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

Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications