Geometric structures and substructures on uniruled projective manifolds

Abstract

With J.-M. Hwang the speaker has developed a geometric theory of uniruled projective manifolds (X) modeled on varieties of minimal rational tangents (mathcal{C}_x(X) subset Bbb PT_x(X)), alias VMRTs. Generalizing works of Hwang-Mok, Hong-Mok considered pairs ((X_0;X)) of uniruled projective manifolds, and established a non-equidimensional Cartan-Fubini Extension Principle (2010) in terms of a certain non-degeneracy condition on the second fundamental form for a pair ((mathcal B subsetmathcal A)) consisting of a VMRT (mathcal A ,) and a linear section (mathcal B ,) of (mathcal A). The latter has led to the characterization of standard embeddings (i: G_0/P_0 hookrightarrow G/P) between rational homogeneous manifolds of Picard number 1 by Hong-Mok (2010) in the long-root and non-linear cases and by Hong-Park (2011) in the short-root cases and in the cases of linear subspaces with identifiable exceptions. The argument therein involving parallel transport of VMRTs has also been applied by Hong-Mok (2013) to establish homological rigidity for certain smooth Schubert cycles. Recently in a joint work with Y. Zhang we have established a stronger rigidity phenomenon for sub-VMRT structures, where in place of a germ of mapping (f: (X_0;0) o (X;0)) we consider a germ of submanifold ((S;0) subset (X;0)) for a uniruled projective manifold (X) equipped with a minimal rational component (mathcal K). Defining a sub-VMRT structure by taking intersections (mathcal C_x(X) cap Bbb PT_x(S)) we have obtained sufficient conditions for (S) to extend to a rationally saturated projective subvariety (Z subset X). In the rational homogeneous case the method yields a strengthening of the results of Hong-Mok and Hong-Park. For instance, if a germ of submanifold ((S;0) subset (X;0)) inherits by intersecting VMRTs with projectivized tangent subspaces a Grassmann structure of rank (ge 2), then (S) in fact extends to a sub-Grassmannian in its standard embedding.