  Let $M$ be the Shimura variety associated to the group of spinor similitudes
of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a
conjecture of Bruinier and Yang, relating the arithmetic intersection
multiplicities of special divisors and CM points on $M$ to the central
derivatives of certain $L$-functions. Each such $L$-function is the
Rankin-Selberg convolution associated with a cusp form of half-integral weight
$n/2 +1 $, and the weight $n/2$ theta series of a positive definite quadratic
space of rank $n$.
  When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve,
and our result is a variant of the Gross-Zagier theorem on heights of Heegner
points.
