Guokuan Shao，Institute of Mathematics, Academia Sinica
Middle Lecture Room, Math Building
In this talk, we will present new versions of index theorems and Morse inequalities on complex manifolds with boundary. Let M be a relatively compact open subset with connected smooth boundary X of a complex manifold M’. Assume that M admits a holomorphic S^1-action preserving the boundary X and the S^1-action is transversal and CR on X. We claim that the m-th Fourier component of the q-th Dolbeault cohomology group H^q_m(\overline M) is of finite dimension. By using Poisson operator, we prove a reduction theorem which shows that the formulas about H^q_m(\overline M) in our main theorems involve only integrations over X. This talk is based on the joint work with Chin-Yu HSIAO, Rung-Tzung HUANG and Xiaoshan LI.