Special lines on contact manifolds

TL;DR

This paper investigates singular and special splitting type contact lines on contact manifolds, extending previous work on smooth lines and exploring restrictions on their families and implications for the structure of contact Fano manifolds.

Contribution

It generalizes results about contact lines to singular and special splitting types, including non-Fano contact manifolds, and establishes restrictions on their families.

Findings

Restrictions on families of singular contact lines.

Extension of results to non-Fano contact manifolds.

Dimension bounds for families of singular lines.

Abstract

In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and has a fixed standard splitting type of the restricted tangent bundle. In this paper we study singular contact lines and those with special splitting type. We provide restrictions on the families of such lines, and on contact Fano manifolds which have reducible varieties of minimal rational tangents. We also show that the results about singular lines naturally generalise to complex contact manifolds, which are not necessarily Fano, for instance, quasi-projective contact manifolds or compact contact manifolds of Fujiki class C. In particular, in many cases the dimension of a family of singular lines is at most 2 less than the dimension of the contact manifold.