{
  "id": "1211.3259",
  "version": "v4",
  "published": "2012-11-14T10:07:28.000Z",
  "updated": "2015-06-04T18:41:03.000Z",
  "title": "Symmetries and stabilization for sheaves of vanishing cycles",
  "authors": [
    "Christopher Brav",
    "Vittoria Bussi",
    "Delphine Dupont",
    "Dominic Joyce",
    "Balazs Szendroi"
  ],
  "comment": "77 pages, LaTeX. (v4) corrections, new Appendix by Joerg Schuermann",
  "journal": "Journal of Singularities 11 (2015), 85-151",
  "doi": "10.5427/jsing.2015.11e",
  "categories": [
    "math.AG",
    "math.CV",
    "math.DG"
  ],
  "abstract": "Let $U$ be a smooth $\\mathbb C$-scheme, $f:U\\to\\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\\mathbb C$-subscheme of $U$. Then one can define the \"perverse sheaf of vanishing cycles\" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $\\Phi:U\\to U$ is an isomorphism with $f\\circ\\Phi=f$ and $\\Phi\\vert_X=$id$_X$. Then $\\Phi$ induces an isomorphism $\\Phi_*:PV_{U,f}\\to PV_{U,f}$. We show that $\\Phi_*$ is multiplication by det$(d\\Phi\\vert_X)=1$ or $-1$. (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-order thickening of $X$ in $U$, and $f^{(3)}=f\\vert_{X^{(3)}}:X^{(3)}\\to\\mathbb A^1$. (c) If $U,V$ are smooth $\\mathbb C$-schemes, $f:U\\to\\mathbb A^1$, $g:V\\to\\mathbb A^1$ are regular, $X=$Crit$(f)$, $Y=$Crit$(g)$, and $\\Phi:U\\to V$ is an embedding with $f=g\\circ\\Phi$ and $\\Phi\\vert_X:X\\to Y$ an isomorphism, there is a natural isomorphism $\\Theta_\\Phi:PV_{U,f}\\to\\Phi\\vert_X^*(PV_{V,g})\\otimes_{\\mathbb Z_2}P_\\Phi$, for $P_\\Phi$ a natural principal $\\mathbb Z_2$-bundle on $X$. (d) If $(X,s)$ is an oriented d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural perverse sheaf $P_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\\to\\mathbb A^1)$ then $P_{X,s}$ is locally modelled on $PV_{U,f}$. We also generalize our results to replace $U,X$ by complex analytic spaces, and $PV_{U,f}$ by $\\mathcal D$-modules, or mixed Hodge modules. We discuss applications of (d) to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to defining a 'Fukaya category' of Lagrangians in a complex symplectic manifold using perverse sheaves. This is the third in a series of papers arXiv:1304.4508, arXiv:1305.6302, arXiv:1305.6428, arXiv:1312.0090, arXiv:1403.2403, arXiv:1404.1329, arXiv:1504.00690.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-12-09T21:14:56.000Z",
      "abstract": "Let $U$ be a smooth $\\mathbb C$-scheme, $f:U\\to\\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\\mathbb C$-subscheme of $U$. Then one can define the \"perverse sheaf of vanishing cycles\" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $\\Phi:U\\to U$ is an isomorphism with $f\\circ\\Phi=f$ and $\\Phi\\vert_X=$id$_X$. Then $\\Phi$ induces an isomorphism $\\Phi_*:PV_{U,f}\\to PV_{U,f}$. We show that $\\Phi_*$ is multiplication by det$(d\\Phi\\vert_X)=1$ or $-1$. (b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-order thickening of $X$ in $U$, and $f^{(3)}=f\\vert_{X^{(3)}}:X^{(3)}\\to\\mathbb A^1$. (c) If $U,V$ are smooth $\\mathbb C$-schemes, $f:U\\to\\mathbb A^1$, $g:V\\to\\mathbb A^1$ are regular, $X=$Crit$(f)$, $Y=$Crit$(g)$, and $\\Phi:U\\to V$ is an embedding with $f=g\\circ\\Phi$ and $\\Phi\\vert_X:X\\to Y$ an isomorphism, there is a natural isomorphism $\\Theta_\\Phi:PV_{U,f}\\to\\Phi\\vert_X^*(PV_{V,g})\\otimes_{\\mathbb Z_2}P_\\Phi$, for $P_\\Phi$ a natural principal $\\mathbb Z_2$-bundle on $X$. (d) If $(X,s)$ is an oriented d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural perverse sheaf $P_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\\to\\mathbb A^1)$ then $P_{X,s}$ is locally modelled on $PV_{U,f}$. We also generalize our results to replace $U,X$ by $\\mathbb K$-schemes over other fields $\\mathbb K$, or by complex analytic spaces, and $PV_{U,f}$ by $\\mathcal D$-modules, or mixed Hodge modules. We discuss applications of (d) to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to defining a 'Fukaya category' of Lagrangians in a complex symplectic manifold using perverse sheaves. This is the third in a series of papers arXiv:1304.4508, arXiv:1305.6302, arXiv:1305.6428, arXiv:1312.0090.",
      "comment": "74 pages, LaTeX. (v3) minor changes, new references",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-06-04T18:41:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S30",
      "32S60",
      "14C30"
    ],
    "keywords": [
      "vanishing cycles",
      "isomorphism",
      "symmetries",
      "stabilization",
      "complex symplectic manifold"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 77,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1288940,
      "adsabs": "2012arXiv1211.3259B"
    }
  }
}