TITLE: Faltings height in short exact sequences
QUESTION [12 upvotes]: Let $K$ be a number field and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ a short exact sequence of abelian varieties over $K$. Let $h(A)$ denote the logarithmic Faltings height (normalized so that it is invariant upon base change to any finite extension $K'/K$; thus, due to this normalization, one may assume that $A$, $B$, and $C$ have semiabelian reduction everywhere). Is $h(B) = h(A) + h(C)$?

REPLY [6 votes]: Proposition 3.3 of Ullmo's paper "Hauteur de Faltings de quotients de J_0(N) " (American Journal of Math., 2000) seems to answer your question.