Some quasihomogeneous Legendrian varieties

Relatively Minimal Quasihomogeneous Projective 3-Folds

In the present work we classify the relatively minimal 3-dimensional quasihomogeneous complex projective varieties under the assumption that the automorphism group is not solvable. By relatively minimal we understand varieties X having at most Q-factorial terminal singularities and allowing an extremal contraction X to Y, where dim Y <3.

在勒让德簇中刻画次伴随簇

Characterizing subadjoint varieties among Legendrian varieties

本文利用迷向表示以及射影第三基本形式和勒让德簇上直线的特殊性质，刻画了具有正维自同构群的非奇异勒让德簇中的次伴随簇。

Legendrian varieties

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 point is a maximal isotropic subspace. We establish basic facts about their geometry and exhibit examples of inhomogeneous smooth Legendrian varieties, the first examples of such in dimension greater than one.

Algebraic 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 properties. The most remarkable case is the Legendrian subvarieties of projective space and prior to the author's research only few smooth examples of these were known. The first series of results of this thesis is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable m…

Cluster varieties from Legendrian knots

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact La…

Legendrian Fronts for Affine Varieties

In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed exact Lagrangian submanifolds. In addition, we prove that the Koras--Russell cubic is Stein deformation equivalent to affine complex 3-space and ve…