{
  "id": "2309.17418",
  "version": "v1",
  "published": "2023-09-29T17:27:16.000Z",
  "updated": "2023-09-29T17:27:16.000Z",
  "title": "Calabi-Yau structures on the complexifications of rank two symmeric spaces",
  "authors": [
    "Naoyuki Koike"
  ],
  "comment": "20pages. arXiv admin note: substantial text overlap with arXiv:2003.04118",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this paper, we prove that there exists a $C^{\\infty}$-Calabi-Yau structure on the complexification $G^{\\mathbb C}/K^{\\mathbb C}$ of a rank two symmetric space $G/K$ of compact type. The proof is performed by deriving a relation (which differs from the known relation somewhat) between the complex Hessian of a real-valued $C^{\\infty}$-function $\\psi$ on $G^{\\mathbb C}/K^{\\mathbb C}$ and the Hessian of the function $\\rho:=\\psi|_{{\\rm Exp}_o(\\mathfrak a)}\\circ{\\rm Exp}_o|_{\\mathfrak a}$ on a maximal abelian subspace $\\mathfrak a$ of the normal space $T_o^{\\perp}(G\\cdot o)(=T_o(G^d\\cdot o)\\subset\\mathfrak g^d)$ of the orbit $G\\cdot o(=G/K)$ in $G^{\\mathbb C}/K^{\\mathbb C}$ at the base point $o$, where ${\\rm Exp}_o$ is the exponential map of $G^{\\mathbb C}/K^{\\mathbb C}$ at $o$ and $\\mathfrak g^d$ is the Lie algebra of the dual $G^d$ of $G$. This relation is derived in a new calculation method by using the explicit descriptions of the shape operators of the orbits of the isotropy action $K\\curvearrowright G^d/K$ and the Hermann type action $G\\curvearrowright G^{\\mathbb C}/K^{\\mathbb C}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-29T17:27:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "calabi-yau structure",
      "symmeric spaces",
      "complexification",
      "hermann type action",
      "maximal abelian subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}