Singularities of special Lagrangian fibrations and the SYZ Conjecture

TL;DR

This paper investigates the singularities of special Lagrangian fibrations related to the SYZ Conjecture, constructing explicit examples and discussing their implications for mirror symmetry and the nature of singular fibers.

Contribution

The paper constructs explicit examples of special Lagrangian fibrations with singularities and discusses their properties, providing insights into the singularity structure relevant to the SYZ Conjecture.

Findings

Examples of special Lagrangian fibrations on open subsets of C^3 are constructed.

Singular fibers are typically of codimension 1 and form 'ribbons' in the base.

The results suggest fibrations are piecewise-smooth with singularities affecting the SYZ Conjecture.

Abstract

The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One of the main problems in proving the SYZ Conjecture (or even in finding the right statement of it) is that the singularities of special Lagrangian 3-folds and fibrations are poorly understood. This paper studies the singularities of special Lagrangian fibrations. Our main rigorous results are the construction of examples of special Lagrangian fibrations on open subsets of C^3. The simplest are given explicitly, and the rest are constructed using analytic existence results from the author's three papers math.DG/0111324, math.DG/0111326, math.DG/0204343 on U(1)-invariant special Lagrangian 3-folds in C^3. We then argue, without full proofs, that some features of our examples should also hold for special Lagrangian fibrations f : M --> B of (almost) Calabi-Yau 3-folds, especially in the generic case. In particular, f will not be smooth but only piecewise-smooth, and the discriminant (set of singular fibres) of f will be of codimension 1 in B, and will typically be composed of 'ribbons'. Finally we draw some conclusions on the SYZ Conjecture, which contradict some stronger statements of it.