Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures

TL;DR

This paper characterizes the varieties of minimal rational tangents (VMRT) of unbendable rational curves subordinate to contact structures, establishing a correspondence with Legendrian submanifolds and constructing explicit examples using contact geometry and symplectic techniques.

Contribution

It proves that Legendrian submanifolds can be realized as VMRTs of unbendable rational curves in contact manifolds, providing a constructive method.

Findings

VMRTs of unbendable rational curves are Legendrian when tangent to contact distributions.

Constructs examples of contact manifolds with prescribed Legendrian VMRTs.

Links contact geometry, symplectic geometry, and rational curve deformation theory.

Abstract

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${\mathcal O}_{{\mathbb P}^1}(1)^{\oplus p} \oplus {\mathcal O}_{{\mathbb P}^1}^{\oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C,$ which is the germ of submanifolds ${\mathcal C}^C_x \subset {\mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT ${\mathcal C}^C_x \subset {\mathbb P} D_x$. When $D \subset TX$ is a contact distribution, a well-known necessary condition is that ${\mathcal C}_x^C$ should be Legendrian with respect to the induced contact structure on ${\mathbb P} D_x$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D \subset TX$ and an unbendable rational curve $C \subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT ${\mathcal C}^C_x \subset {\mathbb P} D_x$ at some point $x\in C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.