Variation of canonical heights of subvarieties for polarized endomorphisms

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.