Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case

Details Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case

2003-02-28

arxiv.org/abs/math/0302355v2

Annals of Global Analysis and Geometry 26 (2004), 1-58.

Mathematics

Differential Geometry

54 pages. (v2) new reference, changed notation

Scientific paper

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.

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