Formally Integrable Structures III. Levi Flat Structures

Abstract: In this paper, we utilize formal integrability to construct a class of differential complexes, thereby providing a resolution for the sheaf of basic sections of a basic vector bundle, as well as a global realization of the Morse-Novikov-Treves complex. In the context of Levi flat structures, we introduce a notion of convexity for each Hermitian metric on a basic line bundle, which enables us to establish vanishing theorems and the global solvability of the Morse-Novikov-Treves complex. In the special case of elliptic structures, we further obtain results concerning singular cohomology and the extension problem for canonical forms.