Variation of canonical heights of subvarieties for polarized endomorphisms

When an endomorphism $f:X\to X$ of a projective variety which is polarized by an ample line bundle $L$, i.e. such that $f^{\ast }L\simeq L^{\otimes d}$ with $d\ge 2$, 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 ${\widehat{h}}_{f_t}(P(t))\mathrm{/}h(t)$ 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.