Infinitesimal deformations of Levi flat hypersurfaces

Abstract

In order to study the deformations of foliations of codimension 1 of a smooth manifold L, de Bartolomeis and Iordan defined the DGLA \( \mathcal {Z}^{*}\left( L\right) \), where \(\mathcal {Z}^{*}\left( L\right) \) is a subset of differential forms on L. In another paper, de Bartolomeis and Iordan studied the deformations of foliations of a smooth manifold L by defining the canonical solutions of Maurer–Cartan equation in the DGLA of graded derivations \(\mathcal {D}^{*}\left( L\right) \). Let L be a Levi flat hypersurface in a complex manifold. Then the deformation theories in \(\mathcal {Z}^{*}\left( L\right) \) and \(\mathcal {D }^{*}\left( L\right) \) lead to the moduli space for the Levi flat deformations of L. In this paper we discuss the relationship between the infinitesimal deformations of L defined by the solutions of Maurer–Cartan equation in \(\mathcal {Z}^{*}\left( L\right) \) and the infinitesimal deformations of L obtained by means of the canonical solutions of Maurer–Cartan equation in the DGLA of graded derivations \(\mathcal {D}^{*}\left( L\right) \).