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

Chin-Yu Hsiao, Rung-Tzung Huang, Xiaoshan Li, Guokuan Shao

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 $H^q_m(\overline M)$.

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

Abstract

Let be a complex manifold of dimension with smooth connected boundary . Assume that admits a holomorphic -action preserving the boundary and the -action is transversal on . We show that the -Neumann Laplacian on is transversally elliptic and as a consequence, the -th Fourier component of the -th Dolbeault cohomology group is finite dimensional, for every and every . This enables us to define the -th Fourier component of the Euler characteristic on and to study large -behavior of . In this paper, we establish an index formula for and Morse inequalities for .

Paper Structure

This paper contains 18 sections, 40 theorems, 262 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Introduction
  3. Main results
  4. Preliminaries
  5. Some standard notations
  6. Set up
  7. $S^1$-equivariant $\overline\partial$-Neumann problem
  8. The operators $\overline\partial_\beta$, $\Box^{(q)}_\beta$ and reduction to the boundary
  9. Poisson operator
  10. $\overline\partial_\beta$-operator
  11. Reduction to boundary
  12. Technical lemmas for $\hat{\Box}^{(q)}_\beta$
  13. Index theorem
  14. The scaling technique
  15. Holomorphic Morse inequalities on complex manifolds with boundary
  16. ...and 3 more sections

Key Result

Theorem 1.4

With the notations and assumptions above, there is an $m_0>0$ such that for every $m\in\mathbb Z$ with $m\geq m_0$, we have where $\mathrm{Td_b\,}(T^{1,0}X)$ denotes the tangential Todd class of $T^{1,0}X$ (see Definition d-gue150516).

Theorems & Definitions (61)

  • Remark 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Theorem 1.11
  • Definition 2.1
  • ...and 51 more