The Betti map associated to a section of an abelian scheme

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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 $$\le 3$$ , 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 $$A\rightarrow S$$ 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 $$\,\xi ^{-1}A_{\mathrm{tors}}$$ is dense in $$S({\mathbb {C}})$$ .