Holomorphic Legendrian curves in projectivised cotangent bundles

TL;DR

This paper investigates holomorphic Legendrian curves in the projectivised cotangent bundle of complex manifolds, providing approximation, deformation, and h-principle results, especially when the base manifold has special properties like being Stein or Oka.

Contribution

It introduces new deformation and approximation theorems for Legendrian curves, including parametric and h-principle results, in the setting of complex contact manifolds.

Findings

Any vertical holomorphic curve can be deformed into a horizontal Legendrian curve.

Stronger results hold when the base manifold is Stein or Oka.

Established basic and parametric h-principles for Legendrian curves.

Abstract

We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a complex manifold $Z$ of dimension at least $2$. We provide a detailed analysis of Legendrian curves degenerating to vertical curves and obtain several approximation and general position theorems. In particular, we prove that any vertical holomorphic curve $M\to X$ from a compact bordered Riemann surface $M$ can be deformed to a horizontal Legendrian curve by an arbitrarily small deformation. A similar result is proved in the parametric setting, provided that all vertical curves under consideration are nondegenerate. Stronger results are obtained when the base $Z$ is an Oka manifold or a Stein manifold with the density property. Finally, we establish basic and 1-parametric h-principles for holomorphic Legendrian curves in $X$.