Integral cohomology of the generalized Kummer fourfold

Menet, Grégoire

doi:10.14231/ag-2018-014

Alg. Geom.

2018

DOI: 10.14231/ag-2018-014

|View full text |Cite

Integral cohomology of the generalized Kummer fourfold

Grégoire Menet¹

Abstract: We describe the integral cohomology of the generalized Kummer fourfold, giving an explicit basis, using Hilbert scheme cohomology and tools developed by Hassett and Tschinkel. Then, we apply our results to an irreducible holomorphic symplectic variety with singularities, obtained by a partial resolution of the generalized Kummer fourfold quotiented by a symplectic involution. We calculate the Beauville-Bogomolov form of this new variety, presenting the first example of such a form that is odd.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2022

2023

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

(1 citation statement)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, S is smooth by[11, Lem. 4.1], which completes the proof.See[30] for further discussion of these fixed-point loci in hyperkählers of Kummer type. Let k now be arbitrary.…”

mentioning

confidence: 99%

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Frei

Honigs

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

We describe the Galois action on the middle $\ell $ -adic cohomology of smooth, projective fourfolds $K_A(v)$ that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surface A with Mukai vector v. We show this action is determined by the action on $H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$ and on a subgroup $G_A(v) \leqslant (A\times \hat {A})[3]$ , which depends on v. This generalizes the analysis carried out by Hassett and Tschinkel over ${\mathbb C}$ [21]. As a consequence, over number fields, we give a condition under which $K_2(A)$ and $K_2(\hat {A})$ are not derived equivalent. The points of $G_A(v)$ correspond to involutions of $K_A(v)$ . Over ${\mathbb C}$ , they are known to be symplectic and contained in the kernel of the map $\operatorname {\mathrm {Aut}}(K_A(v))\to \mathrm {O}(H^2(K_A(v),{\mathbb Z}))$ . We describe this kernel for all varieties $K_A(v)$ of dimension at least $4$ . When $K_A(v)$ is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed-point locus on fourfolds $K_A(0,l,s)$ over ${\mathbb C}$ where A is $(1,3)$ -polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of $K_A(0,l,s)$ .

show abstract

“…Finally, S is smooth by[11, Lem. 4.1], which completes the proof.See[30] for further discussion of these fixed-point loci in hyperkählers of Kummer type. Let k now be arbitrary.…”

mentioning

confidence: 99%

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Frei

Honigs

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

show abstract

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

Markman¹

2022

J. Eur. Math. Soc.

View full text Add to dashboard Cite

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Integral cohomology of the generalized Kummer fourfold

Cited by 2 publications

References 8 publications

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

Contact Info

Product

Resources

About