Lines on contact manifolds II

TL;DR

This paper investigates the deformation of Legendrian subvarieties in complex contact manifolds, providing new insights into minimal rational curves and advancing the classification of contact manifolds.

Contribution

It studies deformations of Legendrian subvarieties generated by moving base points and answers a key question about tangent vectors and minimal rational curves in contact manifolds.

Findings

A positive answer to Hwang's question for contact manifolds.

The normalization of minimal rational curves through a point is a projective cone.

Sufficiently general tangent vectors are contained in at most one minimal rational curve.

Abstract

Complex contact manifolds have recently received considerable attention. Many of the newer publications approach contact manifolds via the covering family of minimal rational curves. This short note furthers the study of these curves. It is known that for any point x in X, the subvariety, which is covered by those curves which contain x, is Legendrian. We will now study the deformations of these subvarieties which are generated by moving the base point. As a main application, we give a positive answer to a question of J.M. Hwang in the case of contact manifolds: a sufficiently general tangent vector is contained in at most a single minimal rational curve. The author believes that this is a necessary step towards a full classification of contact manifolds. We give a second application by showing that the normalization of the subvariety of minimal curves through x is isomorphic to a projective cone.