Smith theory and irreducible holomorphic symplectic manifolds

# Smith theory and irreducible holomorphic symplectic manifolds

## Abstract.

We study the cohomological properties of the fixed locus of an automorphism group of prime order acting on a variety whose integral cohomology is torsion-free. We obtain a precise relation between the mod cohomology of and natural invariants for the action of on the integral cohomology of . We apply these results to irreducible holomorphic symplectic manifolds of deformation type of the Hilbert scheme of two points on a K3 surface: the main result of this paper is a formula relating the dimension of the mod cohomology of with the rank and the discriminant of the invariant lattice in the second cohomology space with integer coefficients of .

###### Key words and phrases:
Smith theory, holomorphic symplectic manifolds, automorphisms
###### 1991 Mathematics Subject Classification:
Primary 14J50; Secondary 14C50, 55T10

## 1. Introduction

Smith theory is the study of the cohomological properties of a group of prime order acting on a topological space . The first important results were obtained by Smith in the late 1930’s by the introduction of the so-called Smith cohomology groups and sequences (see Bredon [10]). The use of equivariant cohomology to reformulate Smith theory was begun by Borel [8] in the 1950’s and further formalized as the “localisation theorem” of Borel–Atiyah–Segal–Quillen in the 1960’s (see Dwyer–Wilkerson [14]).

In this paper, we use these ideas to relate the dimension of the mod cohomology of the fixed point set to natural invariants for the action of on the integral cohomology for (see Corollaries 5.11 & 5.12). This applies nicely to the study of prime order automorphisms on some symplectic holomorphic varieties, particularly those in the deformation class of the Hilbert scheme of two points on a K3 surface . The first main result of this paper is a degeneracy condition for the spectral sequence of equivariant cohomology

 Er,s2:=Hr(G;Hs(X,Fp))⟹Hr+sG(X,Fp).
###### Theorem 1.1.

Let be a group of prime order acting by automorphisms on an irreducible holomorphic symplectic variety . The spectral sequence of equivariant cohomology with coefficients in  degenerate at the -term in the following cases:

1. is deformation equivalent to the Hilbert scheme of two points on a K3 surface and .

2. , acts by natural automorphisms (induced by automorphisms of the surface ) and .

This result is proven in Proposition 6.12 as a consequence of Deligne’s criterium (see Section 4.3) applied to specific geometrical objects in the cohomology of (Lemma 6.9, 6.10 & 6.11).

For deformation equivalent to , denote by the invariant lattice and by its orthogonal complement for the Beauville–Bogomolov bilinear form. We define (see Definitions 5.5 & 5.9) two integers with the property that

 H2(X,Z)TG(X)⊕SG(X)≅(ZpZ)aG(X),rankSG(X)=mG(X)(p−1).

The second main result of this paper is the following formula:

###### Theorem 1.2.

Let be deformation equivalent to and be a group of automorphisms of prime order on with , . Then:

 dimH∗(XG,Fp) =324−2aG(X)(25−aG(X))−(p−2)mG(X)(25−2aG(X)) +12mG(X)((p−2)2mG(X)−p)

with

 2 ≤(p−1)mG(X)<23, 0 ≤aG(X)≤min{(p−1)mG(X),23−(p−1)mG(X)}.

This formula is proven in Corollary 6.15. The proof uses first the localisation theorem as presented in Allday–Puppe [1] (see Proposition 4.2), secondly the degeneracy conditions for the spectral sequence of equivariant cohomology with coefficients in , then the determination of the -module structure of the cohomology space (Proposition 5.1), and finally the computation of the quotient (Proposition 6.6). The relation with the discriminant of the invariant lattice and its orthogonal is given in Lemma 6.5.

As an application of our results, we show in Section 6.5 that there are no free actions by finite groups on deformations of , and we study an order eleven automorphism on a Fano variety of lines of a cubic fourfold constructed by Mongardi [31].

Aknowledgements. We thank Olivier Debarre, Alexandru Dimca, William G. Dwyer, Viacheslav Kharlamov, Giovanni Mongardi, Kieran O’Grady and Volker Puppe for useful discussions and helpful comments.

## 2. Terminology and notation

Let be a prime number and a finite cyclic group of order . We fix a generator  of . Put and .

Let be a finite-dimensional -vector space equipped with a linear action of (a -module for short). The minimal polynomial of , as an endomorphism of , divides the polynomial hence admits a Jordan normal form. We can thus decompose as a direct sum of some -modules of dimension for , where acts on by a matrix (in a suitable basis) of the following form:

Observe that is isomorphic to as a -module. Throughout this paper, the notation will always denote the -module defined by the Jordan matrix of dimension above. We define the integer as the number of blocks of length  in the Jordan decomposition of the -module , in such a way that .

Let be a finite-dimensional graded -vector space, where each graded component is equipped with a linear action of . We define similarly, for any and , the integer as the number of blocks of length  in the Jordan decomposition of the -module .

For any topological space with the homotopy type of a finite CW-complex and any field , we set and .

Let be a smooth connected orientable compact real even-dimensional manifold, with a smooth orientation-preserving action of . Denote by the fixed locus of for the action of ; then is a smooth submanifold of . We define the integers for and as the number of blocks of length in the Jordan decomposition of the -modules and we set .

## 3. Some useful computations in group cohomology

There is a projective resolution of considered as a -module with a trivial action, given by:

 (1) ⋯⟶Z[G]τ→Z[G]σ→Z[G]τ→Z[G]ϵ→Z⟶0

where is the summation map: and , act by multiplication.

Let considered as a trivial -module. The cohomology groups of with coefficients in are the cohomology groups of the complex:

 0→HomG(Z[G],Fp)τ∗→HomG(Z[G],Fp)σ∗→HomG(Z[G],Fp)τ∗→⋯

Observe that by identifying a -homomorphism with its image , so and are identically zero and we get for all .

Let now be as before a -module of finite dimension over . The cohomology of with coefficients in can be computed in a similar way as the cohomology of the complex:

 0→M¯τ→M¯σ→M¯τ→⋯

where denote the reduction modulo of and . Observe that . To compute as an -vector space it is enough to compute the groups .

###### Lemma 3.1.

1. If then for all .

2. and for all .

###### Proof.

The case is clear since as a trivial -module. Assume now that . Let be a basis of such that and for all . It is easy to compute that, as endomorphisms of , one has and for all . Using that we get:

 ker¯σ={Nqif q

If the result is clear. If it follows from:

 ker(¯τ)Im(¯σ)≅⟨v1⟩,ker(¯σ)Im(¯τ)≅⟨vq⟩.

Recall (see [11, Ch. V]) that the cohomology cross-product:

 Hr(G;Fp)⊗ZHs(G;M)⟶Hr+s(G×G;Fp⊗ZM)

followed by a diagonal approximation:

 Hr+s(G×G;Fp⊗ZM)Δ∗−→Hr+s(G;Fp⊗ZM)≅Hr+s(G;M)

defines a cup-product and a graded -module structure on , where is considered as a -module for the diagonal action (and is isomorphic to as a -module since acts trivially on ). Here the diagonal approximation is induced by the maps given by:

 Δr,s(1)=⎧⎪⎨⎪⎩1⊗1for r even1⊗gfor r odd, s even∑0≤i

Let and . Using again the natural identifications and one computes easily the cup-product as follows:

(i) If is even, .

(ii) If is odd and is even, one has (see the proof of Lemma 3.1) so and .

(iii) If is odd and is odd,

 α∪β=α⋅((g+2g2+⋯+(p−1)gp−1)β).

We study the action of on for .

###### Lemma 3.2.

As an endomorphism of , with , one has:

 g+2g2+⋯+(p−1)gp−1=⎧⎨⎩0if q≤p−2,−¯τq−1if q=p−1,−¯τq−1−¯τq−2if q=p.
###### Proof.

One computes:

 p−1∑i=1igi =p−1∑i=1i∑j=0i(ij)¯τj=p−1∑j=0(p−1∑i=ji(ij))¯τj=p−1∑j=0(p−1−j∑k=0(j+k)(jj+k))¯τj =p−1∑j=0(j(pj+1)+(j+1)(pj+2))¯τj

where the last equality follows from an easy induction on (for any integer ). By reduction modulo , all binomial coefficients vanish for so:

 p−1∑i=1igi=−¯τp−1−¯τp−2.

Since on , the result follows. ∎

In the special case , in case (iii) one obtains if and if . It follows that, as a graded commutative algebra:

where , , and denotes the exterior algebra over generated by (see [1, Proposition 1.4.2]).

###### Proposition 3.3.

is a trivial -module. For , is a free -module generated by .

###### Proof.

This follows from Lemma 3.1 and the discussion above. The cases or are clear. In the case and , for and with odd and odd, following the notation used in the proof of Lemma 3.1,  can be represented by a class with . Since , using Lemma 3.2 one gets in case . The result follows. ∎

We denote by the polynomial part of (that is: for and for ). We consider as a -module by evaluating at zero (setting for and for ). For any , we consider as a -module by the inclusion .

###### Corollary 3.4.

1. For and , one has and for , .

2. For and , one has and for , .

3. For , one has:

 dimFpTorR0(H∗(G;Np),Fp)=1=dimFpTorR1(H∗(G;Np),Fp)=1,

and for , .

###### Proof.

There is a length 2 projective resolution of as a -module given by:

 0⟶Rϕ⟶R⟶Fp⟶0

where is the multiplication by for and by for , so for and .

(a) Assume that . By Proposition 3.3, is a free -module so for . Recall that:

 TorR0(H∗(G;Nq),Fp)≅H∗(G;Nq)⊗RFp.

For , is generated by any non zero element as a -module so ; for , is again generated by any non zero as a -module, so is generated by and  as a -module: this gives .

(b) Take . From the length 2 resolution of as a -module, using Proposition 3.3 one gets:

 TorR1(H∗(G;Np),Fp)≅ker(ϕ:H∗(G;Np)→H∗(G;Np))=H∗(G;Np).

By Lemma 3.1 this space is one-dimensional so:

 TorR0(H∗(G;Np),Fp)≅H∗(G;Np)⊗RFp≅Fp.

## 4. Equivariant cohomology

### 4.1. Basic facts on equivariant cohomology

Let be a universal -bundle in the category of CW-complexes. Denote by the orbit space for the diagonal action of on the product and the map induced by the projection onto the first factor. The map is a locally trivial fibre bundle with typical fibre and structure group . The equivariant cohomology of the pair with coefficients in is defined by , naturally endowed with a graded -module structure. Note that there is an isomorphism of graded algebras . The Leray–Serre spectral sequence associated to the map gives a spectral sequence converging to the equivariant cohomology with coefficients in :

 Er,s2:=Hr(G;Hs(X,Fp))⟹Hr+sG(X,Fp).
###### Remark 4.1.

By assumption has the homotopy type of a finite -CW-complex. Denote by the cellular cochain complex of with coefficients in . The spaces are finitely dimensional -vector spaces. Recall that denotes the projective resolution of as a trivial -module, and define the double complex . As is quasi-isomorphic to in the derived category of -modules, the cohomology of the total complex, the cohomology of the total complex computes the equivariant cohomology (see Allday–Puppe [1, Theorem 1.2.8]): . This yields a concrete description of the first quadrant spectral sequence converging to the equivariant cohomology.

### 4.2. Cohomology of the fixed locus

Recall that denotes the polynomial part of . We prove the following formula (see Allday–Puppe [1] for related results):

###### Proposition 4.2.

For one has:

 h∗(XG,Fp)=ν⋅(dimFpTorR0(H∗G(X,Fp),Fp)−dimFpTorR1(H∗G(X,Fp),Fp))

with for and for .

###### Proof.

The graded -module is of finite type so it admits a minimal free resolution [1, Proposition A.4.12]:

 0⟶L1⟶L0⟶H∗G(X,Fp)⟶0

such that . Write (with of degree one if and of degree two if ). For , define . This is consistent with the previous description as an -module. For , the functor is exact [1, Lemma A.7.2] so:

 dimFpH∗G(X,Fp)⊗RFp,α =dimFpL0⊗RFp,α−dimFpL1⊗RFp,α =rankRL0−rankRL1 =dimFpTorR0(H∗G(X,Fp),Fp) −dimFpTorR1(H∗G(X,Fp),Fp).

For , one has (this cohomology is computed with the total differential). We now use the following analogue of the localisation theorem in equivariant cohomology [1, Theorem 1.3.5, Theorem 1.4.5]: for , one has

 H∗(βG(X)⊗RFp,α)≅{H∗(XG,Fp)if p=2H∗(XG,Fp)⊗FpΛ(s)if p≥3.

The result follows. ∎

If the spectral sequence of equivariant cohomology with coefficients in degenerates at the -term, it induces an isomorphism of graded -modules:

 H∗(G;H∗(X,Fp))≅H∗G(X,Fp).

Using Corollary 3.4, Proposition 4.2 gives immediately:

###### Corollary 4.3.

If the spectral sequence of equivariant cohomology with coefficients in degenerates at the -term, then for one has:

 h∗(XG,Fp)=∑1≤q

This formula can be stated differently, using only the parameter , that will appear to be the most important in the sequel:

###### Corollary 4.4.

If the spectral sequence of equivariant cohomology with coefficients in degenerates at the -term, then for one has:

 h∗(XG,Fp)=dimFpH∗(X,Fp)G−ℓ∗p(X).
###### Proof.

Since each Jordan block of contains a one-dimensional invariant subspace, one gets . One conludes by using Corollary 4.3. ∎

### 4.3. Degeneracy condition of the spectral sequence

Even under very nice conditions, one can not expect the collapsing of the spectral sequence in general. For instance, take a non-singular, real projective algebraic variety and the involution of complex conjugation, the order two group acting on . Then is called a GM-variety if the spectral sequence of equivariant cohomology with degenerates. See Krasnov [22, 23] for some examples of and non- varieties. In this section, we prove some degeneracy conditions that will be useful for certain symplectic holomorphic varieties.

###### Proposition 4.5.

Assume that and . If has a fixed point for the action of , then the spectral sequence of equivariant cohomology with coefficients in degenerates at the -term.

###### Proof.

Let be a fixed point for . It induces a section of the projection . Denote by

 u:=s∗1∈H4(XG,Fp)

the proper push-forward of the unit in . We can view as a morphism in the derived category of sheaves of -vector spaces over . Pushing down yields a morphism

 q:=Rf∗u:Rf∗Fp→Rf∗Fp[4]

in the corresponding derived category of sheaves over . From Deligne [13, Proposition 2.1] modified by the arguments of [13, Remarque (1.9), ] (where we use the assumption ) we get that if is an isomorphism, then satisfies the Lefschetz condition relative to , that is:

 Rf∗Fp≅⨁iRif∗Fp[−i]

and the spectral sequence of equivariant cohomology with coefficients in degenerates at the -term.

In order to show that is an isomorphism, note that its source and target, being higher direct images of a constant sheaf along a locally trivial fibration, are locally constant sheaves. Thus it is enough to show that is an isomorphism fibre-wise. This follows from base change, as the fibre of at a point is just and the fibre of the morphism at is the multiplication by the fundamental class of the fixed point . ∎

Let be a vector bundle on . Recall that a -linearisation of is given by the data of homomorphisms for all such that the cocycle condition is fulfilled for all . A -linearised vector bundle is a vector bundle together with the data of a -linearisation. Note that -equivariant resolutions exist, see Elagin [15]. Natural examples are the (co)tangent bundle on (where the -linearization is given by pullback along the action of ) or the sheaf of section for any divisor on  that is globally invariant for the action of .

A -linearisation on a vector bundle induces an ordinary -action on the étale space of , which we denote by again, such that the natural projection becomes a -equivariant map. We can then form the space , which has a natural map to , making it canonically into a vector bundle over . If we restrict to a fibre of (all of which are isomorphic to ), it becomes the vector bundle over again.

If has the additional structure of a complex vector bundle and the -linearisation of is compatible with this structure, the induced bundle inherits this structure as a complex vector bundle. Given two -linearised (complex) vector bundles and over , there is the obvious notion of a -equivariant homomorphism between and . It induces naturally an ordinary homomorphism between and . This construction is compatible with the notion of exact sequences, so we get in fact a group homomorphism from the -equivariant Grothendieck group of to the ordinary Grothendieck group of (complex) vector bundles. By forgetting the -linearisations, one defines another group homomorphism from .

This allows one to construct classes in the equivariant cohomology. Let be a characteristic class of complex -theory with values in (in the sequel, we will use reductions modulo of integral characteristic classes like integral linear combinations of Chern classes). Let be a -equivariant vector bundle, or more generally a class in the -equivariant Grothendieck group . Then . By the naturality of characteristic classes, the restriction of to a fibre of is just .

###### Proposition 4.6.

Assume that and . Let be a class in the equivariant complex -theory of and a characteristic class. Assume that the multiplication maps

 H2(X,Fp) →H6(X,Fp),β↦c∪β H0(X,Fp) →H8(X,Fp),β↦c2∪β

are isomorphisms. Then the spectral sequence of equivariant cohomology with coefficients in degenerates at the -term.

###### Proof.

The proof is virtually the same as for proposition 4.5. Denoting as above and , we use again Deligne [13, Proposition (2.1)] modified by the arguments of [13, Remarque (1.9), ]: if and are isomorphisms, then satisfies the Lefschetz condition relative to and the spectral sequence degenerates at the -term. Again this can be checked fibrewise, where these maps are the multiplications by and respectively. ∎

###### Remark 4.7.

As an example, assume that is a smooth complex algebraic variety of complex dimension two that possesses a -fixed point . The skyscraper sheaf to this point defines a class in (after a finite -equivariant resolution). For take the second Chern class . It follows that is a class whose restriction to each fibre of is just the fundamental class of the point since . This is the class used in Proposition 4.5.

###### Remark 4.8.

The preceding two propositions are valid for any finite group acting on , not only .

## 5. Two integral parameters

Assume that is torsion-free. By the universal coefficient theorem, one has and the homomorphisms of reduction modulo , denoted by

 κk:Hk(X,Z)⟶Hk(X,Fp)

are surjective for all .

Let be a primitive -th root of the unity, and the ring of algebraic integers of . By a classical theorem of Masley–Montgomery [30], is a PID if and only if . The -module structure of is defined by for . For any , we denote by the module whose -module structure is defined by .

###### Proposition 5.1.

Assume that is torsion-free and . Then for one has:

1. for .

2. .

3. .

4. .

###### Proof.

By a theorem of Diederichsen and Reiner [12, Theorem 74.3], is isomorphic as a -module to a direct sum:

 (A1,a1)⊕⋯⊕(Ar,ar)⊕Ar+1⊕⋯⊕Ar+s⊕Y

where the