{
  "id": "math/0302355",
  "version": "v2",
  "published": "2003-02-28T14:48:04.000Z",
  "updated": "2003-03-21T12:26:47.000Z",
  "title": "Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case",
  "authors": [
    "Dominic Joyce"
  ],
  "comment": "54 pages. (v2) new reference, changed notation",
  "journal": "Annals of Global Analysis and Geometry 26 (2004), 1-58.",
  "categories": [
    "math.DG",
    "hep-th"
  ],
  "abstract": "This is the third in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302356, math.DG/0303272 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and applies the results to prove some conjectures. The first two papers math.DG/0211294, math.DG/0211295 studied the regularity of X near its singular points, and the moduli space of deformations of X. In this paper and the fourth math.DG/0302356 we construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds \\tilde N^t in M for small t>0. Suppose L_1,...,L_n are Asymptotically Conical SL m-folds in C^m, with L_i asymptotic to the cone C_i at infinity. We shrink L_i by a small t>0, and glue tL_i into X at x_i for i=1,...,n to get a 1-parameter family of compact, nonsingular Lagrangian m-folds N^t for small t>0. Then we show using analysis that when t is sufficiently small we can deform N^t to a compact, nonsingular SL m-fold \\tilde N^t via a small Hamiltonian deformation. This \\tilde N^t depends smoothly on t, and as t --> 0 it converges to the singular SL m-fold X, in the sense of currents. This paper studies the simpler cases, where by topological conditions on X and L_i we avoid various obstructions to existence of \\tilde N^t. The sequel math.DG/0302356 will consider more complex cases when these obstructions are nontrivial, and also desingularization in families of almost Calabi-Yau m-folds.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-03-21T12:26:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isolated conical singularities",
      "compact special lagrangian submanifolds",
      "unobstructed case",
      "desingularization",
      "nonsingular sl m-fold"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 615673,
      "adsabs": "2003math......2355J"
    }
  }
}