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

Abstract

We prove two gluing theorems for special Lagrangian (SL) conifolds in 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 three or more planar ends and the first (non-trivial) examples of SL conifolds which have a conical singularity (CS) 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 CSs with non-compact asymptotically conical (AC) ends; (iii) a proof that there is no upper bound on the number of AC ends of an SL conifold and (iv) the possibility of replacing a given collection of CSs with a completely different collection of CSs and of AC ends.As a corollary of (i), we improve a result by Arezzo and Pacard (Comm. Pure Appl. Math. 56 (2003) 283-327) concerning minimal desingularizations of certain configurations of SL planes in C m , intersecting transversally.

Special Lagrangian conifolds, I: moduli spaces

“…The 'special' condition rigidifies them considerably: the corresponding gluing and desingularization processes require much 'harder' techniques. We refer the reader to [6,13,14,22] for recent gluing results and [7] for local desingularization issues. The main goal of this paper is to 'quantify' this notion of rigidity by examining the problem of SL deformations and calculating the corresponding degrees of freedom.…”

“…In particular, they might also substitute AC SLs in Joyce's gluing results: one could try to cut out a conical singularity of the CS SL and replace it with a different singular conifold, thus jumping from one area of the boundary of the compactified moduli space, containing certain CS SLs, to another.The paper at hand is Part I of a multi-step project aiming to set up a general theory of SL conifolds. Two other papers related to this project are currently available: [21,22]. Further work is in progress.…”

“…In this sense, this paper also serves the purpose of surveying and unifying those results. More importantly, it lays down the geometric foundations for [22]; the analytic foundations are provided by Pacini [21].…”

“…The paper at hand is Part I of a multi-step project aiming to set up a general theory of SL conifolds. Two other papers related to this project are currently available: [21,22]. Further work is in progress.…”

Gluing and deformation of asymptotically cylindrical Calabi–Yau manifolds in complex dimension three

“…However, once we have a compact Lagrangian submanifold that is close to special Lagrangian he gives a general result on perturbing it to become special Lagrangian [21,Theorem 5.3]. Pacini extended this result to the case of asymptotically conical Lagrangian submanifolds in C n given by patching together special Lagrangians with conical singularities and asymptotically conical ends (see [33,Theorem 6.3]). In the asymptotically cylindrical setting, where there is a change of Calabi-Yau structure arising from the Calabi-Yau gluing, it is much harder to remain Lagrangian and so the analysis is rather different.…”