2022 Volume 74 Issue 2 Pages 571-590
A nonsingular rational curve πΆ in a complex manifold π whose normal bundle is isomorphic to πͺ_{β1}(1)βπ β πͺ_{β1}^{βπ} for some nonnegative integers π and π is called an unbendable rational curve on π. Associated with it is the variety of minimal rational tangents (VMRT) at a point π₯ β πΆ, which is the germ of submanifolds ππΆπ₯ β βππ₯π consisting of tangent directions of small deformations of πΆ fixing π₯. Assuming that there exists a distribution π· β ππ such that all small deformations of πΆ are tangent to π·, one asks what kind of submanifolds of projective space can be realized as the VMRT ππΆπ₯ β βπ·π₯. When π· β ππ is a contact distribution, a well-known necessary condition is that ππΆπ₯ should be Legendrian with respect to the induced contact structure on βπ·π₯. We prove that this is also a sufficient condition: we construct a complex manifold π with a contact structure π· β ππ and an unbendable rational curve πΆ β π such that all small deformations of πΆ are tangent to π· and the VMRT ππΆπ₯ β βπ·π₯ at some point π₯ β πΆ 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.
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