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 𝒪_{ℙ¹}(1)^⊕𝑝 ⊕ 𝒪_{ℙ¹}^{⊕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 ∈ 𝐶, which is the germ of submanifolds 𝒞^𝐶_𝑥 ⊂ ℙ𝑇_𝑥𝑋 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 𝒞^𝐶_𝑥 ⊂ ℙ𝐷_𝑥. When 𝐷 ⊂ 𝑇𝑋 is a contact distribution, a well-known necessary condition is that 𝒞^𝐶_𝑥 should be Legendrian with respect to the induced contact structure on ℙ𝐷_𝑥. 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 𝒞^𝐶_𝑥 ⊂ ℙ𝐷_𝑥 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.