Abstract
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 Hmq(M‾) is finite dimensional, for every m∈Z and every q=0,1,…,n. This enables us to define ∑j=0n(−1)jdimHmj(M‾) the m-th Fourier component of the Euler characteristic on M and to study large m-behavior of Hmq(M‾). In this paper, we establish an index formula for ∑j=0n(−1)jdimHmj(M‾) and Morse inequalities for Hmq(M‾).
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