Cartan-Fubini type extension of holomorphic maps respecting varieties of minimal rational tangents

Abstract

A fundamental result in the theory of minimal rational curves on projective manifolds is Cartan-Fubini extension theorem proved by Hwang and Mok, which describes the extensibility of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents (VMRT), under a mild geometric assumption on the second fundamental forms of VMRT’s. Hong and Mok have developed Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends VMRT’s onto linear sections of VMRT’s and it satisfies a mild geometric condition formulated in terms of second fundamental forms on VMRT’s. In the current paper, we give a generalization of Hong and Mok’s result, under the same condition on second fundamental forms, assuming only that the holomorphic immersions send VMRT’s to VMRT’s. Our argument is different from Hong and Mok’s and is based on the study of natural foliations on the total family of VMRT’s. This gives a substantially simpler proof than Hong and Mok’s argument.