Characterization of distinguished uniruled projective subvarieties in terms of geometric substructures and applications to Kähler geometry

Abstract

Together with Jun-Muk Hwang we introduced in the late 1990s a geometric theory of uniruledprojective manifolds based on the variety of minimal rational tangents (VMRT), i.e., the collec-tion of tangents to minimal rational curves on a uniruled projective manifold (X,K)equippedwith a minimal rational component. This theory provides differential-geometric tools for thestudy of uniruled projective manifolds, especially Fano manifolds of Picard number 1. Associ-ated to (X,K) is the fibered spaceπ:C(X)→Xof VMRTs called the VMRT structure on(X,K). More recently, taking (X,K) as an ambient space, with collaborators the author has beenstudying the geometry of germs of complex submanifolds on them in analogy to the geometry ofsubmanifolds in Riemannian manifolds. We focus on germs of complex submanifolds (S;x0)on(X,K) inheriting geometric substructures, to be called sub-VMRT structures, obtained from in-tersections of VMRTs with tangent subspaces, i.e., from π:C(S)→S,C(S):=C(X)∩PT(S).Central to our study is the characterization of certain classical Fano manifolds such as flag man-ifoldsG/Por distinguished uniruled projective subvarieties on them such as Schubert cycles interms of VMRTs and sub-VMRTs. As applications I will relate the theory to the existence and5 uniqueness of certain classes of holomorphic isometries into bounded symmetric domains. Foruniqueness results parallel transport (holonomy), a notion of fundamental importance both inK ̈ahler geometry and in the study of sub-VMRT structures, plays an important role.