Communications in Analysis and Geometry
Volume 11 (2003)
Singularities of special Lagrangian fibrations and the SYZ Conjecture
Pages: 859 – 907
The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M, \hat M in terms of special Lagrangian fibrations f : M → B and \hat f : \hat 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 C3. The simplest are given explicitly, and the rest use analytic existence results for U(1)-invariant special Lagrangian 3-folds in C3. 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.