Dual cones of varieties of minimal rational tangents

Abstract

The varieties of minimal rational tangents play an important role in the geometry of uniruled projective manifolds. The goal of this paper is to exhibit their role in the symplectic geometry of the cotangent bundles of uniruled projective manifolds. More precisely, let $X$ be a uniruled projective manifold satisfying the assumption that the VMRT at a general point is smooth. We show that the total family of dual cones of the varieties of minimal rational tangents is a coisotropic subvariety in $T^* (X)$. Furthermore, the closure of a general leaf of the null foliation of this coisotropic subvariety is an immersed projective space of dimension $\delta + 1$ where $\delta$ is the dual defect of the variety of minimal rational tangents at a general point. When $\delta = 0$, the symplectic reduction of the coisotropic variety can be realized as a subbundle of the cotangent bundle $T^* (\mathcal{K})$ of the parameter space $\mathcal{K}$ of the rational curves.