Integral cohomology of the Generalized Kummer fourfold

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.