Erratum to the paper: Integral cohomology of the Generalized Kummer fourfold
Simon Kapfer, Gr\'egoire Menet

TL;DR
This paper corrects a previous proof regarding the integral cohomology of generalized Kummer fourfolds, establishing that it is torsion free in dimension 4.
Contribution
It provides a corrected proof confirming the torsion-free nature of the integral cohomology for generalized Kummer fourfolds.
Findings
Integral cohomology of generalized Kummer fourfolds is torsion free.
Corrected proof of the previous theorem in dimension 4.
Clarification of the cohomological properties of these fourfolds.
Abstract
We provide a correct proof of arXiv:1607.03431, Theorem 5.2 in dimension 4. More precisely, we show that the integral cohomology of the generalized Kummer fourfold is torsion free.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Geometric and Algebraic Topology
Erratum: Integral cohomology of the Generalized Kummer fourfold
Simon Kapfer; Grégoire Menet
It was pointed out by B. Totaro that the reference used for [7, Theorem 5.2] is inappropriate. The latter concern the torsion of the integral cohomology of the generalized Kummer. Here we show that [7, Theorem 5.2] holds at least in dimension 4. All the other results of [7] remain unaffected. It would also be interesting to find out whether the generalized Kummer varieties of higher dimension have torsion-free cohomology or not.
**The integral cohomology of the generalized Kummer fourfold is torsion free **
Let be a 2-dimensional complex torus. Let be the Hilbert scheme of 3 points on and the summation morphism. The generalized Kummer fourfold is defined by . We can also consider the following embedding: . The action of the symmetric group on provides an action on via the embedding . Then can also be seen as a resolution of .
Theorem 1**.**
The cohomology is torsion free.
Remark 1*.*
We denote by the torsion of groups. Because of the Poincaré duality and the universal coefficient theorem, we have:
[TABLE]
Thus, it suffices to prove that and are torsion free. Moreover, since Theorem 1 is only a topological result, without loss of generality, we can assume that is an abelian surface.
Let be the locus of subschemes supported at . As it is explained in [6, Section 4], we have:
[TABLE]
Let be the singular point and . Put
[TABLE]
Lemma 1**.**
We have and an injection .
Proof*.*
The generalized Kummer is smooth in . Hence, applying Thom’s isomorphism to the long exact sequence of the relative cohomology of the pair , we obtain:
[TABLE]
Moreover:
[TABLE]
By Thom’s isomorphism, and is torsion free. Furthermore, since, by (1), is the quotient of by an automorphism of order 3. It follows from (3):
[TABLE]
Then (2) concludes the proof. ∎
Hence, it remains to prove that and are torsion free. To do so, we consider
[TABLE]
where is embedded in diagonally: .
Let be the blow-up of in
[TABLE]
As explained in [2, Section 7], we have:
[TABLE]
Lemma 2**.**
The groups and can only have 3-torsion.
Proof*.*
Let
[TABLE]
The surface is isomorphic to the blow-up of in . We consider the blow-up of in . For , we denote
[TABLE]
The surfaces are Hirzebruch surfaces. We also consider
[TABLE]
We have:
[TABLE]
Indeed, if we denote by the blow-up of in the diagonal, it is well known that . For , we denote . Then, if we consider the blow-up of in , we have . Therefore by [5, Corollary II 7.15], we obtain a commutative diagram:
[TABLE]
This provides (5). Then, it follows from (4) a triple cover:
[TABLE]
Hence, if we prove that and are torsion free, Lemma 2 will be proven by [1, Theorem 5.4 ].
We know that is torsion free from [9, Theorem 2.2]. Then, we deduce from [10, Theorem 7.31] (or from [4, Theorem 4.1] which is more general) that:
[TABLE]
Consider the exact sequence:
[TABLE]
By Thom’s isomorphism:
[TABLE]
Since is an Hirzebruch surface, and , are torsion free. It follows from (6) and (7) that and are torsion free. ∎
We can also consider the double cover:
[TABLE]
Applying [1, Theorem 5.4 ] and Lemma 2, it is suffices to prove that and are torsion free to conclude the proof of Theorem 1.
Lemma 3**.**
The groups and are torsion free.
First, we show that . Since the action of on is free, it can be realized using the equivariant cohomology as explained in [8, Section 4]. The computation of the equivariant cohomology can be done using the Boissière–Sarti–Nieper-Wisskirchen invariants defined in [3, Section 2]. We recall their definition in our specific case. Let be a -torsion-free -module of finite rank equipped with a linear action of . We consider the action of on . Then the matrix of the endomorphism on admits a Jordan normal form. We can decompose as a direct sum of some -modules of dimension with , where acts on the in a suitable basis, respectively by matrices of the following form:
[TABLE]
Definition 1*.*
We define the integer as the number of blocks of size in the Jordan decomposition of the -module , so that .
Notation 1*.*
Let being or . For all and all , we denote:
[TABLE]
Proposition 1**.**
The Boissière–Sarti–Nieper-Wisskirchen invariants for the -action on are:
- (i)
* and ;*
- (ii)
, and ;
- (iii)
, and ;
- (iv)
, and .
Proof.
We can start by calculating the , . Applying the same idea as [3, Lemma 6.14], we deduce the other invariants using the fact that:
[TABLE]
for all .
- (i)
For and a generator of , It follows (i).
- (ii)
We have
- (iii)
We have
Moreover,
- (iv)
We have
Moreover:
[TABLE]
∎
Proof of Lemma 3.
Since for all , for all and . Hence by [8, Corollary 4.2] and Proposition 1, we have for all , , and , .
Since acts freely on , the spectral sequence of equivariant cohomology provides that and are torsion free (see [8, Section 4] for a reminder about this spectral sequence). Moreover, is the blow-up of in the image of in . Since and are smooth, by [4, Theorem 4.1] we have an isomorphism of graded -modules:
[TABLE]
where is a class of degree 2. Hence, in degree 3 and 5, we obtain:
[TABLE]
and
[TABLE]
It remains to show that is torsion free. Since , it can be done considering the following exact sequence:
[TABLE]
where is torsion free and by Thom’s isomorphism. ∎
Acknowledgements. This work was supported by the ERC-ALKAGE, grant No. 670846.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 6[6] B. Hassett and Y. Tschinkel, Hodge theory and Lagrangian planes on generalized Kummer fourfolds , Moscow Math. Journal, 13, no. 1, 33-56, (2013).
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- 8[8] G. Menet, On the integral cohomology of quotients of complex manifolds , Journal de Mathématiques pures et appliquées, 119 (2018), no.9, 280-325.
