The integral monodromy of isolated quasihomogeneous singularities

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 $\mathbb{Z}$-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.