and which has b = 16. Moreover, its Beauville–Bogomolov form has been computed in [37, Theorem 2.5] and its fourth Betti number in [37, Proposition 2.40]. It has been proved that M is symplectic irreducible in [38, Proposition 3.8]. 5.12 b = 23 The Hilbert square of K3 surface [2]. 123 Betti numbers of 4-dimensional symplectic orbifolds 225 5.13 Summary b X b b Singularities B–B form Irreducible 2 3 4 ⎛ ⎞ ⎛ ⎞ 2 −13 21 0 1 2 ⎝ ⎠ ⎝ ⎠ 3 M ; M 0 26 a = 5 −18 −1 ; 160 No 11 11 3 −16 002 4 ? ? ? ? ? ? 4 −3 5 M 0 42 a = 9 U ⊕ ⊕ (−14) No 7 7 −34 6 K 0 55 a = 45, a = 2 ? No 2 4 7 K ( A) 8 108 Smooth U ⊕ (−6) Yes 7 K 0 92 a = 12 ? ? −5 −4 8 K 0 90 a = 36 U (3) ⊕ Yes −4 −5 9 Y ( D ) 0 66 a = 35 ? No K 3 3 3 10 Y (Z /4 Z) 0 118 a = 10, a = 6 ? ? K 3 2 4 2 2 11 M 0 102 a = 27 U (3) ⊕ U ⊕ A ⊕ (−6) No 3 3 12-13 ? ? ? ? ? ? ! " 14 Y (Z /2 Z) 0 150 a = 36 ? ? K 3 2 15 ? ? ? ? ? ? 3 2 16 M 0 178 a = 28 U (2) ⊕ E (−1) ⊕ (−2) Yes 2 8 17-22 ? ? ? ? ? ? [2] 3 2 23 K 3 0 276 Smooth U ⊕ E (−1) ⊕ (−2) Yes Here,wedenoteby a the number of isolated cyclic quotient singularities of order k. The singularities of the Fujiki examples are described in [15, Section 13, table 1]. We can then compute their fourth Betti numbers using Proposition 3.6 and Example 3.8.Samething for all the orbifolds with known singularities and b . Acknowledgements We would like to thank Roland Bacher, Arnaud Beauville, Daniel Huybrechts and Radu Laza for useful discussions. We also want to thank the referee for his or her careful reading and helpful comments. We are very grateful to the Second Japanese-European Symposium on Symplectic Varieties and Moduli Spaces where our collaboration was initiated. Funding L. Fu is supported by the Radboud Excellence Initiative programme. G. Menet was ﬁnanced by the Fapesp grant 2014/05733-9, the Marco Brunella grant of Burgundy University and the ERC-ALKAGE, grant No. 670846. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A: Salamon’s relation for orbifolds In this Appendix, we extend some of Salamon’s results [48] to the setting of complex orbifolds. Let us ﬁrst recall some notation. Given a complex orbifold X,wedenoteby [ p] p p ∨∨ := ( ) ι ( ) the sheaf of reﬂexive p-forms, where ι : X → X is the ∗ reg X X X reg 123 226 L. Fu, G. Menet inclusion of the smooth part. The main result of “Appendix” is the following generalization of [48, Corollary 3.4]. Theorem A.1 Let X be a compact complex orbifold of dimension n. Then p 2 orb c c = (−1) 6 p − n(3n + 1) χ , (36) 1 n−1 p=0 [ p] orb where χ := ch · td( X ). X X [ p] [n− p] Proof Using the fact that = ⊗ ω , we easily see from the deﬁnition that X X orb n orb χ = (−1) χ . (37) p n− p We deﬁne the orbifold χ -genus of X to be the following polynomial in t: n n orb p n orb p χ (t ) := χ t = (−1) χ t . (38) orb p n− p p=0 p=0 Let c(T ) = (1 + x ) be the formal factorization of the Chern class of X,where X i i =1 x ,..., x are the Chern roots. Following [48], we deﬁne a new polynomial in t with coef- 1 n ﬁcients characteristic classes of X: K (t ) := x + t · . −x 1 − e i =1 ∗ k Denote by K ∈ H ( X , Q) the coefﬁcient of t . Clearly, K = c .Moregenerally,asshown k 0 n in [48, Proposition 3.2], K has the nice property that K − c is in the ideal generated by k k n−k c ,..., c for any k ≥ 1. For example, n−k+1 n K = c + nc ; (39) 1 n−1 n 1 1 K = c + (n − 1)c + (2c c + n(3n − 5)c ). (40) 2 n−2 n−1 1 n−1 n 2 24 It is easy to relate K (t ) and χ (t ): orb −x K (t ) = td( X ) · (t + 1 − e ) X X i =1 p n− p −x −···−x j j n− p = td( X ) · (t + 1) (−1) e p=0 1≤ j <···< j ≤n 1 n− p [n− p] p n− p = td( X ) · (t + 1) (−1) ch p=0 n p orb = (−1) (−1 − t ) χ n− p p=0 = χ (−1 − t ). orb 123 Betti numbers of 4-dimensional symplectic orbifolds 227 Therefore for any k ≥ 0, 1 (−1) (k) (k) K = K (0) = χ (−1). (41) orb k! k! X X Taking k = 0in (41), we obtain p orb c = (−1) χ . (42) p=0 (This is essentially Blache’s Gauss–Bonnet Theorem 2.13.) Taking k = 1in (41), combined with (39), we ﬁnd p orb c = (−1) pχ . (43) p=0 (This is equivalent to (42) by taking into account the symmetry (37).) Taking k = 2in (41), combined with (40), it yields that 1 n(3n − 5) (−1) orb c c + c = p( p − 1)χ . (44) 1 n−1 n 12 24 2 X X p=0 We deduce (36) by combining (42), (43), and (44). [ p] p,q q Remark A.2 We are mainly interested in the Hodge numbers h := dim H ( X, ) and their alternating sum [ p] q p,q χ := χ( X, ) := (−1) h , q=0 orb rather than χ . Therefore, in practice, Theorem A.1 is often combined with orbifold Riemann–Roch formula relating these two quantities: when X has only isolated singularities, orb Blache’s Riemann–Roch theorem 2.12 implies that for any integer p ≥ 0, χ = χ − S p p with tr(ρ [ p] (g)) x , S := (45) |G | det(id −ρ (g)) x x ,T x ∈Sing X g =id being the correction term determined by the singularities of X, where for any singular point x ∈ X, G is the local fundamental group and for any locally V-free bundle F, ρ is the x x , F associated representation of G ; see Sect. 2. In the general case where the singularities are not necessarily isolated, we can use Kawasaki’s Riemann–Roch formula [32], by replacing in the above deﬁnition of S by + , [ p] 1 tr S := ch · td(T ), (46) p X • ∨ m tr ∧ N μ X μ μ where μ runs over all connected components of the inertia orbifold IX except the component X. We refer to [32,52] for the explanation of the notation as well as more details. Remark A.3 Givenaninteger p, from (37) and the fact χ = (−1) χ ,wesee that p n− p S = (−1) S . p n− p 123 228 L. Fu, G. Menet In fact, if X has only isolated singularities, it is a simple exercise in linear algebra to see more precisely that for any x ∈ Sing X, S = (−1) S , p,x n− p,x where tr(ρ [ p] (g)) 1 x , S := . p,x |G | det(id −ρ (g)) x x ,T g =id When specializing Theorem A.1 to the symplectic case, we get the following orbifold analogue of [48, Theorem 4.1]. Theorem A.2 Let X be a compact Kähler orbifold of even complex dimension n = 2msuch p,q n− p,q that the Hodge numbers satisfy the “mirror symmetry” h = h for any p, q ≥ 0. Then 2n n 1 1 k 2 p 2 c c = (−1) 3k − n(6n + 1) b − (−1) 6 p − n(3n + 1) S , 1 n−1 k p 2 2 p=0 k=0 (47) where S is deﬁned in (45) when X has only isolated singularities and in (46) in general. The most important case we have in mind is when X is a primitively symplectic orbifold (Deﬁnition 3.1), hence the left-hand side of (47) vanishes and we get a linear relation among the Betti numbers and contributions of singularities. orb Proof We keep the same notation as above. Since χ = χ − S , the equations (36)and p p (42) imply that n n 1 1 p+q 2 p,q p 2 p 2 orb (−1) p h − (−1) p S = (−1) p χ = c c + n(3n + 1) c . p 1 n−1 n 6 12 X X p,q=0 p=0 (48) p,q q, p Using the Hodge symmetry h = h ,wehave 1 1 p+q 2 p,q p 2 (−1) q h − (−1) p S = c c + n(3n + 1) c . (49) p 1 n−1 n 6 12 X X p,q=0 p p,q p,n−q The mirror symmetry relation h = h implies that (remember that n is even) n n p+q p,q p+q p,q (−1) pq h = (−1) p(n − q)h . p,q=0 p,q=0 Hence n n n p+q p,q p+q p,q p 2 (−1) pq h = (−1) nph = n (−1) pχ . p,q=0 p,q=0 p=0 orb Writing χ = χ + S , together with (43), we obtain p p n n n n p+q p,q p orb p p 2 (−1) pq h = n (−1) pχ + n (−1) pS = c + n (−1) pS . p n p p,q=0 p=0 p=0 p=0 (50) 123 Betti numbers of 4-dimensional symplectic orbifolds 229 Combining (48), (49), and (50), we ﬁnd n n 1 1 p+q 2 p,q p 2 (−1) ( p + q) h − (−1) p(2 p + n)S = c c + n + n c , p 1 n−1 n 3 6 X X p,q=0 p=0 which is nothing else but 2n n 1 1 k 2 p 2 (−1) k b − (−1) p(2 p + n)S = c c + n + n c , (51) k p 1 n−1 n 3 6 X X k=0 p=0 On the other hand, (42)saysthat 2n n k p c = (−1) b − (−1) S . (52) n k p k=0 p=0 Putting (51)and (52) together, we obtain the following formula, 2n n 1 1 k 2 p 2 2 c c = (−1) 3k − n(6n + 1) b − (−1) 6 p + 3np − 3n − n S . 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Keywords: Hyper-Kähler varieties; Orbifolds; Betti numbers; singularities; Riemann–Roch theorem; 57R18; 14J42; 14J28; 53C26; 19L10; 14B05

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On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

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On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

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