arXiv subjects

Paolo de Bartolomeis

Publications and source records attributed to Paolo de Bartolomeis.

4 recordsLinked to original sources

arXiv2018

Maurer-Cartan equation in the DGLA of graded derivations

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.

math.CV↗

arXiv2014

Deformations of Levi flat hypersurfaces in complex manifolds

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.

math.CV↗

arXiv2014

Deformations of Levi-flat structures in smooth manifolds

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.

math.CV↗

arXiv2011

Some Remarks on Nijenhuis Bracket, Formality, and Kähler Manifolds

One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $δ$ such that $dδ$-lemma holds. We first show that such derivation $δ$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $δ$ (which is a natural, essentially necessary condition to get a $dδ$-lemma) is equivalent to vanishing of the Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that ensure the $dδ$-lemma holds: we consider the cases when $R$ is self adjoint with respect to a Riemannian metric or compatible with an almost symplectic structure. Finally, we show that if $R$ is scew-symmetric with respect to a Riemannian metric, has constant determinant, and if its Nijenhujis torsion vanishes, then the orthogonal component of $R$ in its polar decomposition is a complex structure compatible with the metric, which gives us a new characterization of Kähler structures

math.DG↗