Singularities of special Lagrangian fibrations and 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. This paper studies the singularities of special Lagrangian fibrations. We construct many examples of special Lagrangian fibrations on open subsets of C 3 . The simplest are given explicitly, and the rest use analytic existence results for U(1)-invariant special Lagrangian 3-folds in C 3 . We then argue that some features of our examples should also hold for generic special Lagrangian fibrations of (almost) Calabi-Yau 3-folds, and draw some conclusions on the SYZ Conjecture.