2000/11/30 by Dominic Joyce · 1 citation
Mathematics · Physics and Astronomy · #math.DG #hep-th #math.AG
published as Communications in Analysis and Geometry 11 (2003), 859-907. · 42 pages, LaTeX. (v2) Completely rewritten, shorter, with new proofs of conjectures in (v1). (v3) References added
arxiv created 2003/06/17 · arxiv updated 2009/11/30
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 C3. 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 C3. 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.