$S^1$-equivariant Index theorems and Morse inequalities on complex manifolds with boundary

TL;DR

This paper develops $S^1$-equivariant index theorems and Morse inequalities for complex manifolds with boundary, linking the geometry of the boundary with the spectral properties of the $ar{ ext{d}}$-Neumann Laplacian and cohomology groups.

Contribution

It introduces an index formula and Morse inequalities for Fourier components of Dolbeault cohomology on complex manifolds with boundary under $S^1$-action.

Findings

Finite dimensionality of Fourier components of cohomology groups.

Establishment of an index formula relating geometry and spectral data.

Derivation of Morse inequalities for cohomology Fourier components.

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 $H^q_m(\overline M)$ 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 $H^q_m(\overline M)$.