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Oberseminar SoSe 2011 Zusammenfassung des Vortrags von W. Ebeling und S. M. Gusein-Zade (Moskau, RUS)

Mirror symmetry of orbifold Landau-Ginzburg models

V.I. Arnold discovered a strange duality between the 14 exceptional unimodal singularities. This duality embraces a correspondence between Dolgachev and Gabrielov numbers. It was observed by K. Saito that there is also a duality between the characteristic polynomials of the monodromy of dual singularities. The aim of the talk is to generalize this duality to orbifold Landau-Ginzburg models ( f,G), where f is an invertible polynomial and G a finite group of symmetries of f containing the exponential grading operator, together with their Berglund-Hübsch transposes ( fT,GT). In the first part we consider non-degenerate invertible polynomials of three variables. We show that there is 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 GT-equivariant Milnor number of the mirror cusp singularity. This is joint work of W. Ebeling with A. Takahashi. In the second part, we show how Saito's duality generalizes to orbifold Landau-Ginzburg models with an arbitrary number of variables.