Related papers: Some quasihomogeneous Legendrian varieties

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…

We investigate the geometry of Legendrian complex projective manifolds $X\subset\PP V$. By definition, this means $V$ is a complex vector space of dimension $2n+2$, endowed with a symplectic form, and the affine tangent space to $X$ at each…

In this note, we classify smooth equivariant compactifications of $\mathbb{G}_a^n$ which are Fano manifolds with index $\geq n-2$.

We prove that a general hyperplane section of a smooth Legendrian subvariety in a projective space admits Legendrian embedding into another projective space. This gives numerous new examples of smooth Legendrian subvarieties, some of which…

We classify smooth Fano manifolds X with the Picard number $\rho_X \geq 3$ such that there exists an extremal ray which has a birational contraction that maps a divisor to a point.

In this article, we determine all equivariant compactifications of the three-dimensional vector group $\mathbf{G}_a^3$ which are smooth Fano threefolds with Picard number greater or equal than two.

I prove that every smooth legendrian variety generated by quadrics is a homogeneous variety and further I give a list of all such legendrian varieties. A review of the subject is included, illustrated by examples. Another result is that no…

We construct an explicit semistable degeneration of a Fano eightfold of index three and deduce its Hodge numbers, in particular we show that it has Picard rank one. The Fano variety is of K3 type and it is defined as a connected component…

Let $X$ be a complex smooth Fano variety of dimension at least four. In this paper, we classify such $X$ when the pseudoindex is at least $n-2$ and the Picard number greater than one. We also discuss the relations between pseudoindex and…

For $n\geq 4$, let $X$ be a complex smooth Fano $n$-fold whose minimal anticanonical degree of non-free rational curves on $X$ is at least $n-2$. We classify extremal contractions of such varieties. As an application, we obtain a…

In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal…

We give an $h$--principle type result for a class of Legendrian embeddings in contact manifolds of dimension at least $5$. These Legendrians, referred to as loose, have trivial pseudo-holomorphic invariants. We demonstrate they are…

We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded…

We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds…

We study the varieties of reductions associated to the variety of rank one matrices in $\fgl\_n$. These varieties are defined as natural compactifications of the different ways to write the identity matrix as a sum of $n$ rank one matrices.…

We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically…

Let X be an $n$-dimensional Fano manifold of Picard number 1. We study how many different ways X can compactify the complex vector group C^n equivariantly. Hassett and Tschinkel showed that when X = P^n with n \geq 2, there are many…

For a symplectic vector space $V$, a projective subvariety $Z \subset {\bf P} V$ is a Legendrian variety if its affine cone $\widehat{Z} \subset V$ is Lagrangian. In addition to the classical examples of subadjoint varieties associated to…

Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image H of N_1(D) in N_1(X) under the natural push-forward of 1-cycles. We show that the codimension c of H in N_1(X) is at most 8.…

In this short note we discuss certain examples of Legendrian submanifolds, whose linearized Legendrian contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated…