The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao)

Abstract

Given a point $\xi$ 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 $\xi$. When $(A, \xi)$ 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 $\xi$ 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 $\xi$ 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 $(A, \xi)$ (assuming $A$ without fixed part, and $\mathbb{Z} \xi$ Zariski-dense in $A$), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension $\leq 3$, 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 $A\to S$ 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 $\mathcal{A}_g$ has dimension at least $g$, then the Betti map of any non-torsion section $\xi$ is generically a submersion, so that $\xi^{-1}A_{tors}$ is dense in $S(\mathbb{C})$.