The integral monodromy of isolated quasihomogeneous singularities

Hertling, Claus; Mase, Makiko

doi:10.2140/ant.2022.16.955

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The integral monodromy of isolated quasihomogeneous singularities

Claus Hertling and Makiko Mase

Vol. 16 (2022), No. 4, 955–1024

DOI: 10.2140/ant.2022.16.955

Abstract

The integral monodromy on the Milnor lattice of an isolated quasihomogeneous singularity is subject of an almost untouched conjecture of Orlik from 1972. We prove this conjecture for all iterated Thom–Sebastiani sums of chain type singularities and cycle type singularities. The main part of the paper is purely algebraic. It provides tools for dealing with sums and tensor products of $ℤ$ -lattices with automorphisms of finite order and with cyclic generators. The calculations are involved. They use fine properties of unit roots, cyclotomic polynomials, their resultants and discriminants.

Keywords

cyclic monodromy, cyclotomic polynomial, quasihomogeneous singularity, Milnor lattice, Orlik block, Thom–Sebastiani sum

Mathematical Subject Classification

Primary: 11C20, 15B36, 32S40, 47A80

Milestones

Received: 17 September 2020

Revised: 16 June 2021

Accepted: 24 July 2021

Published: 5 August 2022

Authors

Claus Hertling
Lehrstuhl für algebraische Geometrie Universität Mannheim Mannheim Germany

Makiko Mase
Lehrstuhl für algebraische Geometrie Universität Mannheim Mannheim Germany

Vol. 16, No. 4, 2022