{
  "id": "1401.4949",
  "version": "v2",
  "published": "2014-01-20T15:50:52.000Z",
  "updated": "2014-09-03T15:55:08.000Z",
  "title": "Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow",
  "authors": [
    "Dominic Joyce"
  ],
  "comment": "63 pages. (v2) new section 4 added, discussing exact Lagrangians, and the Kahler-Einstein case. To appear in EMS Surveys in Mathematical Sciences",
  "categories": [
    "math.DG",
    "math.SG"
  ],
  "abstract": "Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that there should be a notion of \"stability\" for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\\{L^t:t\\in[0,\\infty)\\}$ with $L^0=L$ should exist for all time, and $L^\\infty=\\lim_{t\\to\\infty}L^t$ should be the unique special Lagrangian in the Hamiltonian isotopy class of $L$. This paper is an attempt to update the Thomas-Yau conjectures, and discuss related issues. It is a folklore conjecture that there exists a Bridgeland stability condition $(Z,\\mathcal P)$ on the derived Fukaya category $D^b\\mathcal F(M)$ of $M$, such that an isomorphism class in $D^b\\mathcal F(M)$ is $(Z,\\mathcal P)$-semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique. We conjecture that if $(L,E,b)$ is an object in an enlarged version of $D^b\\mathcal F(M)$, where $L$ is a compact, graded Lagrangian in $M$ (possibly immersed, or with \"stable singularities\"), $E\\to M$ a rank one local system, and $b$ a bounding cochain for $(L,E)$ in Lagrangian Floer cohomology, then there is a unique family $\\{(L^t,E^t,b^t):t\\in[0,\\infty)\\}$ such that $(L^0,E^0,b^0)=(L,E,b)$, and $(L^t,E^t,b^t)\\cong(L,E,b)$ in $D^b\\mathcal F(M)$ for all $t$, and $\\{L^t:t\\in[0,\\infty)\\}$ satisfies Lagrangian MCF with surgeries at singular times $T_1,T_2,\\dots,$ and in graded Lagrangian integral currents we have $\\lim_{t\\to\\infty}L^t=L_1+\\cdots+L_n$, where $L_j$ is a special Lagrangian integral current of phase $e^{i\\pi\\phi_j}$ for $\\phi_1>\\cdots>\\phi_n$, and $(L_1,\\phi_1),\\ldots,(L_n,\\phi_n)$ correspond to the decomposition of $(L,E,b)$ into $(Z,\\mathcal P)$-semistable objects. We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times $T_1,T_2,\\ldots.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-20T15:50:52.000Z",
      "comment": "56 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-03T15:55:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lagrangian mean curvature flow",
      "special lagrangian",
      "bridgeland stability",
      "fukaya category",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.4949J"
    }
  }
}