{
  "id": "1712.03458",
  "version": "v1",
  "published": "2017-12-10T00:35:22.000Z",
  "updated": "2017-12-10T00:35:22.000Z",
  "title": "Inequalities of Chern classes on nonsingular projective $n$-folds of Fano or general type with ample canonical bundle",
  "authors": [
    "Rong Du",
    "Hao Sun"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a nonsingular projective $n$-fold $(n\\ge 2)$ of Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $\\kappa$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \\cdots, c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under Gauss map. Moreover, we show that if characteristic of $\\kappa$ is $0$, then the Chern ratios $(\\frac{c_{2,1^{n-2}}}{c_{1^n}}, \\frac{c_{2,2,1^{n-4}}}{c_{1^n}}, \\cdots, \\frac{c_{n}}{c_{1^n}})$ is contained in a convex polyhedron for all $X$. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt (\\cite{Hun}) to all dimensions. As a corollary, we can get that there exists constants $d_1$ and $d_2$ such that $d_1K_X^n\\le\\chi_{top}(X)\\le d_2 K_X^n.$ If characteristic of $\\kappa$ is positive, $K_X$ is ample and $\\mathscr{O}_X(K_X)$ is globally generated, then the same results hold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-10T00:35:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ample canonical bundle",
      "general type",
      "chern classes",
      "nonsingular projective",
      "inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}