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

Abstract

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.

Recent progress on the Tate conjecture

“…To discuss the proof of the Tate conjecture for K3 surfaces in more detail, we will describe a second-generation proof by Charles [8]. This proof still uses the Kuga-Satake correspondence, but to a lesser extent than in the earlier proofs.…”

“…Applying this result over a finite field k, Charles deduces that there are only finitely many isomorphism classes of K3 surfaces over k, with some restrictions on the degree (but a priori allowing arbitrarily high degrees) [8,Proposition 3.17].…”

Moduli of sheaves: A modern primer

“…Theorem 4.4.14 (Charles,[12]). The Tate conjecture holds for K3 surfaces over finite fields of characteristic at least 5.…”

“…There are now many proofs of this in various forms. We comment on the proof of Charles [12], which builds on ideas developed in [28] that have already appeared in Section 6.6. The idea is this: first, the Tate conjecture for all K3 surfaces over all finite extensions of k is equivalent to the statement that for each such extension there are only finitely many K3 surfaces.…”

Moduli spaces of sheaves on K3 surfaces and Galois representations

“…Let v ∈ N (S) be an effective and geometrically primitive Mukai vector with v 2 ≥ 0, and let H be a v-generic polarization on S. Then M is a non-empty, smooth, projective, geometrically irreducible variety over k of dimension v 2 + 2.This was proved in [1, Thm. 2.4(i)] under the stronger assumption that v satisfy condition (C) given in[1, Def. 2.3], which in particular implies that M is a fine moduli space.…”