Monodromy of functions on isolated cylic quotients
The author studies two invariants of the monodromy of a holomorphic function defined near an isolated cyclic quotient singularity (i.e., of the form \((X,x)=(\mathbb{C}^n/G,0)\), where \(G\) is a finite cyclic group), namely the Lefshetz number and the zeta function. Given a germ of a function \(f:(X,x) \to(\mathbb{C},0)\), its algebraic monodromy is the induced transformation \(h_f:H^\bullet (\Gamma_f, \mathbb{C})\to H^\bullet (\Gamma_f,\mathbb{C})\) where \(\Gamma_f\) is a general fiber of \(f\). The Lefshetz number \(\Lambda(f)\) is defined to be \(\sum(-1)^j\text{trace}[h_f]\), while the zeta function is \(\zeta_f (t)=\prod_{i\geq 0}\det [I-th_f]^{(-1)^{i+1}}\). In the case where \((X,x)=(\mathbb{C}^n,0)\), fundamental results were obtained by A'Campo and Varchenko. Given \(f\) as above one may associate to it a \(G\)-invariant germ of a function \(\widetilde f:=fp\), where \(p:(\mathbb{C}^n,0) \to(X,x)\) is the quotient map. When \(f(x)=0\) and it satisfies a certain condition, called \({\mathcal W}\)-nondegeneracy, by using methods of toric geometry and techniques introduced by Varchenko, the author finds a rather explicit formula for \(\zeta_f(t)\) in terms \(\widetilde f\) (more precisely, of certain numbers obtained from the Newton polyhedron of \(\widetilde f\)). There is a formula of the same type for \(\Lambda(f)\), valid under a weaker nondegeneracy condition, as well as other formulas allowing us to compare (under suitable hypotheses) the invariants of \(f\) and \(\widetilde f\). One must be careful with such a comparison, in the last section (devoted to examples) it is shown that certain ``reasonable formulas'', such as \(\Lambda (h^d_{\widetilde f})=d \wedge(f)\) in general may be false. This is a technical but well written article.
scientific article; zbMATH DE number 1367546