Special Lagrangian conifolds, II: Gluing constructions in C^m

TL;DR

This paper develops new gluing theorems for special Lagrangian conifolds in complex space, enabling the construction and desingularization of these geometric objects with multiple ends and singularities, advancing the understanding of their moduli spaces.

Contribution

It introduces the first examples of smooth SL conifolds with multiple planar ends and non-conical singularities, and provides comprehensive desingularization techniques using Lawlor necks.

Findings

First examples of smooth SL conifolds with ≥3 planar ends

Desingularization of conifold intersections using Lawlor necks

No upper bound on the number of AC ends of SL conifolds

Abstract

We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the first (non-trivial) examples of SL conifolds which have a conical singularity but are not, globally, cones. We also obtain: (i) a desingularization procedure for transverse intersection and self-intersection points, using "Lawlor necks"; (ii) a construction which completely desingularizes any SL conifold by replacing isolated conical singularities with non-compact asymptotically conical (AC) ends; (iii) a proof that there is no upper bound on the number of AC ends of a SL conifold; (iv) the possibility of replacing a given collection of conical singularities with a completely different collection of conical singularities and of AC ends. As a corollary of (i) we improve a result by Arezzo and Pacard concerning minimal desingularizations of certain configurations of SL planes in C^m, intersecting transversally.