Arithmetic volumes of unitary Shimura varieties

The integral model of a GU(n-1,1) Shimura variety carries a natural metrized line bundle of modular forms.  Viewing this metrized line bundle as a class in the codimension one arithmetic Chow group, one can define its arithmetic volume as an iterated self-intersection.  We will show that this volume can be expressed in terms of logarithmic derivatives of Dirichlet L-functions at integer points, and explain the connection with the arithmetic Siegel-Weil conjecture of Kudla-Rapoport.  This is joint work with Jan Bruinier.