Varieties of Minimal Rational Tangents on Uniruled Projective Manifolds

Summary

On a polarized uniruled projective manifold we pick an irreducible component X of the Chow space whose generic members are free rational curves of minimal degree. The normalized Chow space of minimal rational curves marked at a generic point is nonsingular, and its strict transform under the tangent map gives a variety of minimal rational tangents, or VMRT. In this survey we present a systematic study of VMRT by means of techniques from differential geometry (distributions, G-structures), projective geometry (the Gauss map, tangency theorems), the deformation theory of (rational) curves, and complex analysis (Hartogs phenomenon, analytic continuation). We give applications to a variety of problems on uniruled projective manifolds, especially on irreducible Hermitian symmetric manifolds S of the compact type and more generally on rational homogeneous manifolds G/P of Picard number 1, including the deformation rigidity of S and the same for homogeneous contact manifolds of Picard number 1, the characterization of S of rank at least 2 among projective uniruled manifolds in terms of G-structures, solution of Lazarsfeld's Problem for finite holomorphic maps from G/P of Picard number 1 onto projective manifolds, local rigidity of finite holomorphic maps from a fixed projective manifold onto G/P of Picard number 1 other than ℙⁿ , and a proof of the stability of tangent bundles of certain Fano manifolds.

Rational curves play a crucial role in the study of Fano manifolds. By Mori's theory, Fano manifolds are uniruled. We consider more generally uniruled projective manifolds. Fixing an ample line bundle and considering only components of the Chow space whose generic members are free rational curves, we introduce the notion of minimal rational curves by minimizing the degree of a generic member.