arXiv subjects

Costantino Medori

Publications and source records attributed to Costantino Medori.

At least 19 recordsLinked to original sources

arXiv2026

CR Orbits on the complex flag manifolds of $G_2$

We give an explicit description of the orbits of the split real form $G_{2(2)}$ on the complex flag manifolds of $G_2$ and study their induced invariant CR structures. In particular we study their Levi forms and show that all possible cases in the simple homogeneous setting occur, namely Levi nondegeneracy, $k$-nondegeneracy for $2\leq k\leq 3$, and holomorphic degeneracy. Moreover, following the classical approach to orbits on flag manifolds, we give equations for these homogeneous CR manifolds.

math.DG↗

arXiv2026

Classification of homogeneous almost complex $4$-manifolds with non-degenerate torsion bundle

We investigate the local and global geometry of almost complex $4$-manifolds admitting non-degenerate torsion bundle. The rigidity of these structures forces a parallelizable $J$-adapted double cover, which imposes severe topological constraints on the underlying manifold. Exploiting this rigidity, we give a complete classification in the homogeneous setting. We show that such a manifold is diffeomorphic either to a $4$-dimensional Lie group carrying an almost complex structure with non-degenerate torsion bundle, or to a product $L(4,1)\times\mathbb R$ or $L(4,1)\times\mathbb T$, where $L(4,1)$ is a lens space. We also determine exactly which real $4$-dimensional Lie algebras admit such a structure. Constructively, we realize every admissible algebra by an explicit invariant structure, thereby closing the existence question in dimension $4$. We also relate these structures to certain Engel structures that we call Nijenhuis--Engel, and answer the resulting existence questions in the homogeneous case.

math.DG↗

arXiv2026

On contact and finitely Levi-nondegenerate CR algebras

We study CR-manifolds of arbitrary CR codimension, mainly focusing on Levi and contact-nondegeneracy and depth. We investigate these and other invariants in the locally homogeneous case, developing a comprehensive theory which establishes correspondences with related properties of the associated CR-algebras and, in the parabolic case, with the combinatorics of their cross-marked painted root diagrams.

math.DG↗

arXiv2025

On Homogeneous CR Manifolds of Arbitrary Order of Levi Nondegeneracy

This paper present homogeneous CR hypersurfaces satisfying the $CR$-invariant property of being $k$-nondegenerate for an arbitrary integer $k\geq 1$. The construction of such homogeneous manifolds are based on $CR$ algebras defined by irreducible representations of $\mathfrak{su}(2)$. An explicit study of the iterated Levi forms with their respective kernels, along with the local model equation, is given.

math.DG↗

arXiv2022

On finitely nondegenerate closed homogeneous CR manifolds

A complex flag manifold F= G /Q decomposes into finitely many real orbits under the action of a real form of G. Their embedding into F define on them CR manifold structures. We characterize the closed real orbits which are finitely nondegenerate.

math.DG↗

arXiv2021

Families of almost complex structures and transverse $(p,p)$-forms

An {\em almost p-Kähler manifold} is a triple $(M,J,Ω)$, where $(M,J)$ is an almost complex manifold of real dimension $2n$ and $Ω$ is a closed real tranverse $(p,p)$-form on $(M,J)$, where $1\leq p\leq n$. When $J$ is integrable, almost $p$-Kähler manifolds are called $p$-{\em Kähler manifolds}. We produce families of almost $p$-Kähler structures $(J_t,Ω_t)$ on $\C^3$, $\C^4$, and on the real torus $\mathbb{T}^6$, arising as deformations of Kähler structures $(J_0,g_0,ω_0)$, such that the almost complex structures $J_t$ cannot be locally compatible with any symplectic form for $t\neq 0$. Furthermore, examples of special compact nilmanifolds with and without almost $p$-Kähler structures are presented.

math.DG↗

arXiv2020

Higher order Levi forms on homogeneous CR manifolds

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous $CR$ manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal $3$ and the compact ones less or equal~$2$. Finally we construct by Lee extensions weakly nondegenerate $CR$ vector bundles with arbitrary orders of nondegeneracy.

math.DG↗

arXiv2019

L-prolongations of graded Lie algebras

In this paper we translate the necessary and sufficient conditions of Tanaka's theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries, which are defined by assigning a structure algebra on the contact distribution.

math.DG↗

arXiv2019

On some classes of Z-graded Lie algebras

We study finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi-Malčev and Levi-Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we systematically present examples in which simple Lie algebras are obtained as prolongations, for reductive structural algebras of type A, B, C and D, of nilpotent Z-graded Lie algebras arising as their linear representations.

math.DG↗

arXiv2017

Coverings for $4$-dimensional almost complex manifolds with non-degenerate torsion

An almost complex manifolds $(M^4,J)$ of real dimension 4 with non-degenerate torsion bundle admit a double absolute parallelism and it is provided the classification of homogeneous $(M^4,J)$ having an associated non-solvable Lie algebra. We extend such a classification to the analysis of the manifolds having an associated solvable Lie algebra, up-to-coverings. Moreover, for homogeneous $(M^4,J)$ we provide examples with connected and non-connected double covering, thus proving that in general the double absolute parallelism is not the restriction of two absolute parallelisms. Furthermore, it is given the definition of a natural metric induced by the absolute parallelisms on $(M^4,J)$ and an example of an almost complex manifold with non-degenerate torsion endowed with that metric such that it becomes an almost Kähler manifold.

math.DG↗

arXiv2017

On transitive contact and $CR$ algebras

We consider locally homogeneous $CR$ manifolds and show that, under a condition only depending on their underlying contact structure, their $CR$ automorphisms form a finite dimensional Lie group.

math.DG↗

arXiv2016

Structure equations of Levi degenerate CR hypersurfaces of uniform type

We explicitly determine the structure equations of 5-dimensional Levi 2-nondegenerate CR hypersurfaces, using our recently constructed canonical Cartan connection for this class of CR manifolds. We also give an outline of the basic properties of absolute parallelisms and Cartan connections, together with a brief discussion of the absolute parallelisms for such CR manifolds existing in the literature.

math.DG↗

arXiv2013

The equivalence problem for 5-dimensional Levi degenerate CR manifolds

Let M be a CR manifold of hypersurface type, which is Levi degenerate but also satisfying a k-nondegeneracy condition at all points. This might be only if dim M is greater than or equal to 5 and if dim M = 5, then k= 2 at all points. We prove that for any 5-dimensional, uniformly 2-nondegenerate CR manifold M there exists a canonical Cartan connection, modelled on a suitable projective completion of the tube over the future light cone {z\in C^3: (x^1)^2+(x^2)^2-(x^3)^2 = 0, x^3>0}. This determines a complete solution to the equivalence problem for this class of CR manifolds.

math.DG↗

arXiv2012

On non-pure forms on almost complex manifolds

T.-J. Li and W. Zhang defined an almost complex structure $J$ on a manifold $X$ to be {\em \Cpf}, if the second de Rham cohomology group can be decomposed as a direct sum of the subgroups whose elements are cohomology classes admitting $J$-invariant and $J$-anti-invariant representatives. It turns out (see T. Draghici, T.-J. Li and W. Zhang) that any almost complex structure on a 4-dimensional compact manifold is \Cpf. We study the $J$-invariant and $J$-anti-invariant cohomology subgroups on almost complex manifolds, possibly non compact. In particular, we prove an analytic continuation result for anti-invariant forms on almost complex manifolds.

math.SG↗

arXiv2011

Reductive compact homogeneous CR manifolds

We consider a class of compact homogeneous CR manifolds, that we call $\mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration $S^3\to\mathbb{CP}^1$. In general these fibrations are not $CR$ submersions, however they satisfy a weaker condition that we introduce here, namely they are CR-deployments.

math.DG↗

arXiv2010

A characterization of CR quadrics with a symmetry property

We study CR quadrics satisfying a symmetry property $(\tilde S)$ which is slightly weaker than the symmetry property $(S)$, recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics satisfying the $(\tilde S)$ property in terms of their Levi-Tanaka algebras. In many cases the $(\tilde S)$ property implies the $(S)$ property; this holds in particular for compact quadrics. We also give a new example of a quadric such that the dimension of the algebra of positive-degree infinitesimal automorphisms is larger than the dimension of the quadric.

math.CV↗

arXiv2009

On homogeneous and symmetric CR manifolds

We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.

math.DG↗

arXiv2009

Orbits of real forms in complex flag manifolds

We investigate the $CR$ geometry of the orbits $M$ of a real form $G_0$ of a complex simple group $G$ in a complex flag manifold $X=G/Q$. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical $G_0$-equivariant and Mostow fibrations, and topological properties of the orbits.

math.CV↗