{
  "id": "1905.02806",
  "version": "v1",
  "published": "2019-05-07T20:58:43.000Z",
  "updated": "2019-05-07T20:58:43.000Z",
  "title": "Twisted Dolbeault cohomology of nilpotent Lie algebras",
  "authors": [
    "Liviu Ornea",
    "Misha Verbitsky"
  ],
  "comment": "18 pages, v. 1.0",
  "categories": [
    "math.DG",
    "math.AG"
  ],
  "abstract": "It is well known that cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result is due to L. Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault version of Alaniya's theorem, showing that the Dolbeault cohomology of a nilpotent Lie algebra with coefficients in any non-trivial 1-dimensional local system vanishes. Note that the Dolbeault cohomology of the corresponding local system on the manifold is not necessarily zero. This implies that the twisted version of Console-Fino theorem is false (Console-Fino proved that the Dolbeault cohomology of a complex nilmanifold is equal to the Dolbeault cohomology of its Lie algebra). As an application, we give a new proof of a theorem due to H. Sawai, who obtained an explicit description of LCK nilmanifolds. An LCK structure on a manifold $M$ is a K\\\"ahler structure on its cover $\\tilde M$ such that the deck transform map acts on $\\tilde M$ by homotheties. We show that any complex nilmanifold admitting an LCK structure is Vaisman, and is obtained as a compact quotient of the product of a Heisenberg group and the real line.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-07T20:58:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nilpotent lie algebra",
      "twisted dolbeault cohomology",
      "complex nilmanifold",
      "lck structure",
      "deck transform map acts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}