On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type of dimension divisible by four

Salvatore Floccari; Lie Fu

paper

On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type of dimension divisible by four

arXiv:2610.31740

Abstract

We prove the Hodge conjecture for all hyper-Kähler varieties of generalized Kummer type of dimension $2 n$ with $n + 1$ a prime number, and all powers of these varieties. For any variety $X$ of generalized Kummer type of dimension divisible by 4, we show that the Lefschetz standard conjecture for $X$ implies the Hodge conjecture for $X$ and all its powers.

arXiv comment49 pages