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There is a well known class of Calabi-Yau (3 dimensional) pairs constructed by Batyrev. These are resolutions of Calabi-Yau hypersurfaces in reflexive polytops of dimension 4.

Question: Does any body know any other mirror pair, or a family of them, beside this kind of pairs?

For example, how about Calabi-Yau complete intersections in higher dimensional weighted projective spaces or Fano toric varieties?

Warning: My question is only about closed Calabi-Yau 3-folds.

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2 Answers 2

I'm not an expert at this at all, but a couple of observations:

1) Rather than just hypersurfaces I believe Batyrev and Borisov have a more general description for complete intersections in toric fano thing manifolds.

2) This recent paper of Alan Stapledon works on orbifolding the Batyrev-Borisov correspondence, and begins with a series of references to known and conjectured mirror pairs, that I think is probably close to the state of the art. Examining those references will probably help, and a key phrase involved in one is the "pfaffian-Grassmannian correspondence"

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I have not seen this paper, but are you sure that batyrev-Borisov have addressed the mirror of complete intersections. Do you know any reference? –  Mohammad F. Tehrani Apr 27 '11 at 23:20
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Again, not an expert on this -- but the introduction to Alan's paper claims they do, and cites their Inventiones paper, on the Arxiv here: arxiv.org/abs/alg-geom/9509009 The abstract is: We prove in full generality the mirror duality conjecture for string-theoretic Hodge numbers of Calabi-Yau complete intersections in Gorenstein toric Fano varieties. So, from everything I can see, they do. –  Paul Johnson Apr 28 '11 at 9:38
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An answer to this question depends on what you mean by "mirror". As to topological mirror symmetry, the Batyrev-Borisov toric construction is the easiest and the most standard. Other examples would be Borcea-Voisin CY3s. If you want to do more, for example instanton computation etc, you need to work on CY3s with small $h^{1,1}$ (or mirrorly $h^{2,1}$) so that you can compute the mirror map, period integral etc.

CY3s with $h^{1,1}=1$ are typically complete intersections of Grassmannians. Batyrev, Ciocan-Fontanine, Kim and van Straten construct mirror manifolds (with mild singularity) of such CY3s via toric degeneration in this paper. The basic idea is easy to understand; they degenerate the ambient Grassmannians to toric varieties and apply the Batyrev-Borisov toric construction. So the essential part is toric in this case, too.

Apart from toric examples, the best known example would be Rodland's pfaffian-7 CY3. In this paper, he constructs a smooth mirror manifold of the pfaffian-7 CY3 by an orbifold method, i.e. taking a special 1-parameter family and take quotient by a finite group. The interesting observation is that the mirror manifold is also mirror of the complete intersection CY3 of Gr(2,7). Later Borisov and Caldararu prove that Rodland's CY3 and the complete intersection CY3 of Gr(2,7) are derived equivalent as HMS indicates.

Recently Kanazawa constructs several new pfaffian CY3 with $h^{1,1}=1$ in this paper, partially solving van Enckevort and van Straten's conjecture. Mirror manifolds (with mild singularity) are also constructed by an orbifold method. Kanazawa's pfaffian CY3s are interesting in the sense that the mirror manifolds have two large complex structure limits (similar to Rodland's case). This kind of phenomenon is also found by Hosono and Takagi in this paper. Their mirror symmetry is based on the standard Batyrev-Borisov toric method (with mild modification), but a non-trivial Fourier-Mukai partner comes into play.

Apart from toric examples, there seems no standard way to construct mirror manifolds for a given smooth compact CY3. Rodland's and Kanazawa's construction is some what an art. People have been working hard to find an intrinsic and systematic mirror construction. SYZ conjecture is one of such attempts.

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There is another new CY3 with two MUM due to Miura arxiv.org/abs/1301.7632 –  Atsushi Kanazawa Aug 29 '13 at 17:20
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