S¹-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 M‾ admits a holomorphic S¹-action preserving the boundary X and the S¹-action is transversal on X. We show that the ∂‾-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(M‾) is finite dimensional, for every m∈Z and every q=0,1,…,n. This enables us to define ∑_j=0 ⁿ(−1)^jdimH_m ^j(M‾) the m-th Fourier component of the Euler characteristic on M and to study large m-behavior of H_m ^q(M‾). In this paper, we establish an index formula for ∑_j=0 ⁿ(−1)^jdimH_m ^j(M‾) and Morse inequalities for H_m ^q(M‾).