Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties.

*(English)*Zbl 0829.14023Two families of Calabi-Yau 3-folds are called mirror dual, if the assigned conformal field theories are isomorphic. This would imply a certain behavior of the Hodge numbers: If \(V, V'\) are representatives from both families, then \(h^{11} (V) = h^{21} (V')\) and \(h^{21} (V) = h^{11} (V')\). (For the remaining Hodge numbers there is no freedom anyway.) Moreover, the number of rational curves with a fixed degree on \(V\) can then be obtained by regarding the Picard-Fuchs equation on \(V'\). Before the paper under review appeared, some pairs of mirror dual families (in the weaker sense regarding the Hodge numbers only) were known by physicists. These families appeared as hypersurfaces in certain 4-dimensional, weighted projective spaces. However, it turned out that some of those CY-hypersurfaces did not have a mirror partner at all.

The author has generalized the weighted projective approach by regarding hypersurfaces (and in subsequent papers even complete intersections) in projective toric varieties. Then, mirror duality (again in the sense of having equations between Hodge numbers) is provided by duality or polarity (i.e. interchanging the role of facets and vertices) of polytopes.

The paper begins with an introduction to the subject of projective toric varieties (given by some lattice polytope \(\Delta)\). Regarding Laurent polynomials with \(\Delta\) as its Newton polyhedron, one obtains the families of hypersurfaces we want to deal with. Using an approach of Danilov and Khovanskij, it is possible to compute the Hodge numbers of the crepant resolutions of these hypersurfaces. – On the other hand, the author introduces the class of so-called reflexive polyhedra (admitting just one interior lattice point \(P\), and the facets of \(\Delta\) have to have lattice distance 1 of \(P)\). It turns out that this class is invariant under duality (polarity) of polytopes; and it is exactly these polytopes that provide Calabi-Yau manifolds as hypersurfaces.

The paper closes with regarding as an example the family of hypersurfaces of degree 5 in the not even weighted \(\mathbb{P}^4\). Its mirror dual is considered using the toric point of view.

The author has generalized the weighted projective approach by regarding hypersurfaces (and in subsequent papers even complete intersections) in projective toric varieties. Then, mirror duality (again in the sense of having equations between Hodge numbers) is provided by duality or polarity (i.e. interchanging the role of facets and vertices) of polytopes.

The paper begins with an introduction to the subject of projective toric varieties (given by some lattice polytope \(\Delta)\). Regarding Laurent polynomials with \(\Delta\) as its Newton polyhedron, one obtains the families of hypersurfaces we want to deal with. Using an approach of Danilov and Khovanskij, it is possible to compute the Hodge numbers of the crepant resolutions of these hypersurfaces. – On the other hand, the author introduces the class of so-called reflexive polyhedra (admitting just one interior lattice point \(P\), and the facets of \(\Delta\) have to have lattice distance 1 of \(P)\). It turns out that this class is invariant under duality (polarity) of polytopes; and it is exactly these polytopes that provide Calabi-Yau manifolds as hypersurfaces.

The paper closes with regarding as an example the family of hypersurfaces of degree 5 in the not even weighted \(\mathbb{P}^4\). Its mirror dual is considered using the toric point of view.

Reviewer: K.Altmann (Berlin)

##### MSC:

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

14J30 | \(3\)-folds |

52B70 | Polyhedral manifolds |