{
  "id": "1510.02242",
  "version": "v1",
  "published": "2015-10-08T09:06:56.000Z",
  "updated": "2015-10-08T09:06:56.000Z",
  "title": "Some remarks on the uniqueness of the complex projective spaces",
  "authors": [
    "Ping Li"
  ],
  "comment": "7 pages",
  "categories": [
    "math.DG",
    "math.AT"
  ],
  "abstract": "We first notice in this article that if a compact K\\\"{a}hler manifold has the same integral cohomology ring and Pontrjagin classes as the complex projective space $\\mathbb{C}P^n$, then it is biholomorphic to $\\mathbb{C}P^n$ provided $n$ is odd. The same holds for even $n$ if we further assume that $M$ is simply-connected. This technically refines a classical result of Hirzebruch-Kodaira and Yau. This observation, together with a result of Dessai and Wilking, enables us to characterize all $\\mathbb{C}P^n$ in terms of homotopy type under mild symmetry. When $n=4$, we can drop the requirement on Pontrjagin classes by showing that a simply-connected compact K\\\"{a}hler manifold having the same integral cohomology ring as $\\mathbb{C}P^4$ is biholomorphic to $\\mathbb{C}P^4$, which improves on results of Fujita and Libgober-Wood.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-08T09:06:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32Q15",
      "53C55",
      "32Q55"
    ],
    "keywords": [
      "complex projective space",
      "pontrjagin classes",
      "integral cohomology ring",
      "uniqueness",
      "mild symmetry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151002242L"
    }
  }
}