Positivity of the CM line bundle for families of K-stable klt Fano varieties

• 成果类型: Article
• 署名作者: Codogni, Giulio; Patakfalvi, Zsolt
• 署名单位: University of Rome Tor Vergata; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00999-y
• 发表日期: 2021
• 页码: 811-894
• 关键词: kahler-einstein metrics; compact moduli spaces; complex-surfaces; minimal models; STABILITY; EXISTENCE; MANIFOLDS; projectivity; termination; invariant

摘要

The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.