Special Lagrangian conifolds, I: Moduli spaces

TL;DR

This paper develops a deformation theory framework for special Lagrangian conifolds in complex space, addressing their role in moduli space compactification and laying groundwork for future gluing construction studies.

Contribution

It introduces a unifying deformation theory for SL conifolds with singularities and ends, extending previous results and setting geometric foundations for subsequent gluing analyses.

Findings

Established a deformation theory framework for SL conifolds.

Unified treatment of conical singularities and asymptotically conical ends.

Laid geometric groundwork for future gluing constructions.

Abstract

We discuss the deformation theory of 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. This category allows for the simultaneous presence of conical singularities and of non-compact, asymptotically conical, ends. Our main theorem is the natural next step in the chain of results initiated by McLean and continued by the author and by Joyce. We emphasize a unifying framework for studying the various cases and discuss analogies and differences between them. This paper also lays down the geometric foundations for our paper "Special Lagrangian conifolds, II" concerning gluing constructions for SL conifolds in C^m.