Modular Heights of Unitary Shimura Varieties I: Derivatives of Eisenstein Series

This is the first of a series of three papers, in which we prove a formula expressing the modular height of a unitary Shimura variety over a CM number field in terms of the logarithmic derivative of the Hecke L-function associated with the CM extension. The main idea of our proof is to compare the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series and the arithmetic degree of a generating series of divisors on unitary Shimura varieties. In this paper, we compute an explicit expression of the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series.