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

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.


Introduction
A compact Kähler manifold is called holomorphic symplectic if it admits a holomorphic 2form that is nowhere degenerate. In particular, it is even-dimensional and has trivial canonical 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 Riemannian metric [51], 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] [6]: 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, Calabi-Yau varieties and IHS manifolds. We refer to [2], [23] and [17,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 [38] [39] in dimensions 6 and 10. The limitedness of available examples suggests the possibility to bound or even classify IHS manifolds (see [24] 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 [50] [31] [25], it is natural to ask the following question. In dimension 4, Guan [18] 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 2 = 23, the Hodge diamond must be the same as that of the Hilbert square of a K3 surface.
The fact that b 2 ≤ 23 was attributed to Beauville. See [18] and [19] for extra constraints on each cases; see [30] for a related result in dimension 6. When b 2 = 23, let us mention the work [40] [27], 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. [29]), it is important to treat varieties with mild singularities. With recent intensive efforts [16] [10] [14] [11] [22], 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 ( [16,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 [13]. 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 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.
The proof of Theorem 1.3 and Theorem 1.4 is given in the end of Section 3.
The bound for b 2 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 [18], despite of our effort in Section 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 2 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 Section 5 for details of these examples.
Definition-Proposition 2.4 ([42, Proposition 6]). Let X be an n-dimensional complex orbifold. Let x ∈ X. Then there exist a finite subgroup G x of GL n (C) and an open neighbourhood V x ⊂ C n of the origin 0 ∈ C n , stable under the action of G x , with V x /G x isomorphic to an open neighbourhood U x of x, and such that Codim Fix(g) ≥ 2 for all g ∈ G x \{id}.
Such a group G x is unique up to conjugation. Let π x : a local uniformizing system of x and G x the local fundamental group of X at x.
The notion of vector bundles naturally generalizes to orbifolds.
• A V-bundle (or orbibundle) on X is a vector bundle F on X reg := X Sing X such that for any local uniformizing system (U, V, G, π), there exists a vector bundleF V on V endowed with an equivariant action of G such that: where Fix G := g∈G,g =id Fix(g). • Let F be a coherent sheaf on X. The sheaf F is said locally V-free if for any x ∈ X, there exist a local uniformizing system (U, V, G, π), a free coherent sheafF V on V , and a G-action onF V such that F |U ≃ π * F G V . By [5, 4.2], the local V-freeness of 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. 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). Given an orbifold X of dimension n, the sheaf of reflexive differential forms ( [15, 2.D], [41,Section 2.5]) is a locally V-free sheaf for any i ∈ N, where ι : X reg → X is the natural inclusion of the smooth part. The sheaf of reflexive forms of top degree is identified with the dualizing sheaf: ω X ∼ = Ω X . Remark 2.7 (Hodge decomposition). Let X be a compact Kähler orbifold. For any integer k ≥ 0, the rational singular cohomology group H k (X, Q) carries a pure Hodge structure of weight k and in the Hodge decomposition X ), see [41,Section 2.5]. We denote h p,q (X) := dim H p,q (X). We have h p,q = h q,p . 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 andF V be a locally free sheaf on V endowed with an action of G as in Definition 2.5. Hence the fiber ofF V at 0 is endowed with an action of G which provides a representation of G. We denote by ρ x,F the representation of G associated to x and F .

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 ([5, Definition 2.9]). Let F be a V-bundle of rank r on an orbifold X. A metric on F is a metric h on F as bundle on X reg such that for all local uniformizing systems (U, V, G, π), the metric π * (h |Ureg ) extends to a metric onF V .
Definition 2.10 ([5, Definition 1.5]). Let X be an orbifold. A smooth differential k-form ϕ on X is a C ∞ differantial k-form on X reg such that for all local uniformazing system (U, V, G, π), π * (ϕ |Ureg ) extends to a C ∞ -differential k-form on V . (We always mean C-valued forms.) Notation 2.11. We denote by A k the sheaf of differential k-forms.
As explained in [5, 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 reg as in the smooth case. To a hermitian metric h on F , we associate a connection D on F , and to D, we associate the curvature D 2 . Let Ξ be the corresponding r × r matrix of curvature 2-forms, then we set c k (h) = P k ( i 2π Ξ) ∈ Γ(X reg , A 2k ), where P k is the k-th elementary invariant polynomial function C r×r → C.
The same process can be also done on all local uniformizing systems (U, V, G, π). The metric π * (h |Ureg ) extends to a hermitian metricĥ onF V which gives rise to a connectionD onF V and hence the curvatureD 2 . As previously, we can construct c k (ĥ) ∈ Γ(V, A 2k V ). By construction π * (c k (h) |Ureg ) extends to c k (ĥ) on V . Hence, we obtain c k (h) ∈ Γ(X, A 2k ). Then, as in the smooth case, we show that c k (h) is a closed form, and that the cohomology class c k (F ) := [c k (h)] ∈ H 2k (X, C), called the k-th Chern class of F , does 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 [5, Theorems 3.5 and 3.17].
Theorem 2.12 (Blache [5]). Let X be a compact complex orbifold with only isolated singularities and let F be a locally V-free sheaf. Then we have where g is viewed as an endomorphism on T 0 V with (U, V, G x , π) a local uniformizing system of x.
Blache also established the following orbifold version of Gauss-Bonnet theorem.
Theorem 2.13 ([5, 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: where G x is the local fundamental group of X at x.

Bounding Betti numbers and singularities
The aim of this section is to show Theorem 1.3 and Theorem 1.4. Let us first make precise the class of singular symplectic varieties that we consider.

Symplectic orbifolds.
Definition 3.1 (Fujiki [13]). A compact Kähler orbifold X is called primitively symplectic if (i) the smooth locus X reg := X Sing X is endowed with a non-degenerated holomorphic 2-form which is unique up to scalar; and (ii) the singular locus Sing X has codimension at least 4. If moreover X reg is simply connected, X is called an irreducible symplectic orbifold. Remark 3.2. As in the smooth case, a primitively symplectic orbifold X has even (complex) dimension and trivial dualizing sheaf ω X ≃ O X . Moreover, the symplectic form extends to a symplectic form on any local uniformizing system. In particular, the contraction with the symplectic form induces an isomorphism T X ≃ Ω [1] X . 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 [16] [22, Remark 3.4 (Hodge diamond). Let X be a 4-dimensional primitively symplectic orbifold. Fujiki [13,Proposition 6.7] showed that X has vanishing irregularity, hence b 1 (X) = 0. Serre-Grothendieck duality gives that In conclusion, the Hodge diamond of X takes the following form.
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).
Proposition 3.5. Let n > 0 be an integer, we denote ξ n := e 2iπ n the primitive n-th root of unity. For integers 1 ≤ k ≤ n, we denote Let G be a finite subgroup of the compact symplectic group Sp(2). Then, up to conjugation, (i) there exists finite subgroups H 1 , H 2 of SU(2), integers n > 0 and k ∈ {1, ..., n}, and a normal subgroup G ′ of G of index at most 2, such that any element M ′ of G ′ has the form If moreover C 4 /G has only the image of 0 as singularity, then there exists a finite subgroup H of SU(2) and θ an automorphism of H such that any element M ∈ G ′ has the form , Proof. Hanany and He classified in [20] 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 [20, Section 3.1.1] and they are numbered from I to VI. However, none of them are symplectic. Indeed, the following matrices are considered: where w := e 2iπ 3 . The matrices F 1 and F 2 do not preserve any common symplectic form, 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 1 and F ′ 2 which do not fix the same symplectic form.
In [20, 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 [20, Section 3.1.3], they consider group obtained from Kronecker products of matrices of SU(2). Let The following couples of matrices (2) ) both do not fix the same symplectic form. Hence the groups from X 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 1 := diag(1, 1, −1, −1) and A 2 := diag(1, −1, −1, 1) do not fix the same 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: which is not symplectic, hence all the groups from XXXI to XXXIII which are partially generated by ∆ are not symplectic. It only remains 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 [20,Definition 2.1] for the precise definition). 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 1 , e 2 , e 3 , e 4 ) be the canonical basis of C 4 . We have: where ξ is a root of the unity and A ∈ U(3). We can find a basis , the symplectic form has to be We consider now another matrix of G expressed in the basis (e 1 , v 1 , v 2 , v 3 ): where a, b, c, d, e, f, g, h, j, k are in C. Since N is symplectic. It follows: Hence c = d = 0. If h = 0, then f = ej h and g = ek h . This is impossible because, it contradicts −jg + f k = 1. So h = 0. For the same reason e = 0 and G is actually a group composed by matrices of the forms: It only remains to study the case of the group XXXIV. In this case, G = G 0 , T n,k , with G 0 composed of matrices of type We consider The group G ′ is a normal subgroup of G and the class T n,k ∈ G/G ′ has order 2. Now, we prove (ii). Let G be a finite subgroup of Sp(2) such that C 4 /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: So, setting θ := P 2 • P −1 1 finishes the proof. In what follows, X is a primitively symplectic orbifold of dimension 4. For any (necessarily isolated) singular point x ∈ X, G x is the local fundamental group of X at x and ρ x,• is the representation of G x defined in Notation 2.8.
Proposition 3.6 (Orbifold Salamon relation). Let X be a primitively symplectic orbifold of dimension 4. We have: is a correction term determined by the singularities. In particular: Remark 3.7. Proposition 3.6 shows that the knowledge of h 1,1 , h 2,1 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 = x∈Sing X s x with: Let (V, 0) be a local uniformizing system around x, then the action of g ∈ G x on T V,0 is symplectic.
We can therefore write that g = diag(ξ 1 , ξ 2 , ξ −1 1 , ξ −1 2 ), with ξ j = e 2iπk j n j , k j , n j ∈ N for all j ∈ {1, 2}. Hence: . So: Hence for any x ∈ Sing X, we have In particular, s x ≤ −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 x is a finite subgroup of Sp(2). Hence, there exists a normal subgroup G ′ of G x 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 with A ∈ H. As we have noticed previously, if A is a matrix of SU(2) of finite order, we have where on the right-hand side, we write a non-trivial element of G ′ as Reordering the sum of the second term, we obtain the following equation: Example 3.8. We compute explicitly s x in the following cases.
• G x = A n is a cyclic group of order n.
In this case, G x = G ′ and H = g n , with g n = diag(e 2iπ n , e − 2iπ n ). By (14), we have: where we used the identity As a result, (16) s x (A n ) = −(n − 1).
• G x =D n is a binary dihedral group of order 4n. In this case, G x = G ′ and H =D n . The binary dihedral groupD n can be generated by Hence by (14), we have that Since tr(B k P ) = 0 for all k ∈ {1, ..., 2n}, by (15), we obtain Therefore (18) s x (D n ) = −(n + 2).

Orbifold Guan inequality.
In the smooth case, Guan [ . 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: 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 [33,Section 3.4]).
By Proposition 3.5, there exist a normal subgroup G ′ of G x 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 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 x . The finite subgroups of SU(2) are the so-called Kleinian groups corresponding to the A-D-E Dynkin diagrams: the cyclic groups A n , the binary dihedral groupsD n and the three sporadic groups E 6 , E 7 and E 8 . The biggest sporadic group E 8 has order 120. Let us check the maximal size of the group for A-D types.
However by (19), we know that s x (G) ≥ −91. Hence, the biggest possible groups are the groups which have a binary dihedral groupD 178 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 x = A n is a cyclic group of order n. Then by (16), n ≤ 92.
• If G x =D n is a binary dihedral group of order 4n. Then by (18), n ≤ 89.
n is a group with a cyclic group of order n as normal subgroup of index 2. Then by (20), n ≤ 181.
n is a group with a binary dihedral group of order 4n as normal subgroup of index 2. Then by (20), n ≤ 178.

Hitchin-Sawon formula
We can try to improve Theorem 1.3 using the same method as in [18,Section 3]. The method of Guan is based on an equation of Hitchin-Sawon [21]. This section is just an attempt and is not needed in the rest of the paper. First, we recall the following generalized Fujiki formula. depending on β such that for all α ∈ H 2 (X, C), one has X β · α 2(n−p) = N (β) X α 2n  The tangent sheaf T X on X can be defined as the unique reflexive sheaf such that T X|Xreg is the usual holomorphic tangent sheaf on X reg . It is a locally V-free sheaf. We consider g a Kähler metric on T X . As explained in Definition 2.9, this provides a metric g on T Xreg such that for all local uniformizing system (U, V, G, π), π * (g |Ureg ) extends to a metric g V on T V . Then the Riemannian curvature K of (X, g) is obtained on X reg by the Riemannian curvature of (X reg , g) and on all local uniformizing systems by the Riemann curvature of (V, g V ). For the same reason as in the smooth case, we can associated to the curvature a class [Φ] ∈ H 1 (X, Sym 3 Ω 1 X ) (see [21,Section 2] or [48]). From this class [Φ], 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 Γ (X) for Γ a trivalent graph with 2n vertices.
In the smooth case, it is well known that: where K is the curvature of X. Because of our definition of Chern classes in Section 2, (21) is also true in the orbifold case. In the symplectic case (21) gives: Then, using this expression for c 2 exactly as Hitchin and Sawon did in [21, Section 3], we can provide an expression of N (c 2 ) ([21, equations (7) and (8)]): where ω generated H 2,0 (X) and c Θ can be express by: with vol(X) = X (ω+ω) 2n 2 2n (2n)! and Θ the trivalent graph with two vertices. Equation (22) can be rewritten: That is: X (ω + ω) 2n 1/n . Then, with (23), we obtain: In general, we can write: Using these expressions and important results on graphs (see [21,Section 5]), Hitchin and Sawon provide an expression of b Θ n in terms of the Pontryagin classes. The results on graph are not affected by having singularities on X, hence, the same expression can be obtained in the symplectic orbifold case: b Θ n (X) = 48 n n! X Â .
Combined with (24) this equation provides our proposition.
Lemma 4.3. Let X be a primitively symplectic orbifold of dimension 4, then: Proof. The proof of [18,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 [33, Proposition 2.14].
Corollary 4.4. Let X be a primitively symplectic orbifold of dimension 4, then: , where S 0 and S 1 are defined in (1), introduced in Section 3.3.
Example 4.5. We can apply Corollary 4.4 to orbifolds with singularities C 4 / ± id. It provides: where N is the number of singular points.
• If N = 28: • If N = 36: This corresponds exactly to the second Betti numbers of examples in [13,Section 13] (see also Section 5.13).

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 [13,Section 13] for more examples; many additional examples could also be obtained by considering partial resolution in codimension 2 of quotients of K3 [2] -type and Kum 2 -type manifolds. We summarize all the numerical results in a table in Section 5.13. 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 will have isolated singularities and singularities in codimension 2. The singularities in codimension 2 can be resolved crepantly (see [33,Remark 3.2]) and we denote by Y K3 (H) (resp. Y T (H)) the primitively symplectic orbifold obtained when S is a K3 surface (resp. when S is a complex torus of dimension 2). As explained in [13,Section 13], the deformation class of Y K3 (H) (resp. Y T (H)) only depends on H.
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. and (X 2 , σ 2 ) of K3 [2] -type endowed with symplectic automorphisms of order 11 such that We denote the quotients M i 11 = X i /σ i with i ∈ {1, 2}. The primitively symplectic orbifolds M 1 11 and M 2 11 both have second Betti number equal to 3 and have 5 isolated singularities. Moreover their Beauville-Bogomolov forms were computed in [35,Theorem 1.2]. Such automorphism was also discovered in [12] for Fano varieties of lines in cubic fourfolds.
In general, the fourth Betti number of the quotient of a manifold of K3 [2] -type by an automorphism of prime order p = 2, 5 can be computed using the Boissière-Nieper-Wisskirchen-Sarti invariants and the fact that: See [7, Section 2, Proposition 5.1, Proposition 6.6 and Lemma 6.14] for more precisions. We obtain b 4 (M i 11 ) = 26 for all i ∈ {1, 2}.
We denote ν [2] the automorphism induced by ν on K 2 (T ). The relation (25) shows that ν [2] admits one fixed surface Σ which induces a surface of singularities in the quotient K 2 (T )/ν [2] . This surface is isomorphic to the K3 surface obtained after resolving the singularities of A/ν. Because of (25), the fixed points of ν are in A [3], there are 9 points of the form (a, 2a), where a ∈ E [3]. Let x 1 , ..., x 9 be these 9 fixed points. It induces 9×8×1 6 = 12 fixed points of ν [2] of the form for all τ = (a, 2a), with a ∈ E [3]. The action of ν [2] on Z τ fixes 1 point and 1 line. The line is included in the surface Σ. We obtain in total 9+12=21 isolated fixed points by ν [2] . The surface Σ can be resolved (see [13,Section 7]) to obtain a primitively symplectic orbifold K ′ 3 with only 21 isolated fixed points and b 2 (K ′ 3 ) = 6. Moreover, because of the action of ν on Λ and [26, Corollary 6.3], the third Betti number of K ′ 5.6. b 2 = 8. We consider a symplectic involution ι on X a manifold of Kum 2 -type. As it is explained in [26,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 2 = 8. Moreover, its Beauville-Bogomolov form has been computed in [26, Theorem 1.1] and its Betti numbers in [26,Proposition 8.23]. It has been proved that M ′ is irreducible symplectic in [33, Proposition 3.8]. 5.7. b 2 = 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 groupD 3 of order 12. For instance, we consider T = E ξ6 × E ξ6 , where E ξ6 = C / 1, ξ 6 and ξ 6 = e iπ 3 . ThenD 3 is generated by the linear automorphisms: Let N be the center ofD 3 which is generated by − id. We consider the K3 surface S obtained as a resolution of T /N . The group H =D 3 /N is isomorphic to the dihedral group of order 6, denoted by D 3 . 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 [13, Section 13] that we obtain a primitively symplectic orbifold Y K3 (D 3 ) with second Betti number 9.
Since θ = id, the resolution in codimension 2 considered by Fujiki in [13, Section 7] corresponds to S [2] /H → (S × S)/G, where S [2] is the Hilbert scheme of 2 points on S and H the induced automorphisms group. That is Y K3 (D 3 ) = S [2] /H.
To determine the singularities of Y K3 (D 3 ), we start by computing the singularities of 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 2 or A 3 (that is analytically equivalent to C 4 /K, with K a cyclic group of order 2 or 3). We denote by N 3 (resp. N 2 ) the number of singularities of S/H of type A 3 (resp. A 2 ). Then the integers N 3 and N 2 can be computed by Riemann-Roch (a direct computation is also possible, but is more technical). We apply Theorem 2.12 to S/H and the V-bundles O S/H , Ω Finally, Theorem 2.13 provides: Combining (26), (27) and (28), we obtain N 2 = 0 and N 3 = 7. Then, we can deduce the singularities of S [2] /H. There are 7×6 2 = 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 K3 (D 3 ) has 35 singularities of type A 3 . 5.8. b 2 = 10. We have examples of Fujiki, for instance for H = Z /4 Z and S a K3 surface. 5.9. b 2 = 11. Let X be a manifold of K3 [2] -type endowed with a symplectic automorphism σ of order 3. As explained in [36,Section 7.3], there are two possibilities: σ has 27 isolated fixed points or σ has an abelian fixed surface. When Fix σ = {27 points}, we always have rk H 2 (X, Z) σ = 11. We denote M 3 := X/σ which is a primitively symplectic orbifold with second Betti number 11 and 27 isolated singular points. Moreover its Beauville-Bogomolov form has been computed in [ 5.11. b 2 = 16. We consider a symplectic involution ι on X a manifold of K3 [2] -type. As it is explained in [37], 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 2 = 16. Moreover, its Beauville-Bogomolov form has been computed in [32,Theorem 2.5] and its fourth Betti number in [32,Proposition 2.40]. It has been proved that M ′ is symplectic irreducible in [33, Proposition 3.8].  We denote by a k the singularities analytically equivalent to C 4 /A k with A k a cyclic group of order k.
The singularities of the Fujiki examples are described in [13, Section 13, table 1]. We can then compute their fourth Betti numbers using Proposition 3.6 and Example 3.8. Same thing for all the orbifolds with known singularities and b 3 .