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The $\bar\partial$-problem on $Z(q)$-domains

Debraj Chakrabarti, Phillip S. Harrington, Andrew Raich

Abstract

Given a complex manifold containing a relatively compact $Z(q)$ domain, we give sufficient geometric conditions on the domain so that its $L^2$-cohomology in degree $(p,q)$ (known to be finite dimensional) vanishes. The condition consists of the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms.

The $\bar\partial$-problem on $Z(q)$-domains

Abstract

Given a complex manifold containing a relatively compact domain, we give sufficient geometric conditions on the domain so that its -cohomology in degree (known to be finite dimensional) vanishes. The condition consists of the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms.
Paper Structure (26 sections, 16 theorems, 119 equations)

This paper contains 26 sections, 16 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Solvability of the $\bar{\partial}$-problem
  3. Results
  4. Definitions and preliminaries
  5. Hermitian forms and Eigenvalues
  6. The Levi form
  7. Some results from L2-theory
  8. Unweighted L2 result
  9. Weighted L2 result
  10. Preliminaries for L2 methods
  11. The proof of Theorem \ref{['thm:L^2 theory, no weight']}
  12. The proof of Theorem \ref{['thm:L^2 theory, with weights']}
  13. Constructing a metric for a single Hermitian form
  14. Analyticity of spectral projections
  15. Proof of Proposition \ref{['prop-main']}
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $M$ be a complex manifold and let $\Omega\subset M$ be a smoothly bounded relatively compact domain, and let $1 \leq q \leq n-1.$ Suppose that the domain $\Omega$ satisfies condition $Z(q)$. Then the $\bar{\partial}$-operator on $(p,q)$-forms on $\Omega$ satisfies the Folland-Kohn basic estimate

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Proposition 4.1
  • ...and 17 more