Complex multiplication cycles and Kudla-Rapoport divisors

TL;DR

This paper investigates the intersection properties of special cycles on a specific unitary Shimura variety, linking their intersection multiplicities to Fourier coefficients of Eisenstein series, thus providing evidence for conjectures relating to Kudla and a potential Gross-Zagier theorem.

Contribution

It establishes new cases of Kudla's conjectures by connecting intersection multiplicities with Eisenstein series Fourier coefficients on unitary Shimura varieties.

Findings

Intersection multiplicities match Fourier coefficients of Eisenstein series

Provides new evidence for Kudla's conjectures

Suggests a Gross-Zagier type theorem for unitary Shimura varieties

Abstract

We study the intersections of special cycles on a unitary Shimura variety of signature (n-1,1), and show that the intersection multiplicities of these cycles agree with Fourier coefficients of Eisenstein series. The results are new cases of conjectures of Kudla, and suggest a Gross-Zagier theorem for unitary Shimura varieties.