N. Mok - Linearly saturated subvarieties on uniruled projective manifolds: Complex analytic and differential geometry 2017

Let Z and X be uniruled projective manifolds of Picard number 1 such that the respective variety of minimal rational tangents (VMRT) at a general point satisfies a nondegeneracy condition on the second fundamental form. In 2001 Hwang and Mok established the equidimensional Cartan-Fubini extension principle, according to which a germ of VMRT-preserving holomorphic map f : (Z, z0) → (X, x0) must necessarily extend to a biholomorphism F : Z→X. In 2010, Hong and Mok extended this to the nonequidimensional case for germs of holomorphic immersions between uniruled projective manifolds, allowing dim(Z) < dim(X), by proving that f must necessarily extend to a rational map provided that a certain relative version of the nondegeneracy condition on the second fundamental form is satisfied. Very recently, Mok and Zhang developed the theory of geometric substructures by considering germs of complex submanifolds of (S, x0) ↪ (X, x0) and introducing geometric substructures on S by taking intersections of the VMRTs of X with projectivized tangent spaces of S. We introduced a new relative nondegeneracy condition related to thesecond fundamental form and proved the extendibility of the germ (S, x0) to a projective subvariety Y ⊂ X under the assumption that X is uniruled by lines, i.e., by rational curves whose homology classes are the positive generator of H2(X,Z)∼=Z. We achieved this by recoveringYas the image under a tautological map of a certain universal family of chains of minimal rational curves. The existence of the latter family is obtained by means of analytic continuation of the Thullen type for germs of holomorphic substructures