Algebraic cycles on hyper-K\"ahler varieties of generalized Kummer type

TL;DR

This paper proves the Hodge and Tate conjectures for four-dimensional hyper-K"ahler varieties of generalized Kummer type, establishing the algebraicity of certain Hodge classes in their cohomology.

Contribution

It demonstrates the algebraicity of Hodge classes in the subalgebra generated by second cohomology for these specific hyper-K"ahler varieties.

Findings

Hodge conjecture verified for these varieties

Tate conjecture verified for these varieties

All Hodge classes in the generated subalgebra are algebraic

Abstract

We prove the conjectures of Hodge and Tate for any four-dimensional hyper-K\"ahler variety of generalized Kummer type. For an arbitrary variety $X$ of generalized Kummer type, we show that all Hodge classes in the subalgebra of the rational cohomology generated by $H^2(X,\mathbb{Q})$ are algebraic.