Generating series of intersection volumes of special cycles on unitary Shimura varieties

TL;DR

This paper constructs a generating series of intersection volumes of special cycles on unitary Shimura varieties, proves it is a Hilbert modular form via theta integrals, and relates it to hermitian modular forms and Eisenstein series.

Contribution

It introduces a new generating series for intersection volumes on unitary Shimura varieties and establishes its modularity through advanced theta and Eisenstein series techniques.

Findings

The generating series is a Hilbert modular form.

The series converges outside the classical Weil range.

It relates to hermitian modular forms and Siegel Eisenstein series.

Abstract

We form a generating series of regularized volumes of intersections of special cycles on a non-compact unitary Shimura variety with a fixed base change cycle. We show that it is a Hilbert modular form by identifying it with a theta integral, which we show converges even though the parameters lie outside the classical convergence range of Weil. By applying the regularized Siegel-Weil formulas of Ichino and Gan-Qiu-Takeda, we show the modular form is the restriction of a hermitian modular form of degree n related to Siegel Eisenstein series on U(n,n). An essential fact used is a computation showing the Kudla-Millson Schwartz function vanishes under the Ikeda map.