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

Abstract

We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin’s trick 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 fields of characteristic $2$.

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Authors

François Charles

Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Orsay, France