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

Abstract

© 2022 The Mathematical Society of JapanA nonsingular rational curve C in a complex manifold X whose normal bundle is isomorphic to OP1 (1)p OP 1q 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 ∈ C, which is the germ of submanifolds CxC ⊂ PTxX consisting of tangent directions of small deformations of C fixing x. Assuming that there exists a distribution D ⊂ 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 CxC ⊂ PDx. When D ⊂ TX is a contact distribution, a well-known necessary condition is that CxC should be Legendrian with respect to the induced contact structure on PDx. We prove that this is also a sufficient condition: we construct a complex manifold X with a contact structure D ⊂ TX and an unbendable rational curve C ⊂ X such that all small deformations of C are tangent to D and the VMRT CxC ⊂ PDx at some point x ∈ 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.