Variation of canonical heights of subvarieties for polarized endomorphisms

TL;DR

This paper extends the understanding of how canonical heights of subvarieties vary in families of polarized endomorphisms, generalizing previous results from points to higher-dimensional subvarieties.

Contribution

It proves the variation formula for canonical heights of subvarieties in families of polarized endomorphisms, generalizing prior work on points.

Findings

Canonical heights of subvarieties vary continuously in families.

The ratio of the canonical height to a base height converges to the generic fiber height.

Extension from points to higher-dimensional subvarieties.

Abstract

When an endomorphism $f:X\to X$ of a projective variety which is polarized by an ample line bundle $L$, i.e. such that $f^*L\simeq L^{\otimes d}$ with $d\geq2$, is defined over a number field, Call and Silverman defined a canonical height $\widehat{h}_f$ for $f$. In a family $(\mathcal{X},\mathcal{f},\mathcal{L})$ parametrized by a curve $S$ together with a section $P:S\to \mathcal{X}$, they show that $\widehat{h}_{f_t}(P(t))/h(t)$ converges to the height $\widehat{h}_{f_\eta}(P_\eta)$ on the generic fiber. In the present paper, we prove the equivalent statement when studying the variation of canonical heights of subvarieties $Y_t$ varying in a family $\mathcal{Y}$ of any relative dimension.