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

• 成果类型: Article
• 署名作者: Charles, Francois
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2016.184.2.4
• 发表日期: 2016
• 页码: 487-526
• 关键词: moduli spaces; sheaves; models; shafarevich; FINITENESS; monodromy; MANIFOLDS; torelli

摘要

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.