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

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