{
  "id": "2112.14463",
  "version": "v2",
  "published": "2021-12-29T09:08:56.000Z",
  "updated": "2022-02-03T08:59:10.000Z",
  "title": "Monge-Ampère functionals for the curvature tensor of a holomorphic vector bundle",
  "authors": [
    "Jean-Pierre Demailly"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $E$ be a holomorphic vector bundle on a projective manifold $X$ such that $\\det E$ is ample. We introduce three functionals $\\Phi_P$ related to Griffiths, Nakano and dual Nakano positivity respectively. They can be used to define new concepts of volume for the vector bundle $E$, by means of generalized Monge-Amp\\`ere integrals of $\\Phi_P(\\Theta_{E,h})$, where $\\Theta_{E,h}$ is the Chern curvature tensor of $(E,h)$. These volumes are shown to satisfy optimal Chern class inequalities. We also prove that the functionals $\\Phi_P$ give rise in a natural way to elliptic differential systems of Hermitian-Yang-Mills type for the curvature, in such a way that the related $P$-positivity threshold of $E\\otimes(\\det E)^t$, where $t>-1/{\\rm rank} E$, can possibly be investigated by studying the infimum of exponents $t$ for which the Yang-Mills differential system has a solution.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-03T08:59:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "holomorphic vector bundle",
      "curvature tensor",
      "monge-ampère functionals",
      "satisfy optimal chern class inequalities",
      "dual nakano positivity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}