Heights and periods of algebraic cycles in families

Abstract: We consider the Beilinson-Bloch heights and Abel-Jacobian periods of homologically trivial Chow cycles in families. For the Beilinson-Bloch heights, we show that for any $g\ge 3$, we can construct a Zariski open dense subset $\mathcal{M}g^{\mathrm{amp}}$ of $\mathcal{M}_g$, the coarse moduli of curves of genus $g$ over $\mathbb{Q}$, such that the heights of Ceresa cycles and Gross-Schoen cycles over $\mathcal{M}_g^{\mathrm{amp}}$ have a lower bound and satisfy the Northcott property. For the Abel-Jacobi periods, we prove that the Ceresa and Gross-Schoen cycle over every non-$\bar{\mathbb{Q}}$ point in $\mathcal{M}_g^{\mathrm{amp}}$ is non-torsion. We also prove results and make conjectures for any family $X \rightarrow S$ of smooth projective varieties and homologically trivial cocycles $Z$ defined over a number field. Using the Betti strata, we construct a Zariski open subset $S^{\mathrm{amp}}$ of $S{\mathbb{C}}$ over which $Z_s$ is non-torsion for all transcendental points $s$ and prove a criterion for $S^{\mathrm{amp}}$ to be dense. We conjecture the rationality of $S^{\mathrm{amp}}$ and also conjecture that $S^{\mathrm{amp}}$ is precisely the open locus on which the Beilinson-Bloch height of $Z_s$ has a lower bound and satisfies the Northcott property.