# 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.

## Authors

Fabrizio Andreatta

Dipartimento di Matematica Federigo Enriques", Università di Milano, via C.~Saldini 50, Milano, Italia

Eyal Z. Goren

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. West, Montreal, QC, Canada

Benjamin Howard

Department of Mathematics, Boston College, 140 Commonwealth Ave., Chicago, IL, USA