Related papers: Higher order Levi forms on homogeneous CR manifold…

In our earlier articles we studied tube hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we showed that the vanishing of the CR-curvature of such a hypersurface is equivalent…

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of…

We introduce a notion of degenerations of graded modules. In relation to it, we also introduce several partial orders as graded analogies of the hom order, the degeneration order and the extension order. We prove that these orders are…

We study the rigidity of holomorphic mappings from a neighborhood of a Levi-nondegenerate CR hypersurface $M$ with signature $l$ into a hyperquadric $Q_{l'}^{N} \subseteq \mathbb{CP}^{N+1}$ of larger dimension and signature. We show that if…

Applying Lie's theory, we show that any $\mathcal{C}^\omega$ hypersurface $M^5 \subset \mathbb{C}^3$ in the class $\mathfrak{C}_{2,1}$ carries Cartan-Moser chains of orders $1$ and $2$. Integrating and straightening any order $2$ chain at…

We present a novel construction of linear deformations for Lie algebras and use it to prove the non-rigidity of several classes of Lie algebras in different varieties. We consider the family of Lie algebras with an abelian factor showing…

We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we get that if M is a closed irreducible oriented 3-manifold that is not a graph manifold, for example a hyperbolic…

This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the quaternion skew field $\bf H$ and the octonion algebra $\bf O$. These groups are…

We classify all closed non-orientable P2-irreducible 3-manifolds with complexity up to 7, fixing two mistakes in our previous complexity-up-to-6 classification. We show that there is no such manifold with complexity less than 6, five with…

Into this note we collect topics related to homogeneous vector bundles, elliptic adjoint orbits and so forth.

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it…

Let $M^{2n+1}$, $n\ge 1$, be a smooth manifold with a pseudo-convex integrable CR structure of hypersurface type. We consider a sequence of CR invariant subsets $ M=\mathcal S_0 \supset \mathcal S_1 \supset \cdots \supset \mathcal S_{n}, $…

It was recently determined exactly through how many general points a nondegenerate curve with nonspecial hyperplane section can pass. This gives rise to a method of constructing reducible curves $C_1 \cup_\Gamma C_2 \to \mathbb{P}^r$ with…

We study, in some models of Levi flat CR manifolds, the correspondence between the tangential Cauchy-Riemann complex and the complex defined by the associated real foliation.

Oeljeklaus-Toma (OT) manifolds are complex non-K\"ahler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field.…

We consider deformations of left-invariant complex structures on simply connected semisimple compact Lie groups which are a priori non-invariant. Computing their cohomologies, we show that they are not actually biholomorphic to…

We study ample stable vector bundles on minimal rational surfaces. We give a complete classification of those moduli spaces for which the general stable bundle is both ample and globally generated. We also prove that if $V$ is any stable…

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In…

Let $M^3 \subset \mathbb{C}^2$ be a $\mathcal{C}^\omega$ Levi nondegenerate hypersurface. In the literature, Cartan-Moser chains are detected from rather advanced considerations: either from the construction of a Cartan connection…

VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to…