  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 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 $H^q_m(\overline M)$.
