Faltings heights of abelian varieties with complex multiplication

Abstract

Let $M$ be the Shimura variety associated with the group of spinor similitudes of a quadratic space over $\mathbb {Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on $M$ to the central derivatives of certain $L$-functions.

As an application of this result, we prove an averaged version of Colmez’s conjecture on the Faltings heights of CM abelian varieties.