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Journal of Differential Geometry 63 (2003), 279-347.

Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications

Dominic Joyce

Abstract:

This is the last in a series of five papers studying compact {\it special Lagrangian submanifolds} (SL $m$-{\it folds}) $X$ in (almost) Calabi-Yau $m$-folds $M$ with {\it singularities} $x_1,\ldots,x_n$ locally modelled on special Lagrangian cones $C_1,\ldots,C_n$ in ${\mathbb C}^m$ with isolated singularities at $0$. Readers are advised to begin with this paper.\newline \hspace*{1em} We survey the major results of the previous four papers, giving brief explanations of the proofs. We apply the results to describe the {\it boundary} of a moduli space of compact, nonsingular SL $m$-folds $N$ in $M$. We prove the existence of special Lagrangian {\it connected sums} $N_1\# \cdots\# N_k$ of SL $m$-folds $N_1,\ldots,N_k$ in $M$. We also study SL 3-folds with $T^2$-cone singularities, proving results related to ideas of the author on invariants of Calabi-Yau 3-folds, and the SYZ Conjecture.\newline \hspace*{1em} Let $X$ be a compact SL $m$-fold with isolated conical singularities $x_i$ and cones $C_i$ for $i=1,\ldots,n$. The first paper studied the {\it regularity} of $X$ near its singular points, and the the second the {\it moduli space of deformations} of $X$. The third and fourth papers construct {\it desingularizations} of $X$, realizing $X$ as a limit of a family of compact, nonsingular SL $m$-folds $N^t$ in $M$ for small $t>0$. Let $L_i$ be an {\it asymptotically conical\/} SL $m$-fold in ${\mathbb C}^m$ asymptotic to $C_i$ at infinity. We make $N^t$ by gluing $tL_i$ into $X$ at $x_i$ for $i=1,\ldots,n$.