  Let M be the Shimura variety associated with the group of spinor similitudes
of a rational quadratic space over 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.
