{
  "id": "2311.11492",
  "version": "v1",
  "published": "2023-11-20T02:31:13.000Z",
  "updated": "2023-11-20T02:31:13.000Z",
  "title": "Mukai Duality for abelian stacks",
  "authors": [
    "Ajneet Dhillon",
    "Brett Nasserden"
  ],
  "categories": [
    "math.AG",
    "math.KT",
    "math.NT"
  ],
  "abstract": "An abelian stack is a stacky generalization of an abelian variety that was introduced by Brochard. Just as an abelian variety has a dual, an abelian stack $\\mathcal{A}$ has a dual $\\mathfrak{D}(\\mathcal{A})$ which generalizes the classical dual. In general, $\\mathfrak{D}(\\mathcal{A})$ is no longer an abelian stack but a commutative group scheme which is an extension of a finite, flat, and finitely presented commutative group scheme by an abelian scheme. We show that Fourier-Mukai duality holds for tame abelian stacks and their duals. Our approach is as follows. Let $QC_\\infty(\\mathcal{A})$ be the stable infinity category of quasi-coherent sheaves on $\\mathcal{A}$. We define a Poincare bundle on $\\mathcal{A}\\times \\mathfrak{D}(\\mathcal{A})$ and use this to show that $QC_\\infty(\\mathcal{A})$ and $QC_\\infty(\\mathfrak{D}(\\mathcal{A}))$ are dual as objects in the infinity category of stable infinity categories. By a result of Ben-Zvi,Francis and Nadler we have that $QC_\\infty(\\mathcal{A})$ is self dual, giving that $QC_\\infty(\\mathcal{A})\\cong QC_\\infty(\\mathfrak{D}(\\mathcal{A}))$ which gives the statement for the derived categories. In addition we give new examples of tame abelian stacks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-20T02:31:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tame abelian stacks",
      "commutative group scheme",
      "stable infinity category",
      "abelian variety",
      "fourier-mukai duality holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}