Joint PU/IAS Number Theory

In this talk, I will report a recent joint work with Shouwu Zhang about a generic positivity of the Beilinson-Bloch height for the Gross-Schoen and Ceresa cycles of curves of genus at least 3. We also construct a Zariski open dense subset U of the moduli space of curves, such that the canonical Gross-Schoen cycle of any curve parametrized by a transcendental point outside U is non-torsion (same for the canonical Ceresa cycle). Some ingredients in our proof, which I will explain in this talk, are Griffiths’ normal functions, Hain’s metrized biextension line bundles and the Betti form, the Ax-Schanuel theorem for variations of mixed Hodge structures, and adelic line bundles over quasi-projective varieties.