2003/02/28 by Dominic Joyce
Mathematics · Physics and Astronomy · #math.DG #hep-th
published as Annals of Global Analysis and Geometry 26 (2004), 1-58. · 54 pages. (v2) new reference, changed notation
arxiv created 2003/03/21 · arxiv updated 2009/11/30
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 x1,...,xn locally modelled on special Lagrangian cones C1,...,Cn in Cm 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 Nt in M for small t>0. Suppose L1,...,Ln are Asymptotically Conical SL m-folds in Cm, with Li asymptotic to the cone Ci at infinity. We shrink Li by a small t>0, and glue tLi into X at xi for i=1,...,n to get a 1-parameter family of compact, nonsingular Lagrangian m-folds Nt for small t>0. Then we show using analysis that when t is sufficiently small we can deform Nt to a compact, nonsingular SL m-fold Nt via a small Hamiltonian deformation. This Nt 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 Li we avoid various obstructions to existence of Nt. 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.