  We prove two formulas in the style of the Gross-Zagier theorem, relating
derivatives of L-functions to arithmetic intersection pairings on a unitary
Shimura variety. We also prove a special case of Colmez's conjecture on the
Faltings heights of abelian varieties with complex multiplication. These
results are derived from the authors' earlier results on the modularity of
generating series of divisors on unitary Shimura varieties.
