Deformations of Levi-flat structures in smooth manifolds

TL;DR

This paper introduces a cohomological framework to study infinitesimal deformations of Levi-flat structures on smooth manifolds, linking complex geometry and deformation theory.

Contribution

It defines a new cohomology group capturing infinitesimal Levi-flat deformations and relates it to existing cohomologies in the real analytic case.

Findings

Cohomology group of order 1 contains infinitesimal deformations.

In the real analytic case, this group decomposes into d-bar cohomology and DGLA cohomology.

Provides a new tool for studying Levi-flat structure deformations.

Abstract

We define a complex whose cohomology group of order 1 contains the infinitesimal deformations of a Levi flat structure on a smooth manifold. In the case of real analytic Levi flat structures, this cohomology group is the product of the d-bar cohomology group of order 1 of tangent vector fields to the Levi structure and the cohomology group of order 1 of the associated DGLA.