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

Source abstract

We prove the Hodge conjecture for all hyper-Kähler varieties of generalized Kummer type of dimension $2n$ 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.