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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.

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  • $\begingroup$ 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. $\endgroup$ – Allen Dec 31 '12 at 13:39
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    $\begingroup$ 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. $\endgroup$ – Mark Gross Dec 31 '12 at 16:43
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    $\begingroup$ You may be interested in a similar question mathoverflow.net/questions/110613/… $\endgroup$ – Atsushi Kanazawa Jan 1 '13 at 7:12
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That depends on what you mean by 'explicit'. For example, in Some examples of special Lagrangian tori (Adv. Theor. Math. Phys., vol. 3 no. 1 (1999), pp. 83–90, also available at arXiv:math/9902076), I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals, which is a known method for constructing special Lagrangian tori.

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.

That's an old paper by now, though, and someone may have constructed a more explicit example in the meantime that I don't know about.

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.

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  • $\begingroup$ Thank you for the answer. It helps me to know the current state of the field. I thought there were some examples because of tons of papers on the subject. I was not aware of your paper, and it may be interesting to extend the sLag to a family as you pointed out in the second paragraph. $\endgroup$ – T Wong Dec 31 '12 at 20:29
  • $\begingroup$ @T Wong: You are welcome. Actually, though, I am sure that Mark Gross and/or some combination of Dominic Joyce or Mark Haskins and his coworkers can point you to many more cases where special Lagrangian fibrations are either proved or suspected to exist, even though they might be less explicit than the ones I mentioned above. (And please keep in mind that 'explicit' is somewhat in the eye of the beholder, being more a matter of taste and experience than definition.) $\endgroup$ – Robert Bryant Jan 2 '13 at 15:39

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