Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures (Q2134219)

This article is concerned with unbendable rational curves, which play a distinguished role in the study of uniruled projective manifolds in algebraic geometry. A nonsingular rational curve in a complex manifold \(X\) of dimension \(n\) is here called unbendable, if its normal bundle is isomorphic to \[ \mathcal O_{\mathbb P^1}(1)^{\oplus p}\oplus \mathcal O_{\mathbb P^1}^{\oplus(n-1-p)} \] for some nonnegative integer \(p\), which implies that deformations of an unbendable rational curve in \(X\) are unobstructed and small deformations of it are again unbendable rational curves. An important invariant of an unbendable rational curve \(C\subset X\) is its variety of minimal rational tangents (VMRT) at a point \(x\in C\), which is the germ of submanifolds \(\mathcal C^C_x\subset \mathbb P T_xX\) consisting of tangent directions of small deformations of \(C\) fixing \(x\). There are many instances where an unbendable rational curve \(C\subset X\) is subordinate to a distribution \(D\subset TX\) on \(X\), in the sense that all small deformations of \(C\) are tangent to \(D\). This raises the following natural question addressed here: What kind of submanifolds of projective space \(\mathbb P^{r-1}\) may be realised as the VMRT \(\mathcal C^C_x\subset \mathbb P D_x\) at some point \(x\) of a unbendable rational curve \(C\subset X\) subordinate to a distribution \(D\subset TX\) of rank \(r\)? If \(D\subset TX\) is a contact distribution it has been known that a necessary condition is that \(\mathcal C^C_x\) is Legendrian with respect to the induced contact structure on \(\mathbb P D_x\). In this article the author shows that this condition is also sufficient. More precisely, it is shown that, given an arbitrary locally closed Legendrian submanifold \(S\subset \mathbb P V\) in the projectivisation of a symplectic vector space \(V\), then there exist a contact manifold \((X,D)\) and an unbendable rational curve \(C\subset X\) subordinate to \(D\) such that \(\mathcal C^C_x\subset \mathbb P D_x\) at some point \(x\in C\) is projectively isomorphic to an open subset of \(S\subset \mathbb P V\); see Theorem 1.3 or Theorem 4.9 for a more precise version. The proof employs the geometry of contact lines of the Heisenberg group and takes advantage of the symplectic geometry of subbundles of the cotangent bundle of a manifold.