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

TL;DR

This paper develops a new framework for Levi flat structures using structure sheaves, differential complexes, and convexity concepts, leading to results on global solvability and regularity in complex geometry.

Contribution

It introduces a novel approach employing structure sheaves and differential complexes to analyze Levi flat structures, extending the Treves complex and establishing global solvability.

Findings

Global exactness of the differential complex for Levi flat structures

Sobolev regularity in the compact case

Solutions to the extension problem for canonical forms

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.