# The construction of Green currents and singular theta lifts for unitary groups

## TL;DR

This paper constructs Green currents for unitary Shimura varieties using singular theta lifts, establishes their modularity and eigenvalue properties, and compares them with existing constructions.

## Contribution

It introduces new Green forms via singular theta lifts for unitary groups and proves their modularity, eigenvalue equations, and equivalence with prior constructions.

## Key findings

- Green forms satisfy Laplace eigenvalue equations
- The difference of Green forms has a modular generating function
- Green forms coincide with those by Oda and Tsuzuki

## Abstract

With applications in the Kudla program in mind we employ singular theta lifts for the reductive dual pair $U(p,q)\times U(1,1)$ to construct two different kinds of Green forms for codimension $q$-cycles in Shimura varieties associated to unitary groups. We establish an adjointness result between our singular theta lift and the Kudla-Millson lift. Further, we compare the two Greens forms and obtain modularity for the generating function of the difference of the two Green forms. Finally, we show that the Green forms obtained by the singular theta lift satisfy an eigenvalue equation for the Laplace operator and conclude that our Green forms coincide with the ones constructed by Oda and Tsuzuki by different means.