The Betti map associated to a section of an abelian scheme

• 成果类型: Article
• 署名作者: Andre, Y.; Corvaja, P.; Zannier, U.
• 署名单位: Sorbonne Universite; Universite Paris Cite; University of Udine; Scuola Normale Superiore di Pisa
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00963-w
• 发表日期: 2020
• 页码: 161-202
• 关键词: theorem

摘要

Given a point. on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of.. When ( A,.) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often. takes a torsion value (for instance, Manin's theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when. is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira-Spencer map associated to ( A,.) (assuming A without fixed part, and Z. Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension = 3, and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if A. S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space Ag has dimension at least g, then the Betti map of any non-torsion section. is generically a submersion, so that. -1Ators is dense in S(C).