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Levi Flat Structures via Structure Sheaves: Differential Complexes, Convexity, and Global Solvability

Qingchun Ji, Jun Yao

Abstract

This paper investigates Levi flat structures from the perspective of structure sheaves. We employ formal integrability to construct a class of differential complexes, thereby providing a resolution for the structure sheaf and a global realization of the Treves complex. Drawing inspiration from Morse theory and Grauert's convexity, we introduce notions of convexity and positivity that fully exploits Levi flatness, which ensures the global exactness of the differential complex and demonstrates Sobolev regularity in the compact case. As applications, we establish the global solvability of the Treves complex for Levi flat structures, together with results on singular cohomology and the extension problem for canonical forms in the elliptic case.

Levi Flat Structures via Structure Sheaves: Differential Complexes, Convexity, and Global Solvability

Abstract

This paper investigates Levi flat structures from the perspective of structure sheaves. We employ formal integrability to construct a class of differential complexes, thereby providing a resolution for the structure sheaf and a global realization of the Treves complex. Drawing inspiration from Morse theory and Grauert's convexity, we introduce notions of convexity and positivity that fully exploits Levi flatness, which ensures the global exactness of the differential complex and demonstrates Sobolev regularity in the compact case. As applications, we establish the global solvability of the Treves complex for Levi flat structures, together with results on singular cohomology and the extension problem for canonical forms in the elliptic case.

Paper Structure

This paper contains 14 sections, 34 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of the differential complex
  4. Notions of -convexity and -positivity
  5. Setup for -theory
  6. Global existence and regularity of the differential complex
  7. Global existence theorems
  8. Sobolev regularity on compact manifolds
  9. The Morse-Novikov-Treves complex
  10. Global solvability of the Morse-Novikov-Treves complex
  11. A reformulation of the Morse-Novikov-Treves complex
  12. The -leafwise Morse-Novikov complex
  13. The -leafwise Morse-Novikov-Dolbeault complex
  14. Logarithmic forms of elliptic structures

Key Result

Theorem 1.1

Suppose that $M$ admits a $q$-convex exhaustion function $\varphi\in C^\infty(M)$ with respect to a Levi flat structure $\mathcal{V}$, $\vartheta$ is a smooth $1$-form on $M$ satisfying $(mnf)$, and $E$ is a basic vector bundle over $(M,\mathcal{V})$. Then for any $f\in L^2_{loc}(M,\Lambda_\mathcal{

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Corollary 1.3
  • Remark 2
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • Example 1
  • Proposition 2.1
  • ...and 54 more