Monodromy of functions on isolated cyclic quotients

We study two important invariants of the monodromy of a function on an isolated cyclic quotient (Cn/G,0), where G is a finite cyclic group: the Lefschetz number and the zeta-function. Our approach relies on a certain “good” toric modification of Cn inducing a toric resolution of the cyclic quotient. We prove that the Lefschetz number has a sum decomposition into Lefschetz numbers of well-defined weighted-homogeneous “pieces” of the initial function, the weights depending only on the group action. We define a class of nondegenerate functions and prove for them a zeta-function formula, using Varchenkoʹs approach via the Newton polyhedron.