Examples of Special Lagrangian Fibrations

TL;DR

This paper examines various examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds, focusing on their structure, discriminant loci, and implications for mirror symmetry and related theories.

Contribution

It provides new insights into the structure of special Lagrangian fibrations on non-compact Calabi-Yau manifolds, including crepant resolutions and deformations, and discusses their relation to mirror symmetry.

Findings

Examples of fibrations on crepant resolutions of toric singularities.

Analysis of global structure and discriminant loci of these fibrations.

Connections to mirror symmetry and Strominger-Yau-Zaslow conjecture.

Abstract

We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toric singularities (already found by Goldstein), proper versions of these fibrations, and fibrations on flat deformations of canonical toric singularities. We do this with an eye towards understanding the global structure and discriminant loci of such fibrations. The paper ends with some speculation, both about local mirror symmetry, and the connections of this work with the work of W.D. Ruan and D. Joyce. In the last section, we discuss the philosophy of the Strominger-Yau-Zaslow conjecture in the light of a number of recent ideas of myself and Wilson, Kontsevich and Soibelman, and Joyce.