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Mirror symmetry between orbifold curves and cusp singularities with group action

By Wolfgang Ebeling and Atsushi Takahashi

Abstract

We consider an orbifold Landau-Ginzburg model $(f,G)$, where $f$ is an invertible polynomial in three variables and $G$ a finite group of symmetries of $f$ containing the exponential grading operator, and its Berglund-H\"ubsch transpose $(f^T, G^T)$. We show that this defines a mirror symmetry between orbifold curves and cusp singularities with group action. We define Dolgachev numbers for the orbifold curves and Gabrielov numbers for the cusp singularities with group action. We show that these numbers are the same and that the stringy Euler number of the orbifold curve coincides with the $G^T$-equivariant Milnor number of the mirror cusp singularity.Comment: 29 pages, Table 2 corrected, Assumption g=0 added to Theorem 2

Topics: Mathematics - Algebraic Geometry, Mathematics - Representation Theory, 14J33, 32S25, 32S35, 14L30, 53D37
Year: 2011
OAI identifier: oai:arXiv.org:1103.5367
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