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I would like to know for which Calabi-Yau threefolds SYZ conjecture is known to hold. I am aware of works by Gross-Wilson (Borcea-Voisin CY3s) and Ruan (quintic CY3), but they are quite classical works today and I now wonder if there are any more examples of SYZ. Of course, answer depends on what I mean by "SYZ conjecture", so I am happy with SYZ examples at any level. I also wonder what can one conclude from Gross's many works on this subject.

I am also aware of Auroux's (and his collaborators') work on SYZ for Fano manifolds, but I am mainly interested in CY threefold case. This comment does not exclude any information about their work that might be relevant to CY3 case (I just don't know any relation due to my ignorance).

Thank you for your information.

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None which is compact and smooth. –  Mohammad F. Tehrani Oct 25 '12 at 4:11
    
It seems that the OP is asking for a list... shouldn't it be community wiki then ? –  DamienC Oct 25 '12 at 4:52
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@Mohammad I think Gross and Wilson construct SLAG fibration for Borcea-Voisin CY3s (although the metric is a bit degenerate). They are certainly compact and smooth. –  Pooya Oct 25 '12 at 6:12
    
@DamienC I made this community wiki. Thanks. –  Pooya Oct 25 '12 at 6:13
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In the toric calabi yau case, which is not compact, Leung and his gropu have series of papers on this: arxiv.org/pdf/1006.3830.pdf –  Jay Jan 26 '13 at 21:22
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1 Answer 1

Let me go from the weakest to strongest sense in which the conjecture should be true.

First, at the purely topological level, it is true for any Calabi-Yau variety with a toric degeneration whose dual intersection complex is ``simple''. These notions are part of my program with Bernd Siebert: see our paper http://arxiv.org/abs/math/0309070 for the definitions of toric degeneration and simple. In http://arxiv.org/abs/math/0406171 I proved that all Calabi-Yau varieties arising in the Batyrev-Borisov construction as complete intersections in toric varieties have such degenerations.

The problem is that Bernd and I have been putting off writing the paper linking the logarithmic approach to topological fibrations for years now, largely due to lack of motivation. So there is no reference in the literature yet for this result. I do hope we will finally complete this paper next year.

Second, at the Lagrangian level, there are the results of W.-D. Ruan you mentioned. In addition, Castano-Bernard and Matessi in http://arxiv.org/abs/math/0611139 showed that given an affine three-manifold $B$ with ``simple'' singularities, one can construct a symplectic six-manifold along with a Lagrangian fibration to $B$. So one can apply this to the case where $B$ is the intersection complex of a polarized toric degeneration of Calabi-Yau threefolds. One expects this six-manifold to be symplectomorphic to a general fibre of the degeneration, but there is no proof of this at the moment.

Finally, at the special Lagrangian level, I think it is safe to say there are no known examples on compact non-singular Calabi-Yau threefolds with non-degenerate metric. There are some examples for non-compact Calabi-Yau varieties, specifically toric ones, see my paper http://arxiv.org/abs/math/0012002

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