K3 surfaces associated with varieties of generalized Kummer type

TL;DR

This paper explores the geometric relationship between hyper-K"ahler varieties of generalized Kummer type and associated K3 surfaces, proving the Hodge conjecture for their powers and related abelian fourfolds.

Contribution

It establishes a geometric link via moduli spaces of sheaves and proves the Hodge conjecture for these complex varieties and their powers.

Findings

Proves the Hodge conjecture for all powers of associated K3 surfaces.

Establishes a geometric relation between hyper-K"ahler varieties and K3 surfaces.

Extends results to abelian fourfolds of Weil type with discriminant 1.

Abstract

With any hyper-K\"ahler variety $K$ of generalized Kummer type is associated via Hodge theory a K3 surface $S_K$. We show how they are related geometrically through a moduli space of sheaves on $S_K$. As a consequence, building fundamentally on the works of O'Grady, Markman, Voisin, Varesco, we establish the Hodge conjecture for all powers of any of these K3 surfaces as well as for all abelian fourfolds of Weil type with discriminant 1 and their powers, strenghtening a result of Markman.