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Enhanced equivariant Saito duality

Wolfgang Ebeling

Leibniz Universität Hannover, Institut für Algebraische Geometrie, Postfach 6009, D-30060 Hannover, Germany

E-mail Address: [email protected]

Corresponding author.

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and

Sabir M. Gusein-Zade

Faculty of Mechanics and Mathematics, Moscow State University, Moscow, GSP-1, 119991, Russia

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219498818501815Cited by:0 (Source: Crossref)

Abstract

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring $\hat{B} (G)$ of a finite group $G$ is defined. An element of it is represented by a finite $G$ -set with a $G$ -equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic $G$ -manifold with a $G$ -equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of $\hat{B} (G)$ . It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibers of Berglund–Hübsch dual invertible polynomials are enhanced dual to each other up to sign. As a byproduct, this implies the result about the orbifold zeta functions of Berglund–Hübsch–Henningson dual pairs obtained earlier.

Communicated by S. Ishii

Keywords:

AMSC: 14R20, 19A22, 14J33, 57R18, 58K10

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