Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-K\"ahler varieties

TL;DR

This paper proves the Mumford-Tate conjecture for certain hyper-K"ahler varieties, showing its invariance under deformation and linking the Hodge and Tate conjectures for these varieties.

Contribution

It establishes the Mumford-Tate conjecture for hyper-K"ahler varieties with specific Betti number conditions and demonstrates its deformation invariance.

Findings

Proved the conjecture for hyper-K"ahler varieties with $b_2 \,\geq\, 4$.

Hodge conjecture implies Tate conjecture for powers of these varieties.

Mumford-Tate conjecture is invariant under deformation.

Abstract

In this paper, we establish two main results concerning the Mumford-Tate conjecture for hyper-K\"ahler varieties. First, we prove the conjecture for the semisimplified $\ell$-adic Galois representations attached to hyper-K\"ahler varieties with second Betti number $b_2 \geq 4$. As a direct consequence, we deduce that the Hodge conjecture implies the Tate conjecture for powers of hyper-K\"ahler varieties. Second, we show that the Mumford-Tate conjecture for hyper-K\"ahler varieties is invariant under deformation. The proofs rely on comparing the ranks of $\ell$-adic algebraic monodromy groups in higher degrees to those in degree $2$ via the theory of Frobenius tori and the Looijenga-Lunts-Verbitsky Lie algebra.