# Example of special Lagrangian fibration of compact CY3?

I would like to know an explicit example of a special Lagrangian fibration of compact Calabi-Yau 3-folds. Are there any example known among experts? I know that there are some for noncompact Calabi-Yau 3-folds.

• Mark Gross is an expert in this field. By the way, as far as I know consturcting a special Lag submanifold on a compact CY3 is too difficult, not to say you want a special Lag fibration. This is the reason why many people now only consider Lag fibration for the purpose of mirror symmetry. – Allen Dec 31 '12 at 13:39
• As Robert points out, it is easy to find examples where the holonomy group is not all of $SU(3)$, such as a product of a K3 surface and an elliptic curve, or quotients thereof where the metric is an orbifold metric. But there are still no compact examples with full $SU(3)$ holonomy. It would be nice to see further progress on this problem. – Mark Gross Dec 31 '12 at 16:43
• You may be interested in a similar question mathoverflow.net/questions/110613/… – Atsushi Kanazawa Jan 1 '13 at 7:12

By McLean's deformation theorem, when the hypersurface has complex dimension $3$, each such torus lies in a $3$-parameter family of special Lagrangian tori in the Calabi-Yau, though, as far as I know, no one knows how to write down the family explicitly. One certainly expects that, outside some locus of smaller dimension, this $3$-parameter family foliates the entire Calabi-Yau, though, as far as I know, this has never been proved rigorously. One piece of evidence in favor of this, is that, for the corresponding real slices of Calabi-Yau hypersurfaces of complex dimensions $1$ and $2$, the corresponding family of special Lagrangian tori do, indeed, foliate the hypersurface and generate a fibration of exactly the desired kind.
Added Remark: Also, I should say that it depends on what you mean by 'CY3' as well. If by this, you mean a compact Ricci-flat Kähler $3$-fold, then, of course, there are products, such as the product of $3$ elliptic curves or the product of an elliptic curve and a K3 surface. In these cases, one can construct the special Lagrangian fibrations more-or-less explicitly.