arXiv subjects
Publications and source records attributed to Andrei Iordan.
We prove that the normal bundle to the Levi foliation of a smooth Levi flat hypersurface does not admit a Hermitian metric with positive curvature along the leaves in compact Kähler manifolds of dimension at least 3. This represents an answer to a conjecture of Marco Brunella.
Let M be a smooth manifold and $Φ$ a differential 1-form on M with values in the tangent bundle TM. We construct canonical solutions $e_Φ$ of Maurer-Cartan equation in the DGLA of graded derivations D*(M) of differential forms on M by means of deformations of d depending on $Φ$. This yields to a classification of the canonical solutions of the Maurer-Cartan equation according to their type: $e_Φ$ is of finite type r if there exists $r\in N$ such that $Φ^r[Φ,Φ]_{FN} = 0$ and r is minimal with this property, where $[.,.]_{FN}$ is the Frölicher-Nijenhuis bracket. A distribution $ξ\subset TM$ of codimension k > 1 is integrable if and only if the canonical solution $e_Φ$ associated to the endomorphism $Φ$ of TM which is trivial on $ξ$ and equal to the identity on a complement of $ξ$ in TM is of finite type $\leq 1$, respectively of finite type 0 if k = 1.
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.
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.