Prescribed volume forms in Apelli cohomology groups on compact Hermitian manifolds-关波 教授（俄亥俄州立大学）
Title: Prescribed volume forms in Apelli cohomology groups on compact Hermitian manifolds
Speaker: 关波 教授（俄亥俄州立大学）
Abstract：Given a strongly positive (p, p) form Ω on a Hermitian manifold of dimension n ≥ 2, where 1 ≤ p < n is an integer, we study the problem of finding a strongly positive (p, p) form in the Apelli cohomology class of Ω with prescribed volume form. For p = 1, this is equivalent to the classical Calabi conjecture solved by S.T. Yau in the case, while for p = n − 1 it corresponds to the Gauduchon conjecture proved by Szekelyhidi-Tosatti-Weinkove more recently. From the PDE point of view, this leads to a new fully nonlinear elliptic equation which falls outside the framework developed by Cafferlli-Nirenberg-Spruck. We shall treat a general class of PDEs which also arise from other geometric problems. The talk is based on work with my student Mathew George.