The Hodge and Tate conjectures for hyper-K\"{a}hler sixfolds of generalized Kummer type

TL;DR

This paper proves the Hodge and Tate conjectures for six-dimensional hyper-K"ahler varieties of generalized Kummer type, confirming these deep conjectures in this specific geometric context.

Contribution

It establishes the validity of the Hodge and Tate conjectures for a new class of hyper-K"ahler sixfolds, expanding the known cases where these conjectures hold.

Findings

Hodge conjecture proven for these varieties

Tate conjecture proven for these varieties

Validates conjectures in a new geometric setting

Abstract

We prove the conjectures of Hodge and Tate for any six-dimensional hyper-K\"ahler variety that is deformation equivalent to a generalized Kummer variety.