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 S1-action preserving the boundary X and the S1-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.