{
  "id": "1711.06388",
  "version": "v1",
  "published": "2017-11-17T03:41:49.000Z",
  "updated": "2017-11-17T03:41:49.000Z",
  "title": "On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold",
  "authors": [
    "Zhiwei Wang"
  ],
  "journal": "Ann. Polon. Math., 117(1):41--58, 2016",
  "categories": [
    "math.DG",
    "math.CV"
  ],
  "abstract": "This paper divides into two parts. Let $(X,\\omega)$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $\\omega$ satisfies the assumption that $\\partial\\overline{\\partial}\\omega^k=0$ for all $k$, we generalize the volume of the cohomology class in the K\\\"{a}hler setting to the Hermitian setting, and prove that the volume is always finite and the Grauert-Riemenschneider type criterion holds true, which is a partial answer to a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical bundle $K^{-1}_X$ is nef, then for any $\\varepsilon>0$, there is a smooth function $\\phi_\\varepsilon$ on $X$ such that $\\omega_\\varepsilon:=\\omega+i\\partial\\overline{\\partial}\\phi_\\varepsilon>0$ and Ricci$(\\omega_\\varepsilon)\\geq-\\varepsilon\\omega_\\varepsilon$. Furthermore, if $\\omega$ satisfies the assumption as above, we prove that for a Harder-Narasimhan filtration of $T_X$ with respect to $\\omega$, the slopes $\\mu_\\omega(\\mathcal{F}_i/\\mathcal{F}_{i-1})\\geq 0$ for all $i$, which generalizes a result of Cao which plays a very important role in his studying of the structures of K\\\"{a}hler manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-17T03:41:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact hermitian manifold",
      "harder-narasimhan filtration",
      "semi-positive properties",
      "pseudo-effective class",
      "grauert-riemenschneider type criterion holds true"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}