Splitting submanifolds in rational homogeneous spaces of Picard number one

TL;DR

This paper proves that splitting submanifolds in certain rational homogeneous spaces are themselves rational homogeneous or Hermitian symmetric, using global holomorphic vector fields to classify these submanifolds.

Contribution

It introduces a new approach using vector fields to classify splitting submanifolds, extending previous algebraic geometry results to differential geometry.

Findings

Splitting submanifolds are rational homogeneous in large classes of spaces.

In Hermitian symmetric spaces, splitting submanifolds are also Hermitian symmetric.

Provides a differential geometric proof for classification in hyperquadrics.

Abstract

Let $M$ be a complex manifold. We prove that a compact submanifold $S\subset M$ with splitting tangent sequence (called a splitting submanifold) is rational homogeneous when $M$ is in a large class of rational homogeneous spaces of Picard number one. Moreover, when $M$ is irreducible Hermitian symmetric, we prove that $S$ must be also Hermitian symmetric. The basic tool we use is the restriction and projection map $\pi$ of the global holomorphic vector fields on the ambient space which is induced from the splitting condition. The usage of global holomorphic vector fields may help us set up a new scheme to classify the splitting submanifolds in explicit examples, as an example we give a differential geometric proof for the classification of compact splitting submanifolds with $\dim\geq 2$ in a hyperquadric, which has been previously proven using algebraic geometry.