{
  "id": "1804.04716",
  "version": "v1",
  "published": "2018-04-12T20:22:35.000Z",
  "updated": "2018-04-12T20:22:35.000Z",
  "title": "The pre-symplectic geometry of opers and the holonomy map",
  "authors": [
    "Andrew Sanders"
  ],
  "comment": "52 pages",
  "categories": [
    "math.DG",
    "math.SG"
  ],
  "abstract": "Given a connected complex semi-simple Lie group $G$ and a Riemann surface $X,$ a $G$-oper on $X$ is a higher rank generalization of a complex projective structure on $X.$ These objects play an important role in integrable systems and geometric representation theory, a status that was cemented by the seminal work of Beilinson-Drinfeld \\cite{BD91}. For $G$ a connected complex simple Lie group of adjoint type, we study the global deformation theory of $G$-opers on a connected, closed, oriented smooth surface $\\Sigma$ of genus at least two. We exhibit the deformation space of $G$-opers on $\\Sigma$ as a holomorphic fiber bundle over Teichm\\\"{u}ller space, and elucidate the relationship with the deformation space of complex projective structures. Then, we show that there is a family of identifications of the deformation space of $G$-opers with a holomorphic vector bundle $\\mathcal{B}_{G}(\\Sigma)$ over Teichm\\\"{u}ller space whose typical fiber over a Riemann surface $X$ is a sum of spaces of pluri-canonical sections. Finally, we show that the holonomy map from the deformation space of $G$-opers to the deformation space of flat $G$-bundles on $\\Sigma$ is a holomorphic immersion. As a consequence of this result, we show that the deformation space of $G$-opers carries a (pre-symplectic) closed holomorphic differential $2$-form of constant rank, and we prove that a sub-family of the identifications of $\\mathcal{B}_{G}(\\Sigma)$ with the deformation space of $G$-opers is a holomorphic pre-symplectic map for a natural holomorphic pre-symplectic form on $\\mathcal{B}_{G}(\\Sigma).$ These results generalize the fundamental features of the deformation space of complex projective structures on $\\Sigma$ to the setting of $G$-opers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-12T20:22:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "deformation space",
      "holonomy map",
      "complex projective structure",
      "pre-symplectic geometry",
      "complex simple lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}