Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 051, 15 pages      arXiv:1907.11421      https://doi.org/10.3842/SIGMA.2020.051
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic

Wolfgang Ebeling a and Sabir M. Gusein-Zade b
a)  Leibniz Universität Hannover, Institut für Algebraische Geometrie, Postfach 6009, D-30060 Hannover, Germany
b)  Moscow State University, Faculty of Mechanics and Mathematics, Moscow, GSP-1, 119991, Russia

Received July 29, 2019, in final form June 01, 2020; Published online June 11, 2020

Abstract
P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called parity condition (PC). Here we prove that, if the permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.

Key words: group action; invertible polynomial; orbifold Euler characteristic; mirror symmetry; Berglund-Hübsch-Henningson-Takahashi duality.

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