$S^1$-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 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 $H_m^q(\overline{M})$ is finite dimensional, for every $m\in \mathbb{Z}$ and every $q=0,1,\dots ,n$. This enables us to define ${\sum }_{j=0}^n(-1{)}^j\dim H_m^q(\overline{M})$ the $m$-th Fourier component of the Euler characteristic on $M$ and to study large $m$-behavior of $H_m^q(\overline{M})$. In this paper, we establish an index formula for ${\sum }_{j=0}^n(-1{)}^j\dim H_m^q(\overline{M})$ and Morse inequalities for $H_m^q(\overline{M})$.

39 pages

References in corpus (2)

• Morse inequalities for Fourier components of Kohn-Rossi cohomology of CR manifolds with $S^1$-action
• Heat kernel asymptotics, local index theorem and trace integrals for CR manifolds with $S^1$ action

Cited by in corpus (3)

• On Bergman kernel functions and weak Morse inequalities
• On the singularities of the Bergman projections for lower energy forms on complex manifolds with boundary
• $G$-invariant Bergman kernel and geometric quantization on complex manifolds with boundary