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

Abstract

With applications in the Kudla program in mind we employ singular theta lifts for the reductive dual pair $\mathrm {U}(p,q)\times \mathrm {U}(1,1)$ 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.