| Abstract: |
| Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\overline{M}$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal and CR on $X$. We show that the $\bar{\partial}$-Neumann Laplacian on $M$ is transversally elliptic and as a consequence, the $m$-th Fourier component of the $q$-th Dolbeault cohomology group $H^q_m(\overline{M})$ is finite dimensional, for every $m \in \mathbb{Z}$ and every $q=0,1,\cdots,n$. This enables us to define $\sum^{n}_{j=0}(-1)^j{\dim}H^j_m(\overline{M})$ the $m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-behavior of $H^q_m(\overline{M})$. In this talk, we will present an index formula for $\sum^{n}_{j=0}(-1)^j{\dim}H^j_m(\overline{M})$ and Morse inequalities for $H^q_m(\overline{M})$. This is based on a joint work with Chin-Yu Hsiao, Xiaoshan Li and Guokuan Shao. |
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