For a Riemann surface, the classical Abel-Jacobi theory in the 19th century gives a parametrization of degree 0 divisor classes by the so-called Jacobian variety. When this Riamman surface is defined over algebraic numbers,  we have a canonical height pairing on the Jacobian called the Neron—Tate heights. This height pairing is a power ingredient in modern number theory, for example, the proof of the Mordell—Weil theorem, the Moredell conjecture, and the formulation of BSD conjecture. 

In this talk, we consider the analogous problems about homologically trivial cycles on projective varieties following the classical constructions of Weil, Griffiths, Beilinson, and Bloch. One far-reaching conjecture is the positivity of the so-called Beilinson—Bloch heights. In this talk, we report our joint work with Ziyang Gao about a generic positivity for the Ceresa cycles parameterized by moduli of curves of genus at least 3. The proof is a combination of the Ax-Schanuel theorem for the variation of mixed Hodge structures and adelic line bundles over quasi-projective varieties.