Arithmetic Volumes of Unitary Shimura Varieties

The integral model of a G U ( n − 1 , 1 ) \mathrm {GU}(n-1,1) Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet L L -functions. We also determine the arithmetic volumes of Kudla-Rapoport divisors and relate them to coefficients of Eisenstein series.