The Betti map associated to a section of an abelian scheme

Abstract

Given a point ξ 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 ξ. When (A, ξ) 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 ξ 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 ξ 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, ξ) (assuming A without fixed part, and Zξ 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 ξ is generically a submersion, so that ξ −1 Ators is dense in S(C).

Ax-Schanuel for Shimura varieties

“…One such application has been given by Daw-Ren [7]. An application in a different direction is given in [2]. A generalization to variations of Hodge structures is given by Bakker-Tsimerman [4].…”

Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

“…Here N is bounded only in terms of T and thus only in terms of c 1 , c 2 , and n. It is independent of x. If X ⊆ G 1 ∪ · · · ∪ G N , then we are in case (1).…”

“…The assumption dim S = 1 forces that X an ∩A ∆ = b −1 (b(X an ∩ A ∆ )). 1 In other words the fibers X s = π| −1 X (s) do not depend on s ∈ ∆ for the identification A an s = T 2g . So the action of π 1 (S an , s) on A an s leaves X s invariant.…”

“…Ullmo and Yafaev [50] exploited exponential growth in groups in combination with the Pila-Wilkie Theorem to prove their hyperbolic Ax-Lindemann Theorem for projective Shimura varieties. A recent result of André, Corvaja, and Zannier [1] also deals with the rank of the Betti map on the moduli space of principally polarized abelian varieties of a given dimension. More recently, Cantat, Xie, and the authors [11] gave a different approach to the functional constancy problem that does not rely on the Pila-Wilkie Theorem but rather on results from dynamical systems.…”

Torsion Points of Sections of Lagrangian Torus Fibrations and the Chow Ring of Hyper-Kähler Manifolds

“…It uses the easy Proposition 2.1 which says that the torsion points of ν are dense in B if the natural locally defined map f ν : B → H 1 (A t0 , R) obtained from ν by a real analytic trivialization of the family of complex tori A → B is generically submersive. This map is called the Betti map in [2], [13]. A key tool for our work is the following very useful result of André-Corvaja-Zannier [2].…”