Integral cohomology of the Generalized Kummer fourfold

TL;DR

This paper explicitly describes the integral cohomology of the Generalized Kummer fourfold, applies the results to a related singular IHS variety, and computes its Beauville--Bogomolov form, revealing a novel odd form example.

Contribution

It provides an explicit basis for the integral cohomology of the Generalized Kummer fourfold and computes the Beauville--Bogomolov form for a new singular IHS variety.

Findings

Explicit basis for cohomology of the Generalized Kummer fourfold

First example of an odd Beauville--Bogomolov form in such a context

Application to a singular IHS variety with partial resolution

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 a IHS variety with singularities, obtained by a partial resolution of the Generalized Kummer 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.