  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.
