Rigidity properties of holomorphic Legendrian singularities

TL;DR

This paper investigates the rigidity of Legendrian singularities in complex contact manifolds, proving that certain singularities are biholomorphically equivalent to their tangent cones and that normal Legendrian singularities are deformation-rigid.

Contribution

It establishes two new rigidity theorems for Legendrian singularities, advancing understanding of their local and deformation properties in complex contact geometry.

Findings

Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones.

Normal Legendrian singularities exhibit deformation-rigidity, with trivial holomorphic families up to contactomorphisms.

The proofs leverage the relation between infinitesimal contactomorphisms and holomorphic line bundle sections.

Abstract

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 biholomorphic to their tangent cones. This result is partly motivated by a problem on Fano contact manifolds. The second result is the deformation-rigidity of normal Legendrian singularities, meaning that any holomorphic family of normal Legendrian singularities is trivial, up to contactomorphic biholomorphisms of germs. Both results are proved by exploiting the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle on the contact manifold.