Integral cohomology of quotients via toric geometry

We describe the integral cohomology of $X/G$ where $X$ is a compact complex manifold and $G$ a cyclic group of prime order with only isolated fixed points. As a preliminary step, we investigate the integral cohomology of toric blow-ups of quotients of $\mathbb{C}^n$. We also provide necessary and sufficient conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to degenerate at the second page. As an application, we compute the Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.


Background and motivation
Let X be a topological space endowed with the action of a finite automorphism group G. We consider π : X → X/G the quotient map. It is quite easy to compute the cohomology of X/G with rational coefficients since it is isomorphic to the invariant cohomology H * (X/G, Q) H * (X, Q) G . However, switching to integral coefficients, several complications appear. Let Y be a topological space. We denote by H * f (Y , Z) the torsion-free part of the cohomology (see Section 1.4 (vi) for the notation).
(i) The first problem is to determine the torsion of H * (X/G, Z).
(ii) The second problem is to find a basis of H * f (X/G, Z). We are also interested in the ring structure of H * f (X/G, Z) which differs from the one of H * f (X, Z) G . No existing theory solves these problems; nevertheless, we mention a result of Smith which is an important tool for our purpose. Smith in [Smi83] (see also [Bre72,Section III.7] and [AP06]) has constructed a push-forward map π * : H * (X, Z) → H * (X/G, Z) with the following properties: (1.1) π * • π * = d id H * (X/G,Z) , π * • π * = g∈G g * , where d is the order of the group. We approach the problem by considering groups of prime order p as a fundamental case before investigating more complicated frameworks. Then, much information can be obtained from these equations in order to simplify problems (i) and (ii). We denote by H * o−t (Y , Z) the torsion part of the cohomology without the p-torsion part. The first property that we can deduce is:

Organization of the paper
In Section 2, we define and study the basic properties of the Boissière-Nieper-Wisskirchen-Sarti invariants which characterize the Z[G]-modules and the F p [G]-modules when G is a group of prime order. These invariants are one of the main tools of this paper. In particular, they allow to describe the second page of the spectral sequence of equivariant cohomology with coefficients in Z (cf. Proposition 2.19) and in F p (cf. Proposition 2.4). In Section 2.5, we also remark that all the results of [Men18] can be generalized to the case when p > 19.
In Section 3, we examine the integral cohomology of toric blow-ups of C n /G and P n /G, for G a linear action of prime order (cf. Proposition 3.13 and Theorem 3.17). As an application we show how toric blow-ups of isolated quotient singularities in complex spaces modify the integral cohomology (cf. Corollaries 3.19, 3.20 and 3.21).
In Section 4, we apply the previous results to compute the integral cohomology of quotients. In Section 4.2, we compute explicitly the odd coefficients of surjectivity. In Section 4.4, we provide a general expression of the even coefficients of surjectivity and provide an upper bound in terms of the Boissière-Nieper-Wisskirchen-Sarti invariants (cf. Proposition 4.8). Section 4.5 is devoted to the proof of Theorem 4.9 and Section 4.6 to the proof of Theorem 1.1.
In Section 5, we give examples of applications. In Section 5.1, we describe the integral cohomology of X/G for X a K3 surface and G an automorphism group of prime order with only isolated fixed points. In Section 5.3, we give examples of spectral sequences which degenerate at the second page and Section 5.4 is devoted to the proof of Theorems 1.2 and 1.3. Finally, in Section 5.5, we show that Theorem 1.1 can be applied to the quotients of complete intersections.

Notation about Z-modules and integral cohomology
Let T be a Z-module and G a finite group of prime order p acting on T linearly.
(i) We denote by T G the invariant submodule of T , (ii) by tors T the submodule of T generated by the torsion elements, (iii) by tors p T the submodule of T generated by the elements of p-torsion, (iv) by T f the torsion-free part of T , that is T f := T tors T . (v) We define the rank of T as the rank of T f and we denote rk T := rk T f . Let Y be a topological space and G an automorphism group. Let k ∈ N.
(vi) We denote by H k f (Y , Z) the torsion-free part of H k (Y , Z). (vii) H * (Y , Z) denotes the direct sum of all the cohomology groups, H 2 * (Y , Z) the direct sum of all the cohomology groups of even degrees and H 2 * +1 (Y , Z) the direct sum of all the cohomology groups of odd degrees. We adopt the same notation for coefficients in F p . (viii) h * (Y , Z) := rk H * (Y , Z), h 2 * (Y , Z) := rk H 2 * (Y , Z) and h 2 * +1 (Y , Z) := rk H 2 * +1 (Y , Z). We adopt the same notation for coefficients in F p replacing rk by dim F p .
(ix) When tors p H k (Y , Z) is a F p -vector space of finite dimension, we denote: t k p (Y ) := dim F p tors p H k (Y , Z).

Convention on automorphisms
Let C be a category. Let X be an object of C. In this paper, an automorphism φ on X refers to an invertible morphism of C from X to itself. For instance, if X is a C ∞ -manifold, an automorphism φ on X is a C ∞ -diffeomorphism.

Small reminder on lattices
We recall some basic notions regarding lattices which are used in this paper (see for example [Dol12,Chapter 8.2.1] for more details). A lattice T is a free Z-module endowed with a non-degenerate bilinear form T × T → Z (symmetric or skew-symmetric). Without an explicit mention, the bilinear form of the lattice will be denoted by "·". We denote by discr T the discriminant of T , which is the absolute value of the determinant of the bilinear form of T . We say that T is unimodular if its discriminant is 1. A sublattice N of T is said to be primitive if T /N is torsion-free.
Let N ∨ := Hom(N , Z) be the dual lattice of N ; it can be seen as a sub-module of N ⊗ Q via the natural isomorphism N ⊗ Q → Hom(N , Q) : x → (y → x · y). We also denote by A N := N ∨ /N the discriminant group of N . Let N be a sublattice of a lattice T ; the saturation of N in T is the primitive sublattice N of T containing N and such that N /N is finite.
If N is a sublattice of a lattice T of the same rank, we have the basic formula: If T is an unimodular lattice and L is a primitive sublattice, then (1.4) discr L = discr L ⊥ .
We still assume that T is unimodular. Then, the map defined by the composition T ⊂ (L ⊕ L ⊥ ) ⊗ Q → L ⊗ Q provides the following isomorphism: (1.5) T L ⊕ L ⊥ → A L .

Main idea and sketch of the proof of Theorem 1.1
Let X be a complex manifold and G an automorphism group of prime order. Let M := X/G. Let π : X → M be the quotient map. Let r : M → M be a resolution of singularities of M. To simplify the exposition, we assume in this section that X is a surface. The main idea is to deduce the integral cohomology of M from the integral cohomology of M via the Poincaré duality.

The exceptional lattice.
By Poincaré duality, the group H 2 f M, Z endowed with the cup-product is a unimodular lattice. From this fundamental information, the objective will be to obtain information on H 2 (M, Z). We define the second exceptional lattice of r by: N 2,r := r * π * (H 2 (X, Z)) ⊥ , see [Men18, Section 5.1]. The lattice N 2,r is the primitive lattice generated by the exceptional divisors of r and the lattice N ⊥ 2,r is the saturation of r * π * (H 2 (X, Z)) . The unimodularity of H 2 f M, Z connects N 2,r and N ⊥ 2,r by the following equation (see (1.4)): (1.6) discr N 2,r = discr N ⊥ 2,r . The objective will be to compute discr N 2,r and discr N ⊥ 2,r separately to obtain information on the coefficient of degeneration α 2 (X) (see (1.2)).

Toric blow-up.
Now, we explain how to compute discr N 2,r . We assume that G has only isolated fixed points. Let V := X Fix G and U := V /G. We have the following exact sequence: We are going to connect N 2,r with Im g. To do so, we want a good understanding of g and f . We remark that this is possible if we choose for r a toric blow-up (see Section 3.2). The main objective of Section 3 is to understand Im g and Im f . Corollary 3.19 shows that g is injective and f is surjective. Moreover, Corollary 3.20 leads to the following exact sequence: By (1.3) it follows that: discr N 2,r = discr Im g p 2N .
By Corollary 3.21, we have that: where η(G) is the number of fixed points by G. Moreover from Corollary 3.20, we can expressed N in terms of the torsion of H 2 M, Z and H 2 (U , Z). We have: (see Section 1.4 (ix) for the notation). Hence:

Spectral sequence of equivariant cohomology.
The torsion of H 2 (U , Z) can be computed using the spectral sequence of the equivariant cohomology. This is one of the objective of Section 2. The second page of the spectral sequence of the equivariant cohomology can be expressed in terms of the Boissière-Nieper-Wisskirchen-Sarti invariants (see Proposition 2.4 and 2.19).
In Section 4.1, we define the coefficient of resolution β 2 (X) which allows to complete the computation: .

Definition of the Boissière-Nieper-Wisskirchen-Sarti invariants
We recall here the definition of invariants introduced by Boissière, Nieper-Wisskirchen and Sarti in [BNS13]. Let p be a prime number, T a F p -vector space of finite dimension and G = φ an automorphism group of prime order p. The minimal polynomial of φ, as an endomorphism of T , divides X p − 1 = (X − 1) p ∈ F p [X], hence φ admits a Jordan normal form. We can decompose T as a direct sum of some F p [G]-modules N q of dimension q for 1 ≤ q ≤ p, where φ acts on N q in a suitable basis by a matrix of the following form: Definition 2.1. We define the integer q (T ) as the number of blocks of size q in the Jordan decomposition If T is a finitely generated Z-module endowed with the action of an automorphism group of prime order, we define q (T ) := q (T ⊗ F p ). We call the q (T ) the Boissière-Nieper-Wisskirchen-Sarti invariants.
One of the uses of these invariants is the computation of the cohomology of the group G with coefficients in a F p -vector space of finite dimension.
Notation 2.3. Let X be a topological space endowed with the action of an automorphism group G of prime order p. Assume that H k (X, F p ) has finite dimension for all k ≥ 0. Then, for all k ∈ N and all 1 ≤ q ≤ p, we denote: k q (X) := q (H k (X, F p )), and k q,t (X) := q (tors p H k (X, Z)). We also set * q (X) := k≥0 k q (X).
From Lemma 2.2, we can express the cohomology of G with coefficients in H * (X, F p ) as follows.

Boissière-Nieper-Wisskirchen-Sarti invariants of a free Z-module
In this section we are going to provide properties of the Boissière-Nieper-Wisskirchen-Sarti invariants when T is free Z-module (Theorem 2.10). To do so, we introduce some additional invariants + (T ) and − (T ) which are intrinsically defined from the Z-module structure of T and we show that there are equal to the Boissière-Nieper-Wisskirchen-Sarti invariants 1 (T ) and p−1 (T ) apart for p = 2. The invariants + (T ) and − (T ) are needed in the case p = 2; for this reason they will be more convenient to use as notation.
Let ξ p be a primitive p th root of unity. We set K := Q(ξ p ) and O := Z[ξ p ]. Let G = φ be a group of prime order p. Let A be an O-ideal in K, the Z[G]-module structure of A is defined by φ · x = ξ p x for x ∈ A. For any a ∈ A, we denote by (A, a) the Z-module A ⊕ Z whose Z[G]-module structure is defined by φ · (x, k) = (ξ p x + ka, k).
The following result is a slight adaptation of [CR88,Theorem 74.3].
Theorem 2.5. Let T be a free Z-module of finite rank and G an automorphism group of prime order p. Then, we have an isomorphism of Z[G]-modules: Notation 2.6. We set − (T ) := s and + (T ) := t.
Lemma 2.7. Let A be an ideal of O and a ∈ A such that a (ξ p − 1)A. Then: Proof. Let σ = φ p−1 + · · · + φ + id. If p = 2, we have σ · (0, 1) = (a, 2). Since a 2 Z, the image σ · (0, 1) in (A, a) ⊗ F p is non-zero. Since N 1 ⊂ Ker σ , necessarily (A, a) ⊗ F p = N 2 . Now we assume that p ≥ 3. We can compute that: If σ · (0, 1) was divisible by p, we would have: for y ∈ A. However, we have ( Hence, we would have a ∈ (ξ p − 1)A which is a contradiction. Therefore, σ · (0, 1) is not divisible by p. So its image σ · (0, 1) in (A, a) ⊗ F p is non-zero. Moreover, we can apply the Jordan decomposition to (A, a) ⊗ F p . Since for all q < p, we have N q ⊂ Ker σ ,  Proof. If p = 2, then O = Z and the prime ideals of O are given by p Z with p a prime number. If 2 divides N (P ) then P = 2 Z. Now we assume that p ≥ 3. First note that P ∩ Z is a prime ideal of Z. Hence: is the reduction modulo p of the p th cyclotomic polynomial. In particular: Since p O ⊂ P , we have that N (P )| N (p O). Since p divide N (P ), we have by (2.3) that p = p. Therefore Φ p = (X − 1) p−1 . In particular, (Z /p Z[X])/(Φ p ) has only one prime ideal which is (X − 1), with X − 1 the reduction modulo p of X − 1. By (2.2), it means that O/pO has only one prime ideal which is (ξ p − 1)/pO. Since P /pO is a prime ideal of O/pO, it follows that (ξ p − 1)/pO = P /pO. Since both (ξ p − 1) and P are ideals containing pO, we obtain that: Proof. When p = 2, we have O = Z and A is principal, so the result follows immediately. We assume that p ≥ 3. We know from the proof of [BNS13, Proposition 5.1] that: Therefore, there exists 1 ≤ j ≤ i such that p divides N (P j ). Then by Lemma 2.8, P j = (ξ p − 1). We can consider A = P 1 · · · P j−1 P j+1 · · · P i . Since P j is principal, we have an isomorphism of Z[G]-modules A A . By induction, we will obtain A an ideal of O isomorphic to A as a Z[G]-module and such that p does not divide N (A ). Then, we are back to the first case.
As a consequence, we obtain the following generalization of [BNS13, Proposition 5.1] for p > 19. The main improvement is that Lemmas 2.7 and 2.9 are proven without assuming Z[ξ p ] being a PID.
Theorem 2.10. Let T be a free Z-module endowed with the action of an automorphism group of prime order p. Let r be the integer obtained by applying the decomposition of Theorem 2.5 to T . Then: Proof. Tensorizing (2.1) by F p , we obtain (i), (ii), (iii) and (iv) from Lemmas 2.7 and 2.9. Statement (v) is proved in the proof of [MTW18, Lemma 1.8].
Remark 2.11. Let T be a free Z-module. From the previous theorem, we see that the integers + (T ) and − (T ) are more relevant in the case p = 2; for this reason, it will be use preferentially in the rest of the paper. The integers + (T ) and − (T ) are also designated as Boissière-Nieper-Wisskirchen-Sarti invariants of T .
When T is a lattice, there is an easy technique to compute the Boissière-Nieper-Wisskirchen-Sarti invariants in practice. Assume that we know rk T , rk T G , discr T G and discr(T G ) ⊥ , then the Boissière-Nieper-Wisskirchen-Sarti invariants can be computed using (1.3), Theorem 2.10 (v) and the proposition below.
As previously, these invariants can be used to compute the cohomology of G. The following proposition is given in [Men18,Proposition 4.1] for p ≤ 19.
Lemma 2.13. Let T be a p-torsion-free Z-module of finite rank endowed with the action of an automorphism group G of prime order p. Then for all i ∈ N * : Proof. The proof of (i) and (ii) is identical to the one given in [Men18,  We adopt the specific following notation when T is a cohomology group.
Notation 2.14. Let X be a topological space endowed with the action of an automorphism group G of prime order p. We assume that H k (X, Z) is finitely generated for all k ≥ 0. Then, for all k ∈ N, we denote Remark 2.17. Note that q (H k f (X, Z) ⊗ F p ) = 0 for all 1 < q < p − 1 because of Theorem 2.10 (iv). Hence there is no interest to define k q,f (X) for 1 < q < p − 1. Also the terms are not relevant because they coincide when p = 2 leading to a loss of information. This is why we use k − (X) and k + (X) instead (see the proposition below for the different relations between the invariants).
The universal coefficient theorem and Theorem 2.10 provide the following proposition.
Proposition 2.18. For all k ∈ N, we have: Proof. By the universal coefficient theorem, we have, for all k ∈ N, the following two exact sequences: Moreover, the maps of these exact sequences are morphisms of G-module (see for instance [Hat01,p. 196]). Furthermore, we have the canonical isomorphisms (which respect the G-module structure): Moreover from (2.5), we have the following isomorphisms of G-module: . So, we obtain with (2.6) the following isomorphism of G-module: . Then, the results follow from Proposition 2.10.
We can also express the cohomology of G with coefficients in H k (X, Z).
Proposition 2.19. For all k ≥ 0 and i > 0, we have: Proof. Since H k f (X, Z) and tors p H k (X, Z) are stable under the action of G, we have: ), for all k and i in N. Then, the result follows from Lemmas 2.2 and 2.13.

Boissière-Nieper-Wisskirchen-Sarti invariants and Lefschetz fixed point theorem
The following proposition is due to Simon Brandhorst. Among others, it has allowed to simplify the proof of Theorem 4.9 via Corollary 2.23.
Proposition 2.20. Let X be a compact complex manifold and G an automorphism group of prime order p. Then Proof. Let n = dim C X and g ∈ G be a generator, the Lefschetz fixed point theorem provides: By Theorem 2.5, we have an isomorphism of Z[G]-modules: where r, s, t are integers, A i , A j are ideals of O and a j (ξ p − 1)A j , for all 1 ≤ j ≤ r and 1 ≤ i ≤ s. By definition of the Boissière-Nieper-Wisskirchen-Sarti invariants (Notation 2.6, 2.14 and Theorem 2.10 (i)), it can be rewritten: Hence, after tensorizing by R, we obtain an isomorphisms of R[G]-module: Furthermore, the minimal polynomial of g |O⊗R is X p−1 + X p−2 + ... + 1 and the minimal polynomial of Because of their degrees the previous polynomials are also the characteristic polynomials and we obtain: tr(g |O⊗R ) = −1 and tr(g |(O,a j )⊗R ) = 0. Then (2.8) becomes: Remark 2.21. Note that Proposition 2.20 remains true if we assume that X is a 2n-dimensional compact connected orientable C ∞ -manifold and G an automorphism group of prime order p with only isolated fixed points such that X behaves around the fixed points of G = g as a complex manifold and G as a biholomorphic morphism group (i.e.: for each fixed point x there exists an open set x ∈ U x and a diffeomorphism f : Remark 2.22. Note that we can find again the generalized Hurwitz formula χ(X) = pχ(X/G)+(1−p)χ(Fix G) by combining Proposition 2.20 and Theorem 2.10 (v).
The next corollary is a direct consequence of Proposition 2.20. It will be very useful in practice to deal with the spectral sequence of equivariant cohomology (see Section 4.5).
Corollary 2.23. Let X be a compact complex manifold and G an automorphism group of prime order p with only η(G) isolated fixed points. Then:

Application to the degeneration of the spectral sequence of equivariant cohomology
Let X be a CW-complex and G an automorphism group on X (G permutes the cells). Let EG → BG be an universal G-bundle in the category of CW-complexes. Denote by X G = EG × G X the orbit space for the diagonal action of G on the product EG × X and f : X G → BG the map induced by the projection onto the first factor. The map f is a locally trivial fiber bundle with typical fiber X and structure group G. We define the G-equivariant cohomology of X with coefficients in a ring Λ (in this paper Λ is Z or F p with p a prime number) by H * G (X, Λ) := H * (EG × G X, Λ). In particular, observe that if G acts freely on X, then the canonical map EG × G X → X/G is a homotopy equivalence and so we obtain H * G (X, Λ) = H * (X/G, Λ).
Moreover, the Leray-Serre spectral sequence associated to the map f gives a spectral sequence converging to the equivariant cohomology (see [Bro82, Chapter VII, Section 7]): . In particular, the degeneration of this spectral sequence at the second page has interesting consequences (see for instance [Men18, Theorem 1.1]). We also recall the following result of Boissière, Nieper-Wisskirchen and Sarti which will be used several time in this paper.
. Let X be a compact connected orientable C ∞ -manifold of dimension n and G an automorphism group of prime order p. If the spectral sequence of equivariant cohomology with coefficients in F p degenerates at the second page, then: It is interesting to compare the previous equation with the equation provided by the Lefschetz fixed point theorem (Proposition 2.20).
Corollary 2.25. Let X be a compact complex manifold and G an automorphism group of prime order p. We assume that the spectral sequence of equivariant cohomology with coefficients in F p degenerates at the second page. Then: Adding or subtracting the equation of Proposition 2.20, we obtain our result.
We have the following particular case when G has only isolated fixed points (this result will be useful in Section 4.5).
Corollary 2.26. Let X be a compact complex manifold and G an automorphism group of prime order p with only isolated fixed points. We assume that the spectral sequence of equivariant cohomology with coefficients in F p degenerates at the second page. Then: 2 * +1 + (X) = 2 * − (X) = 0. Using Propositions 2.4, 2.19 and 2.18, we can compare the spectral sequences with coefficients in Z and F p . The end of this section is devoted to the proof of the following proposition.
Proposition 2.27. Let X be a CW-complex endowed with the action of an automorphism group G of prime order p. We assume that H k (X, Z) is finitely generated for all k ∈ N.The following statements are equivalent.
-The spectral sequence of equivariant cohomology of (X, G) with coefficients in F p degenerates at the second page. -The spectral sequence of equivariant cohomology of (X, G) with coefficients in Z degenerates at the second page.
From now and until the end of this section, X is a CW-complex endowed with the action of an automorphism group G of prime order p, with H k (X, Z) finitely generated for all k ≥ 0. We first introduce a tool to measure the degeneration of our spectral sequence. We define the dimensions of degeneration as Definition 2.28. We define the k th dimension of degeneration for Z-coefficients (resp. for F p -coefficients) by the non-negative integer u k (X): (resp. u k (X) such that): Remark 2.29. As, we will see below, the role of these integers will be to measure the distance of the spectral sequence from being degenerate at the second page. Note that the terms with d = 0 is slightly different between u k (X) and u k (X). It is because, the E 0,k r are not F p -vector spaces. We can write Hence, it is more practical to avoid these terms in the definition of u k (X).
For a better understanding, we recall the shape of the second page of our spectral sequence.
Lemma 2.30. The spectral sequence of equivariant cohomology with coefficients in Z (resp. in F p ) degenerates at the second page if and only if u k (X) = 0 (resp. u k (X) = 0) for all k ∈ N.
Proof. We prove the result for the spectral sequence of equivariant cohomology with coefficient in Z (the proof is identical for the spectral sequence with coefficient in F p ). One direction is trivial, if the spectral sequence degenerates at the second page then u k (X) = 0 for all k ∈ N.
Now, we assume that the spectral sequence does not degenerate at the second page and we will show that there exist k ∈ N such that u k (X) > 0. First note that (dim F p E d,q r ) r is a decreasing sequence for all d > 0 and q ≥ 0. If the spectral sequence does not degenerate at the second page then there exists a differential at a page E r , r ≥ 2, which is not trivial: is not trivial it means that Im δ We also mention the following property which will be used in Section 4.5.
Of course it is true for r = 2. Assume that it is true for r, we will prove it for r + 1. We have: . Hence, we would have: This contradicts u k−1 (X) = 0. With exactly the same argument, we have that δ l = 0. It follows that E d,q r = E d,q r+1 , which ends the proof.
Proof. Let r ≥ 2. The proof is exactly the same as the one of Lemma 2.31 with an additional complication which imposes k ≥ n + 2. Indeed, as explain in Remark 2.29, knowing that u k−1 (X) = 0 does not imply anything on E 0,k−1 r / tors p E 0,k−1 r since this group is not "encoded" in u k−1 (X). In particular, we can have u k−1 (X) = 0 and the differential δ : E 0,k−1 r → E r,k−r r which is not trivial (note that even if we would have added the term rk(E 0,k−1 2 / tors p E 0,k−1 2 ) in the definition of u k−1 (X) it would not have solved this problem). To avoid this problem, we need that E 0,k−1 2 = 0. This is the case if k ≥ n + 2 by Proposition 2.19 (i).
We can express the difference between the two kinds of dimensions of degeneration using our invariants.
Lemma 2.33. We have: Proof. The equation of the lemma is a consequence of the universal coefficient theorem and our computations of the cohomology of the group G. By Proposition 2.4: By Proposition 2.19: where the notation t k p is defined in Section 1.4 (ix). Moreover the universal coefficient theorem provides a relation between the two kinds of dimensions of degeneration. Indeed, by (2.7), we have for k ∈ N: . By Proposition 2.19 (i) and Theorem 2.10 (v), we have: . It follows from (2.11), (2.12) and Proposition 2.18 that: We obtain the same result for odd degrees, hence we have for all k ≥ 0: Then, we obtain our result by (2.10).
Finally, Proposition 2.27 is a direct consequence of Lemmas 2.33 and 2.30.

Application to the cohomology of quotients when p > 19
Because of Theorem 2.10, all the statements of [Men18] can be stated without any restriction on the prime number p. In particular, we have the following propositions which will be used several times in the paper.
Proposition 2.34. Let X be a compact connected orientable manifold and G be an automorphism group of prime order p. We denote by π : X → X/G the quotient map. Let T be a unimodular sublattice of H * (X, Z) stable under the action of G; then: discr π * (T ) f = p + (T ) .
. Moreover, if X/G is smooth the previous inequality becomes an equality.
Proof. As before, it is a direct consequence of [Men18, Corollary 3.10], since we have shown in Theorem 2.10 for all prime number p. Remark 2.36. More generally, in all the statements [Men18, Propositions 2.9, 3.14 and 5.2], the assumption on the prime number p can be removed.
The main results of [Men18] can also be stated without any restriction on the prime number p. We recall the definition of simple fixed points of an automorphism group of a complex manifold [Men18, Definition 5.9]. A fixed point x is said to be simple if the local action of G around x corresponds to the action of one of the diagonal matrices diag(1, ..., 1, ξ p , ..., ξ p ) in 0 ∈ C n with ξ p a p th root of the unity. If Fix G has several connected components, we denote by Codim Fix G the codimension of the component of higher dimension.
Theorem 2.37. Let X be a compact complex manifold of dimension n and G an automorphism group of prime order p. Let c := Codim Fix G. Assume that  Remark 2.39. By Corollary 2.25, the numerical condition (ii) of the previous theorems can be replaced by "the spectral sequence of equivariant cohomology with coefficients in F p degenerates at the second page".

Reminders on toric geometry
Our main references are [Dan78], [Ful93] and [Oda88]. Let M be a lattice. A set σ in M Q := M ⊗ Q is called a cone, if there exist finitely many vectors v 1 , . . . , v n ∈ M such that σ = Q + v 1 + · · · + Q + v n . The dimension of σ is defined to be the dimension of the subspace Vect(σ ). If H ⊂ M Q is a hyperplane which contains the origin 0 ∈ M Q such that σ lies in one of the closed half-spaces of M Q bounded by H, then the intersection σ ∩ H is again a cone which is called a face of σ . If {0} is a face of σ , we say that σ has a vertex at 0. Let σ be a cone, we denote by -every cone σ ∈ Σ has a vertex at 0; -if τ is a face of a cone σ ∈ Σ, then τ ∈ Σ; -if σ , σ ∈ Σ, then σ ∩ σ is a face of both σ and σ . -We define the support of Σ by |Σ| = σ ∈Σ σ and we say that Σ is complete if |Σ| = M Q . -Let Σ be another fan such that every cone of Σ is contain in some cone of Σ and |Σ | = |Σ|; we call Σ a subdivision of Σ. -Let σ ∈ Σ be a cone. We say that σ is regular according to M if it is generated by a subset of a basis of M. We say that Σ is regular according to M if every σ ∈ Σ is regular according to M. -Let Σ(1) be the cones of Σ of dimension 1. The Stanley-Reisner ring R Σ of Σ is the commutative ring generated by elements x σ with σ ∈ Σ(1) and the relations x σ 1 x σ 2 · · · x σ r = 0 for all distinct σ 1 , σ 2 , . . . , σ r ∈ Σ(1) that do not generate a cone of Σ.

Definition 3.4 (Toric variety).
Let M and N be lattices dual to one another, and let Σ be a fan in N Q . With each cone σ ∈ Σ we associate an affine toric variety X σ ∨ = Spec C[σ ∨ ∩ M]. By [Dan78, Section 2.6.1], if τ is a face of σ , then X τ ∨ can be identified with an open subvariety of X σ ∨ . These identifications allow us to glue together the X σ ∨ (as σ ranges over Σ) to form a variety, which is denoted by X Σ and is called the toric variety associated to Σ and N . (1) X Σ is complete if and only if Σ is complete.
(2) X Σ is smooth if and only if Σ is regular according to N .
In this section we will be interested in the integral cohomology of toric varieties, for this reason we recall the following well known result. It can be shown that this property characterizes toric varieties: if a normal variety X contains a torus T as dense open subvariety, and the action of T on itself extends to an action on X then X is of the form X Σ . Considering this action, an important tool is the equivariant cohomology of X Σ under the action of T denoted by H * T (X Σ , Z) (see Section 2.4 for the definition of equivariant cohomology). This tool is used to prove the next proposition which will be needed in Section 3.3.
Proposition 3.7. Let Σ and Σ be two regular fans such that Σ ⊂ Σ. Let X Σ and X Σ be the associated toric varieties. Let j : X Σ → X Σ be the natural embedding. Assume that H * (X Σ , Z) and H * (X Σ , Z) are concentrated in even degrees, then j * : Proof. The proof is based on a well known result that we will recall here. For each 1-dimensional cone σ ∈ Σ, we can construct a T -invariant divisor V (σ ) in X Σ (see for instance [Ful93, Section 3]). Let R(Σ) be the Stanley-Reisner ring of Σ. Then there is a natural ring morphism: T (X Σ , Z) defined on the 1-dimensional cones, by sending x σ to the equivariant cohomology class associated to the divisor V (σ ). When X Σ is smooth, c Σ is an isomorphism (this is due to [Bri96, Section 2.2] where the result is stated with rational coefficients, however the proof is still true considering integral coefficients; see [Fra06, Sections 2.3 and 3] for the same result in a more general setting). Then, we have the following commutative diagram: The map R(Σ) → R(Σ ) is surjective because Σ ⊂ Σ. Moreover, the maps H * T (X Σ , Z) → H * (X Σ , Z) and H * T (X Σ , Z) → H * (X Σ , Z) are surjective because the cohomologies of X Σ and X Σ are concentrated in even degrees (see for instance [Fra10, Lemma 5.1]). By commutativity of the diagram, it follows that j * : H * (X Σ , Z) → H * (X Σ , Z) is surjective.

Definition of toric blow-ups For toric varieties.
Let X Σ be a toric variety. By Theorem 3.5 (2), to resolve the singularities of X Σ , we only need to consider Σ a regular subdivision of Σ. In [Dan78, Section 8.2], Danilov explains that we can always find a regular subdivision Σ such that f : X Σ → X Σ verifies the following properties: f is an isomorphism over the smooth locus of X Σ ; f is a projective morphism.
Such a transformation f is called a toric blow-up of X Σ .
Example 3.8. The variety C n is an affine toric variety given by the lattice M := Z n and the cone σ = (Q + ) n . Let G ⊂ GL(n, C) be a finite abelian group with n > 1. Since G is finite abelian group, there exists a basis of C n in which all the elements of G can be expressed as diagonal matrices. Let g ∈ G, we have g = diag(ξ a 1 , ..., ξ a 1 ), with ξ an m th root of unity and 0 ≤ a i ≤ m. Let M g := (x 1 , ..., x n ) ∈ M | n i=0 x i a i ≡ 0 mod m and M G := ∩ g∈G M g . The quotient C n /G is also an affine toric variety given by the lattice M G and the cone σ . In particular, C n /G is the toric variety associated to the fan Σ in (M G ) ∨ Q containing the cone σ ∨ and all its faces. Hence, the singularities of C n /G can be resolved by a toric blow-up.  (W x , W , G, h) is called a local uniformizing system of x.
Let X be a topological space. Let x ∈ X be an isolated complex quotient point and (W x , W , G, h) a local uniformizing system of x such that G is an abelian group. Let f : C n /G → C n /G be a toric blow-up of We obtain a map X → X that we call a toric blow-up of X in x according to (W x , W , G, h).
Remark 3.10. As defined here, a toric blow-up is not unique. In order to define the toric blow-up of an orbifold with any singularities, we would need to require some universal properties for our toric blow-up.

Integral cohomology of toric blow-ups of C n /G, with G a cyclic group
In this section, we use the notation of Example 3.8. Let n > 1 and G ⊂ GL(n, C) be a finite group of prime order p with only 0 as fixed point. The action of G extends to an action on P n . Let ξ p be a p th root of the unity. Without loss of generality we can assume that G = φ with φ = diag(ξ α 1 p , . . . , ξ α n p ) and 1 ≤ α i ≤ p − 1 for all i ∈ {1, . . . , n}. It provides: P n / / P n (a 0 : a 1 : · · · : a n ) / / (a 0 : ξ α 1 p a 1 : · · · : ξ α n p a n ).
Then (1 : 0 : · · · : 0) is an isolated fixed point of the action of G on P n ; we denote 0 := (1 : 0 : · · · : 0). If we identify C n with the chart a 0 0, this action on P n is an extension of the action on C n . Hence, if we denote by Σ the fan of P n /G the natural embedding C n /G → P n /G corresponds to the inclusion of the fans Σ ⊂ Σ.
Let f : C n /G → C n /G and f : P n /G → P n /G be toric blow-ups of C n /G and P n /G respectively such that they coincide in 0; that is the cone σ ∨ is subdivided in the same way in Σ and Σ. Let Σ and Σ be the fans of P n /G and C n /G, it follows an inclusion Σ ⊂ Σ and an open embedding j : C n /G → P n /G.
Let Σ * = Σ σ ∨ ; it is the fan of P n /G {0}. We consider Σ * the regular subdivision of Σ * such that Σ * ⊂ Σ ; it is the fan of P n /G * := P n /G f −1 (0) = P n /G j(f −1 (0)). We denote i : P n /G * → P n /G the inclusion. We also denote (C n /G) Remark 3.11. By Theorem 3.6, H * P n /G, Z is torsion-free and concentrated in even degrees.
Proposition 3.12. We have: Proof. The space (C n /G) * is homotopy equivalent to S 2n−1 /G. The space S 2n−1 /G is the lens space L(n − 1, p) and its cohomology is well known.
Proposition 3.13. We have: Proof. The quotient C n /G is contractible, so its cohomology is concentrated in degree 0: Then, the result can be obtained from the exact sequence of relative cohomology of the pair (C n /G, (C n /G) * ): Hence for k ≥ 1, we have: We obtain our result for k > 1 using Proposition 3.12. It only remains to compute H 0 (C n /G, (C n /G) * , Z) and H 1 (C n /G, (C n /G) * , Z). The previous exact sequence provides: Proposition 3.14. The cohomology groups H * P n /G * , Z and H * C n /G, Z are torsion-free and concentrated in even degree.
Proof. We first show that H 2k−1 P n /G * , Z = H 2k−1 C n /G, Z = 0 for all 1 ≤ k ≤ n. The main idea of the proof is to apply Proposition 3.12 and Remark 3.11 to the long exact sequence of relative cohomology of the couples P n /G, P n /G * and C n /G, (C n /G) * .
Assume first that k < n. We consider the following commutative diagram of embeddings: which induces the following commutative diagram on the cohomology: By Remark 3.11, we have: H 2k−1 P n /G, Z = 0 and by Proposition 3.12, we have: By commutativity of the diagram, the map H 2k−1 P n /G * , Z → H 2k P n /G, P n /G * , Z is necessarily 0. It follows that: For the same reason, the map H 2k−1 C n /G, (C n /G) * , Z → H 2k−1 C n /G, Z is also trivial. So When k = n. The exact sequence obtained from (3.1), is slightly different. (The coefficient ring of the cohomology is Z; we do not write it to avoid a too large diagram.) We have H 2n P n /G, Z = Z because P n /G is smooth and compact; C n /G and P n /G * being open submanifolds of a compact complex manifold of dimension n, we get H 2n P n /G * , Z = H 2n C n /G, Z = 0.
So H 2n−1 P n /G * , Z = 0. By commutativity of the diagram, we also have γ = 0. So (3.3) implies that The cohomology is torsion-free. The varieties P n /G * and C n /G are smooth toric varieties. Moreover we have seen that their integral cohomology of odd degrees is trivial. Therefore, [Fra10, Proposition 1.5] shows that H * P n /G * , Z and H * C n /G, Z are torsion-free.
Remark 3.15. In [BBFK98], more general results related to the cohomology with rational coefficients of toric varieties can be found.

G. Menet 24 G. Menet
Proof. Indeed, using Propositions 3.12, 3.13 and 3.14, the relative cohomology exact sequences of the pairs (C n /G, (C n /G) * ) and C n /G, (C n /G) * provide the following commutative diagram: The last result of this section makes precise how a toric blow-up modifies the cohomology.
Theorem 3.17. We consider the following exact sequence: with 1 ≤ k ≤ n − 1. Then: (iii) H 2k P n /G, P n /G * , Z is torsion-free and H 2n P n /G, P n /G * , Z = Z, (v) If n is even, then Im g n is a sublattice of H n P n /G, Z of discriminant p.
Outline of the proof.
The statements (i), (ii) and (iii) will be easily obtained from Proposition 3.14 and the long exact sequence of relative cohomology of P n /G, P n /G * . We provide an outline of the proof of (v); the proof of (iv) being similar.
-Using the long exact sequence of relative cohomology of P n /G, P n /G * and Proposition 3.14, we can show that: Im g n is primitive in H n P n /G, Z .
-However, using long exact sequence of relative cohomology of P n /G, (C n /G) * and Proposition 3.12, we can show that Im g n ⊕ (Im g n ) ⊥ is not primitive in H n P n /G, Z . We have: H n P n /G, Z / Im g n ⊕ (Im g n ) ⊥ = Z /p Z .
-Since H n P n /G, Z is unimodular, we are able to deduce (v) from the lattice results of Section 1.6.
Proof of (i), (ii) and (iii). Let 1 ≤ k ≤ n − 1. The statement (i) is a direct consequence of Proposition 3.14 looking at the following exact sequence: Moreover, from Proposition 3.7 and 3.14, we know that the map i * is surjective. It follows: (3.6) H 2k+1 P n /G, P n /G * , Z = 0.
The previous exact sequence also provides that H 2k P n /G, P n /G * , Z is torsion-free since H 2k P n /G, Z is torsion-free by Remark 3.11. In addition, we have H 2n P n /G, P n /G * , Z = Z by (3.4).
Now, (iv) follows from the following commutative diagram: The zeros in the diagram come from (3.6), (3.7), (3.8), Propositions 3.12 and 3.14, and Remark 3.11. Moreover the diagram shows that the exact sequence splits. Hence the maps p 2k and p 2k are the natural embeddings: By commutativity of the diagram, it follows: (3.10) Im g 2k = Im g 2k ⊕ Im g 2k .
By Proposition 3.14, H 2k P n /G * , Z and H 2k C n /G, Z are torsion-free. It follows from the diagram that Im g 2k and Im g 2k are primitive sub-groups in H 2k P n /G, Z . However, H 2k ((C n /G) * , Z) = Z /p Z according to Lemma 3.12. This means that: So by (3.10): We are considering 1 ≤ k ≤ n − 1, so in particular, the same result is true for Im g 2(n−k) and Im g 2(n−k) in H 2(n−k) P n /G, Z . This means: with the orthogonality which is due to P n /G = P n /G * ∪ C n /G. Indeed the cup-product in relative cohomology is a map: By Poincaré duality, H 2k P n /G, Z ⊕ H 2(n−k) P n /G, Z is unimodular, it follows from (1.3), (1.4) and the primitivity of Im g and Im g that: discr Im g 2k ⊕ Im g 2(n−k) = discr Im g 2k ⊕ Im g 2(n−k) = p 2 .
The statement (v) is a particular case of statement (iv) when n is even. Assume n is even, we also get that Im g n and Im g n are primitive in H n P n /G, Z with Im g n which admits a primitive element divisible by p.

Hence:
H n P n /G, Z Im g n ⊕ ⊥ Im g n = Z /p Z .
As before, the unimodularity of H n P n /G, Z provides: discr Im g n = discr Im g n = p.
Remark 3.18. We can also mention that H 0 P n /G, P n /G * , Z = H 1 P n /G, P n /G * , Z = 0 because the map i * : H 0 P n /G, Z → H 0 P n /G * , Z is an isomorphism.

Application to the integral cohomology of the toric blow-up of isolated quotient singularities
Now, we apply the previous result to understand better how a toric blow-up modifies the cohomology.
Corollary 3.19. Let M be a topological space with an isolated complex quotient point x ∈ M. We assume that x admits a local uniformizing system (W x , W , G, h) with G of prime order. Let r : M → M be a toric blow-up of M in x. We denote U x := M r −1 (x) and n := dim W . We consider the following exact sequence: with 1 ≤ k ≤ n − 1. Then: Proof. The statements (ii) and (iii) are immediate consequence of the excision theorem: and Theorem 3.17 (ii), (iii). The statement (i) is a consequence of the following commutative diagram and Proposition 3.12. (3.11) The previous corollary allows to describe the integral cohomology of a toric blow-up in several isolated points.  Proof. For 1 ≤ k ≤ n − 1, we consider the following exact sequence (the coefficient ring Z is omitted for clarity sake). (3.12) where δ k,n−1 is the Kronecker delta. The relative cohomology groups in the sequence are given by Corollary 3.19 and Proposition 3.13. The following part of (3.12) provides (i): By commutativity of (3.12), the maps r * : H 2k (M) → H 2k M and g 2k are injective. Hence, we can extract from (3.12), the following exact sequence: This means that the map j 2k induces the following isomorphism: By commutativity of (3.12), we obtain the following isomorphism: Hence (ii) follows from (i). By Lemma 3.16, the map f 2k+2 is always an isomorphism. It follows from the commutativity of Diagram (3.12) that: Therefore (iii) follows from (i) and the following part of (3.12): Moreover, since g 2 is injective by Corollary 3.19, by Proposition 3.13 and Remark 3.18, we also obtain the following diagram: which provides (iv). Statement (v) can also be proved using the commutativity of (3.12) and the bijectivity of f 2n looking at this part of the diagram: The d k p will be explicitly computed when M is a quotient in the proof of Theorem 4.12.
x is a sublattice of H 2k M, Z ⊕ H 2n−2k M, Z of discriminant p 2 .
(ii) If n is even, then Im g n x is a sublattice of H n M, Z of discriminant p.
Proof. This corollary is also a consequence of the excision theorem and Theorem 3.17 (iii), (iv) and (v). By construction of the cup product, we have a commutative diagram (see for instance [Hat01,p. 209]): where "∪" is the cup product. Since H 2n P n /G, P n /G * , Z = H 2n M, U x , Z = Z by Theorem 3.17 and Corollary 3.19, the cup product on H 2 * P n /G, P n /G * , Z and on H 2 * M, U x , Z can be seen as a bilinear form. By (3.2) and (3.5), g 2n : H 2n P n /G, P n /G * , Z → H 2n P n /G, Z is an isomorphism. Therefore, the commutativity of the previous diagram shows that g 2 * : H 2 * P n /G, P n /G * , Z → H 2 * P n /G, Z is an isometry.
We show that the same property also holds for g 2 * x : H 2 * M, U x , Z → H 2 * M, Z . As before, we have a commutative diagram: Moreover, we have the following exact sequence: Since M is an 2n-dimensional compact connected orientable manifold, we have H 2n (U x , Z) = 0. Since H 2n M, U x , Z = Z, we have that g 2n x is necessarily bijective. Therefore, g 2 * x : H 2 * M, U x , Z → H 2 * M, Z is an isometry.
Finally, the excision theorem provides an isometry H 2 * P n /G, P n /G * , Z H 2 * M, U x , Z . Hence Theorem 3.17 (iv) and (v) conclude the proof.

Notation, hypothesis and definition
In this section X is a compact complex manifold of dimension n such that H * (X, Z) is p-torsion-free and G an automorphism group of prime order p with only isolated fixed points. Remark 4.2. Actually, all the results of this section remain true if we choose for X a 2n-dimensional compact connected orientable C ∞ -manifold, with H * (X, Z) p-torsion-free, with G an automorphism group of prime order p which respects an orientation which has only isolated fixed points and such that all points of Sing M are isolated complex quotient points (see Definition 3.9).
Proof. Statement (i) follows immediately from the fact that H k (V , Z) = H k (X, Z) for all k ≤ 2n−2. Moreover, we have the following exact sequence: Since H 2n−1 (X, Z) is p-torsion-free and by Thom's isomorphism H 2n (X, V , Z) = Z η(G) , H 2n−1 (V , Z) is also p-torsion-free. We obtain (ii).
From Lemma 4.5 (i) and (ii), we see that the spectral sequences of equivariant cohomology of (X, G) and (V , G) coincide sufficiently to obtain the following expressions for the dimensions of degeneration. We recall that the notation t 2k+1 p (Y ) for a topological space Y is defined in Section 1.4. Lemma 4.6. We have for all 1 ≤ k ≤ n − 1: 2i Proof. We prove (i), the proof of (ii) is identical. By Definition 2.28, we have: 2 is the second page of the spectral sequence of equivariant cohomology of (X, G) with coefficient in Z (see Section 2.4). First note that t p E 0,2k 2 = 0 by Proposition 2.19 since H * (X, Z) is p-torsion-free. Hence also t p E 0,2k ∞ = 0. By Proposition 2.19, Moreover, by convergence of the spectral sequence: In particular the previous proposition can be applied to a K3 surface. Let L 17 be the lattice Corollary 5.3. Let X be a K3 surface endowed with the action of an automorphism group G of prime order p. Assume that Fix G is finite and non-empty. Then for each p, the lattice H 2 (X/G, Z) is given by: For instance, Proposition 5.1 can also be applied to hypersurfaces in P 3 (see Proposition 5.19). We propose an example.
Example 5.4. Let X be the hypersurface of dimension 2 and degree p with equation: Let g be the automorphism on X induced by the following automorphism on P 3 : P 2n+1 / / P 2n+1 (a 0 : a 1 : a 2 : a 3 ) / / (a 0 : ξ p a 1 : ξ l p a 2 : a 3 ), with ξ p a p th root of the unity and l 1 or 0 mod p. We denote G = g . Then: (i) α 2 (X) = 0, (ii) H 2 (X/G, Z) and H 4 (X/G, Z) are torsion-free and H 3 (X/G, Proof. Statements (i) and (ii) are direct consequence of Proposition 5.1. The fixed points of G are given by X ∩ V (x 1 , x 2 ) which consists in p isolated points. The point (iv) follows from Proposition 5. b 2 (X) = p 3 − 4p 2 + 6p − 2.
We obtain statement (iii).

An example with a group of order higher than 19
If Fix G contains a curve then it is a rational curve. Indeed let C be a curve, then rk H 2 * f (C, Z) = 2. Hence, we obtain by (5.7) that H 1 (C, Z) = 0. Moreover in this case, by (5.8), Fix G cannot have any other connected component. If Fix G contains only isolated points, then (5.7) provides: that is Fix G contains only two isolated points. It proves (ii).

Examples of degenerations at the second page of equivariant cohomology spectral sequences
Let S be a K3 surface. We denote by S [m] the Hilbert scheme of m points on S. We recall that an automorphism group G on S [m] is said to be natural if it is induced by an automorphism group on the K3 surface S. Let X be a hyperkähler manifold of K3 [m] -type (equivalent by deformation to a Hilbert scheme of m points on a K3 surface). A pair (X, G) is said to be standard if it is deformation equivalent to a pair (S [m] , G) with G a natural group.
Corollary 5.6. Let X be a hyperkähler manifold of K3 [m] -type and G an automorphism group of prime order 3 ≤ p such that (X, G) is a standard pair. Assume that G has only isolated fixed points, then the spectral sequence of equivariant cohomology with coefficients in F p degenerates at the second page.
Let φ be an automorphism on S. For simplicity, we also denote by φ the induced automorphism on S [d] for all d > 0 and on S [l+q] × S × S [l] .
Proof of Corollary 5.6. Let (X, G) be a standard pair. We are going to prove that 2 * +1 + (X) = 0 and apply Theorem 4.9 (iii). The Z[G]-module structure of H * (X, Z) is invariant under deformation. Hence the k + are also independent under deformation for all integer k. Hence, without loss of generality, we can assume that X = S [m] and G is natural.

Quotients of K3 [m] -type hyperkähler manifolds by symplectic automorphisms of order 5 and 7
If we consider φ a symplectic automorphism of order 5 (resp. 7) on S, then the fixed locus of the induced automorphism φ [m] on S [m] has only isolated fixed points when m ≤ 4 (resp. m ≤ 6). Moreover, it is shown in [Mon13a, Theorem 7.2.7, Section 7.3] and [Mon13b, Theorem 2.5] that all the symplectic automorphisms of order 5 and 7 on a hyperkähler manifold of K3 [m] -type with m ≤ 6 are standard. So by Remark 5.8, we obtain the following result.
Proof. The proof of this proposition is very similar to the one of [BNS13,Proposition 4.6]. We refer to [BNS13,Section 4.3] for the notion of G-linearisation. Since G is a linear automorphism group, the line bundle L := O X (1) is G-linearisable. Hence, we can consider L G := L× G EG, with EG → BG an universal G-bundle. Let f : X G → BG be the projection. We set u := c 1 (L G ) ∈ H 2 (X G , F p ) and q := Rf * u : Rf * F p → Rf * F p [2]. Let n be the dimension of X. By [Del68, Proposition 2.1], if q k : R n−k f * F p → R n+k f * F p is an isomorphism for all k, then the spectral sequence of equivariant cohomology with coefficients in F p degenerates at the second page. Since we are considering direct images of constant sheaves along locally trivial fibrations, we can check that the q k are isomorphisms fibrewise. That is, we need to prove that the Lefschetz maps L k : H n−k (X, F p ) → H n+k (X, F p ) are isomorphisms. This is given by hypothesis.
The condition on Lefschetz maps can easily be verified for smooth complete intersections. Let X be a smooth complete intersection of multidegree (a 1 , ..., a q ). We call q i a i the degree of X. We recall the following well known result.
Proposition 5.19. Let X be a smooth complete intersection of dimension n and degree d. Let D := c 1 (O X (1)). Then: if k is odd and k n, if k is even and k > n.
Moreover H n (X, Z) is torsion-free.
Remark 5.20. Let X be a smooth complete intersection of dimension n and degree d. Note that b n (X) can be computed via Riemann-Roch theorem. For instance if X is a hypersurface b n (X) = n k=0 (−1) k n + 2 k d n+1−k + (−1) n+1 2 n 2 .
It follows from the following corollary.
Corollary 5.21. Let X be a smooth complete intersection of degree d endowed with a finite automorphism group G which is linear according to some embedding. Let p be a prime number which does not divide d. Then the spectral sequence of equivariant cohomology of (X, G) with coefficients in F p degenerates at the second page.
Then, we are going to see that Theorem 2.37 and 2.38 can easily be applied to complete intersections.
Lemma 5.22. Let X be a smooth complete intersection endowed with a finite abelian group G which is linear according to some embedding. Then H * (Fix G, Z) is torsion-free.
Proof. Let X → P N be an embedding such that the automorphism group G extends to an automorphism group G on P N . Then we have: Let Y be a connected component of Fix G. By (5.22), there exists a connected component H of Fix G such that Y ⊂ H ∩X. Since H have degree 1, we can write H as an intersection of hyperplanes H = H 1 ∩· · ·∩H l . Then we can prove recursively that X ∩ H 1 ∩ · · · ∩ H l has its cohomology torsion-free; moreover X ∩ H 1 ∩ · · · ∩ H l is connected or of dimension 0. It is true for X, we assume that it is true for X ∩ H 1 ∩ · · · ∩ H i and we prove it for X ∩ H 1 ∩ · · · ∩ H i+1 . There are two possibilities or X ∩ H 1 ∩ · · · ∩ H i+1 is a hyperplane section of X ∩ H 1 ∩ · · · ∩ H i or X ∩ H 1 ∩ · · · ∩ H i ⊂ H i+1 . In the second case X ∩ H 1 ∩ · · · ∩ H i = X ∩ H 1 ∩ · · · ∩ H i ∩ H i+1 and there is nothing to prove. In the first case, this is a consequence of the hyperplane Lefschetz theorem and the universal coefficient theorem. Hence Y is a point or Y = H ∩ X. In the both cases, H * (Y , Z) is torsion-free. G. Menet So by Theorem 2.10 (v), we have b 5 (X/G) = 0 (we could also have used the generalized Hurwitz formula). Finally, as stated in Theorem 4.12, we have that H 3 (X/G, Z) ⊕ H 9 (X/G, Z) = (Z /43 Z) 7− 2 + (X) = (Z /43 Z) 6 and H 5 (X/G, Z) ⊕ H 7 (X/G, Z) = (Z /43 Z) 7− 4 + (X) = (Z /43 Z) 6 .