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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 of a basic vector bundle. Drawing inspiration from Morse theory and Grauert's convexity, we introduce a notion of convexity 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, and derive results on singular cohomology, as well as the extension problem for canonical forms in the elliptic case.