{
  "id": "2106.16133",
  "version": "v1",
  "published": "2021-06-30T15:36:46.000Z",
  "updated": "2021-06-30T15:36:46.000Z",
  "title": "The d-critical structure on the Quot scheme of points of a Calabi-Yau 3-fold",
  "authors": [
    "Andrea T. Ricolfi",
    "Michail Savvas"
  ],
  "comment": "36 pages, comments welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "The Artin stack $\\mathcal M_n$ of $0$-dimensional sheaves of length $n$ on $\\mathbb A^3$ carries two natural d-critical structures in the sense of Joyce. One comes from its description as a quotient stack $[\\textrm{crit}(f_n)/\\textrm{GL}_n]$, another comes from derived deformation theory of sheaves. We show that these d-critical structures agree. We use this result to prove the analogous statement for the Quot scheme of points $\\textrm{Quot}_{\\mathbb A^3}(\\mathscr O^{\\oplus r},n) = \\textrm{crit}(f_{r,n})$, which is a global critical locus for every $r>0$, and also carries a derived-in-flavour d-critical structure besides the one induced by the potential $f_{r,n}$. Again, we show these two d-critical structures agree. Moreover, we prove that they locally model the d-critical structure on $\\textrm{Quot}_X(F,n)$, where $F$ is a locally free sheaf of rank $r$ on a projective Calabi-Yau $3$-fold $X$. Finally, we prove that the perfect obstruction theory on $\\textrm{Hilb}^n\\mathbb A^3=\\textrm{crit}(f_{1,n})$ induced by the Atiyah class of the universal ideal agrees with the \\emph{critical} obstruction theory induced by the Hessian of the potential $f_{1,n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-30T15:36:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quot scheme",
      "d-critical structures agree",
      "calabi-yau",
      "perfect obstruction theory",
      "derived deformation theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}