Title: $S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary,
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 $\overline\partial$-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 $q=0,1,\ldots,n$. 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
$H^q_m(\overline M)$. In this paper, we establish an index formula for
$\sum^{n}_{j=0}(-1)^j{\rm dim\,}H^q_m(\overline M)$ and Morse inequalities for
$H^q_m(\overline M)$.