Deformations of Levi flat hypersurfaces in complex manifolds

TL;DR

This paper develops a deformation theory for Levi flat hypersurfaces in complex manifolds, characterizing their rigidity and nonexistence in certain classes of manifolds using a DGLA framework.

Contribution

It introduces a deformation framework for Levi flat hypersurfaces via Maurer-Cartan solutions in a DGLA, establishing rigidity results and nonexistence theorems.

Findings

Proves infinitesimal rigidity of certain Levi flat hypersurfaces.

Provides conditions for rigidity in Kahler manifolds.

Shows nonexistence of specific Levi flat hypersurfaces in some complex manifolds.

Abstract

We first give a deformation theory of integrable distributions of codimension 1. We define a parametrization of families of smooth hypersurfaces near a Levi flat hypersurface L such that the Levi flat deformations are given by the solutions of the Maurer-Cartan equation in a DGLA associated to the Levi foliation. We say that L is infinitesimally rigid if the tangent cone at the origin to the moduli space of Levi flat deformations of L is trivial. We prove the infinitesimal rigidity of compact transversally parallelisable Levi flat hypersurfaces in compact complex manifolds and give sufficient conditions for infinitesimal rigidity in Kahler manifolds. As an application, we prove the nonexistence of transversally parallelizable Levi flat hypersurfaces in a class of manifolds which contains the complex projective plane.