{
  "id": "2012.14741",
  "version": "v1",
  "published": "2020-12-29T13:12:38.000Z",
  "updated": "2020-12-29T13:12:38.000Z",
  "title": "Splitting submanifolds in rational homogeneous spaces of Picard number one",
  "authors": [
    "Cong Ding"
  ],
  "comment": "22 pages",
  "categories": [
    "math.AG",
    "math.DG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-29T13:12:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C30",
      "32M10"
    ],
    "keywords": [
      "rational homogeneous spaces",
      "splitting submanifold",
      "picard number",
      "global holomorphic vector fields",
      "hermitian symmetric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}