Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for $K3$ surfaces, and the Tate conjecture

TL;DR

This paper establishes boundedness results for holomorphic symplectic varieties, constructs low-degree line bundles on moduli spaces of stable sheaves, and provides new proofs of the Tate conjecture for K3 surfaces over finite fields.

Contribution

It proves a birational boundedness result for holomorphic symplectic varieties, constructs big line bundles on moduli spaces, and offers new geometric proofs of the Tate conjecture for K3 surfaces.

Findings

Boundedness results for families of holomorphic symplectic varieties.

Construction of low-degree big line bundles on moduli spaces.

New geometric proofs of the Tate conjecture for K3 surfaces.

Abstract

We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin's trick by for $K3$ surfaces by constructing big line bundles of low degree on certain moduli spaces of stable sheaves, and proving birational versions of Matsusaka's big theorem for holomorphic symplectic varieties. As a consequence of these results, we give a new geometric proof of the Tate conjecture for $K3$ surfaces over finite fields of characteristic at least $5$, and a simple proof of the Tate conjecture for $K3$ surfaces with Picard number at least $2$ over arbitrary finite fields -- including characteristic $2$.