{
  "id": "math/0607401",
  "version": "v2",
  "published": "2006-07-18T03:47:31.000Z",
  "updated": "2007-03-11T22:46:51.000Z",
  "title": "Generalized geometry, equivariant $\\bar{\\partial}\\partial$-lemma, and torus actions",
  "authors": [
    "Yi Lin"
  ],
  "comment": "to appear in the Journal of Geometry and Physics, 27 pages, a few typos and small mistakes corrected, added a few more references",
  "doi": "10.1016/j.geomphys.2007.03.004",
  "categories": [
    "math.DG",
    "hep-th",
    "math-ph",
    "math.MP",
    "math.SG"
  ],
  "abstract": "In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant Dolbeault cohomology. If the generalized complex manifold $M$ satisfies the $\\bar{\\partial}\\partial$-lemma, we prove that they are both canonically isomorphic to $(S\\g^*)^G\\otimes H_H(M)$, where $(S\\g^*)^G$ is the space of invariant polynomials over the Lie algebra $\\g$ of $G$, and $H_H(M)$ is the $H$-twisted cohomology of $M$. Furthermore, we establish an equivariant version of the $\\bar{\\partial}\\partial$-lemma, namely $\\bar{\\partial}_G\\partial$-lemma, which is a direct generalization of the $d_G\\delta$-lemma for Hamiltonian symplectic manifolds with the Hard Lefschetz property. Second we consider the torus action on a compact generalized K\\\"ahler manifold which preserves the generalized K\\\"ahler structure and which is equivariantly formal. We prove a generalization of a result of Carrell and Lieberman in generalized K\\\"ahler geometry. We then use it to compute the generalized Hodge numbers for non-trivial examples of generalized K\\\"ahler structures on $\\C\\P^n$ and $\\CP^n$ blown up at a fixed point.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-03-11T22:46:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C15",
      "53C55",
      "53D20"
    ],
    "keywords": [
      "torus action",
      "generalized geometry",
      "generalized complex manifold",
      "compact connected lie group",
      "generalized equivariant dolbeault cohomology"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 737082
    }
  }
}