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

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 ${\mathcal{C}}_x^C\subset \mathbb{P}{T}_{x}X$ 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 ${\mathcal{C}}_x^C\subset \mathbb{P}{D}_x$. 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 ${\mathcal{C}}_x^C\subset \mathbb{P}{D}_x$ 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.

arXiv comment20 pages, to appear in J. Math. Soc. Japan