Geometric substructures and uniruled projective subvarieties of Fano manifolds of Picard number 1

Abstract

In the late 1990s Hwang and Mok initiated a geometric theory of uniruled projective manifolds modeled on their varieties of minimal rational tangents (VMRTs), proving later on Cartan-Fubini extension (2001), according to which a germ of VMRT-preserving biholomorphism f : (X; x0) → (Y ; y0) between two Fano manifolds of Picard number 1 extends necessarily to a biholomorphism F : X → Y whenever their VMRTs are of positive dimension and Gauss maps on VMRTs are generically finite. Hong-Mok (2010) extended Cartan-Fubini extension to the non-equidimensional setting under a relative nondegeneracy condition on second fundamental forms on VMRTs. Recently Mok-Zhang considered the analytic continuation of a germ of complex submanifold (S; x0) ,→ (X; x0) of a Fano manifold of Picard number 1 uniruled by lines, where S inherits a sub-VMRT structure defined by intersections of VMRTs with tangent spaces. Assuming that sub-VMRTs satisfy new non-degeneracy conditions and that the distribution spanned by sub-VMRTs is bracket-generating, we showed that S ⊂ Z for some irreducible subvariety Z ⊂ X, dim(Z) = dim(S) by constructing a universal family of chains of rational curves by an analytic process and proving its algebraicity.