LAUSR

Abstract Let M be a complex manifold of dimension n with smooth connected boundary X. Assume that M ‾ admits a holomorphic S 1 -action preserving the boundary X and the S 1 -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 ) j dim H 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 ) j dim H m j ( M ‾ ) and Morse inequalities for H m q ( M ‾ ) .