Complex multiplication cycles and Kudla-Rapoport divisors, II

Abstract

This paper is about the arithmetic of {\it Kudla-Rapoport divisors} on Shimura varieties of type ${\rm GU}(n-1,1)$. In the first part of the paper we construct a toroidal compactification of N.~Kr\"amer's integral model of the Shimura variety. This extends work of K.-W.~Lan, who constructed a compactification at unramified primes. In the second, and main, part of the paper we use ideas of Kudla to construct Green functions for the Kudla-Rapoport divisors on the open Shimura variety, and analyze the behavior of these functions near the boundary of the compactification. The Green functions turn out to have logarithmic singularities along certain components of the boundary, up to log-log error terms. Thus, by adding a prescribed linear combination of boundary components to a Kudla-Rapoport divisor one obtains a class in the arithmetic Chow group of Burgos-Kramer-K\"uhn. In the third and final part of the paper we compute the arithmetic intersection of each of these divisors with a cycle of complex multiplication points. The computation is quickly reduced to the calculations of the author's earlier work {\it Complex multiplication cycles and Kudla-Rapoport divisors}. The arithmetic intersection multiplicities are shown to appear as Fourier coefficients of the diagonal restriction of the central derivative of a Hilbert modular Eisenstein series.