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.

中文翻译：

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酉群的绿流和奇异θ提升的构造

考虑到 Kudla 程序中的应用，我们对还原对偶 $U(p,q)\times U(1,1)$ 使用奇异 theta 提升来构造两种不同的格林形式，用于 codimension $q$-cycles Shimura 品种与单一群体相关。我们在奇异 theta 提升和 Kudla-Millson 提升之间建立了一个伴随结果。此外，我们比较了两种格林形式，并获得了两种格林形式差异的生成函数的模数。最后，我们证明了通过奇异 theta 提升获得的格林形式满足拉普拉斯算子的特征值方程，并得出结论，我们的格林形式与织田和津木以不同方式构造的形式一致。