$S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary

Let $M$ be a complex manifold of dimension $n$ with smooth connected boundary $X$. Assume that $\bar{M}$ admits a holomorphic $S^1$-action preserving the boundary $X$ and the $S^1$-action is transversal on $X$. We show that the $\bar{\mathrm{\partial }}$-Neumann Laplacian on $M$ is transversally elliptic and as a consequence, the $m$-th Fourier component of the $q$-th Dolbeault cohomology group $H_m^q(\bar{M})$ is finite dimensional, for every $m\in \mathbb{Z}$ and every $q=0,1,\dots ,n$. This enables us to define $\sum^{n}{j=0}(-1)^j{\rm dim,}H^q_m(\overline M)$the$m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-behavior of $H_m^q(\bar{M})$. In this paper, we establish an index formula for $\sum^{n}{j=0}(-1)^j{\rm dim,}H^q_m(\overline M)$andMorseinequalitiesfor$H^q_m(\overline M)$.