{
  "id": "2311.14958",
  "version": "v1",
  "published": "2023-11-25T08:01:52.000Z",
  "updated": "2023-11-25T08:01:52.000Z",
  "title": "Monge-Ampère equation on compact Hermitian manifolds",
  "authors": [
    "Genglong Lin",
    "Yinji Li",
    "Xiangyu Zhou"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:0907.4490, arXiv:1705.05796, arXiv:2303.11584 by other authors",
  "categories": [
    "math.DG",
    "math.CV"
  ],
  "abstract": "Given a cohomology $(1,1)$-class $\\{\\beta\\}$ of compact Hermitian manifold $(X,\\omega)$ such that there exists a bounded potential in $\\{\\beta\\}$, we show that degenerate complex Monge-Amp\\`ere equation $(\\beta+dd^c \\varphi)^n=\\mu$ has a unique solution in the full mass class $\\mathcal{E}(X,\\beta)$, where $\\mu$ is any probability measure on $X$ which does not charge pluripolar subset. We also study other Monge-Amp\\`ere types equations which correspond to $\\lambda>0$ and $\\lambda<0$. As a preparation to the $\\lambda<0$ case, we give a general answer to an open problem about the Lelong number which was surveyed by Dinew-Guedj-Zeriahi \\cite[Problem 36]{DGZ16}. Moreover, we obtain more general results on singular space and of the equations with prescribed singularity if the model potential has small unbounded locus. These results generalize much recent work of \\cite{EGZ09}\\cite{BBGZ13}\\cite{DNL18}\\cite{LWZ23} etc.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-25T08:01:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact hermitian manifold",
      "monge-ampère equation",
      "full mass class",
      "charge pluripolar subset",
      "unique solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}