Birationality of Berglund-Hübsch-Krawitz mirrors.

*(English)*Zbl 1395.14034One very explicit way to construct mirror pairs of Landau-Ginzburg models is due to the construction of Berglund-Hübsch-Krawitz. Given a weighted polynomial \(W\) with \(n+1\) monomials in \(n+1\) variables and a certain finite group \(G\), there is a mirror dual Landau-Ginzburg model \((W^T,G^T)\) which is essentially given by transposing a square matrix whose entries are the exponents of \(W\).

If the weights sum up to the degree of \(W\), then \(\{W = 0\} \subset [ \mathbb P/G]\) and \(\{W^T = 0\} \subset [ \mathbb P^T/ G^T]\) give mirror Calabi-Yau varieties by work of A. Chiodo and Y. Ruan [Adv. Math. 227, No. 6, 2157–2188 (2011; Zbl 1245.14038)].

It is known that two Calabi-Yau varieties given by \((W,G)\) and \((W',G)\) are related by a smooth deformation, if they are homogeneous with respect to the same weights and share the same group \(G\). The main result of this article is that the respective mirrors \(\{W^T = 0\}\) and \(\{W'{}^T = 0\}\) inside \([\mathbb P^T/G^T]\) are birational.

This result is proven describing \([\mathbb P/G]\) and \([\mathbb P^T/G^T]\) as toric orbifolds. Actually, the author describes the underlying coarse moduli spaces which are toric varieties, but for orbifolds the corrisponding lattices and fans are the same, see the work of L. A. Borisov et al. [J. Am. Math. Soc. 18, No. 1, 193–215 (2005; Zbl 1178.14057)]. Using this, the Berglund-Hübsch-Krawitz mirror construction can be understood in terms of the toric mirror construction by V. V. Batyrev [J. Algebr. Geom. 3, 493–535 (1994; Zbl 0829.14023)]. The author gives an explicit way to obtain the fan \(\Sigma\) of \(\mathbb P/G\). Then \(\{W = 0\}\) is given as the zero set of a rational function on \(\mathbb P/G\), again explicitly given by a sum of \(n+1\) points in the (toric) lattice. Applied to the two mirrors given by \((W^T,G^T)\) and \((W'{}^T,G)\), the author shows that these lie in (birational) toric orbifolds given as the zero set of the same equation, hence they are birational.

The author gives as an example three quite differently looking hypersurfaces, which are mirror dual to the Fermat quintic threefold. By the main result they are birational.

If the weights sum up to the degree of \(W\), then \(\{W = 0\} \subset [ \mathbb P/G]\) and \(\{W^T = 0\} \subset [ \mathbb P^T/ G^T]\) give mirror Calabi-Yau varieties by work of A. Chiodo and Y. Ruan [Adv. Math. 227, No. 6, 2157–2188 (2011; Zbl 1245.14038)].

It is known that two Calabi-Yau varieties given by \((W,G)\) and \((W',G)\) are related by a smooth deformation, if they are homogeneous with respect to the same weights and share the same group \(G\). The main result of this article is that the respective mirrors \(\{W^T = 0\}\) and \(\{W'{}^T = 0\}\) inside \([\mathbb P^T/G^T]\) are birational.

This result is proven describing \([\mathbb P/G]\) and \([\mathbb P^T/G^T]\) as toric orbifolds. Actually, the author describes the underlying coarse moduli spaces which are toric varieties, but for orbifolds the corrisponding lattices and fans are the same, see the work of L. A. Borisov et al. [J. Am. Math. Soc. 18, No. 1, 193–215 (2005; Zbl 1178.14057)]. Using this, the Berglund-Hübsch-Krawitz mirror construction can be understood in terms of the toric mirror construction by V. V. Batyrev [J. Algebr. Geom. 3, 493–535 (1994; Zbl 0829.14023)]. The author gives an explicit way to obtain the fan \(\Sigma\) of \(\mathbb P/G\). Then \(\{W = 0\}\) is given as the zero set of a rational function on \(\mathbb P/G\), again explicitly given by a sum of \(n+1\) points in the (toric) lattice. Applied to the two mirrors given by \((W^T,G^T)\) and \((W'{}^T,G)\), the author shows that these lie in (birational) toric orbifolds given as the zero set of the same equation, hence they are birational.

The author gives as an example three quite differently looking hypersurfaces, which are mirror dual to the Fermat quintic threefold. By the main result they are birational.

Reviewer: Andreas Hochenegger (Köln)

##### MSC:

14J33 | Mirror symmetry (algebro-geometric aspects) |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

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

14E05 | Rational and birational maps |

##### References:

[1] | Batyrev, V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebr. Geom., 3, 493-535, (1994) · Zbl 0829.14023 |

[2] | Batyrev, V.V., Borisov, L.A.: Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds, pp. 71-86. Mirror symmetry, II, AMS/IP Studies in Advanced Mathematics, vol. 1. American Mathematical Society, Providence (1997) · Zbl 0927.14019 |

[3] | Berglund, P., Hübsch, T.: A generalized construction of mirror manifolds. Nucl. Phys. B 393, 371-391 (1993) · Zbl 1245.14039 |

[4] | Borisov, L., Berglund-hubsch mirror symmetry via vertex algebras, Commun. Math. Phys., 320, 73-99, (2013) · Zbl 1317.17032 |

[5] | Clarke, P.: Duality for toric Landau-Ginzburg models (preprint). arXiv:0803.0447v1 · Zbl 1386.81130 |

[6] | Chiodo, A.; Ruan, Y., LG/CY correspondence: the state space isomorphism, Adv. Math., 227, 2157-2188, (2011) · Zbl 1245.14038 |

[7] | Cox, D.A., Katz, S.: Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, vol. 68, pp. xxii+469. American Mathematical Society, Providence, RI (1999). doi:10.1090/surv/068. http://dx.doi.org/10.1090/surv/068 · Zbl 0951.14026 |

[8] | Fan, H.; Jarvis, T.; Ruan, Y., The Witten equation, mirror symmetry and quantum singularity theory, Ann. Math., 178, 1-106, (2013) · Zbl 1310.32032 |

[9] | Krawitz, M.: FJRW rings and Landau-Ginzburg mirror symmetry. Ph.D. Thesis, University of Michigan, 67 pp. ProQuest, LLC (2010). ISBN: 978-1124-28080-6 |

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