The maximal ideal cycles over normal surface singularities with ${\Bbb C}^*$-action

Abstract

The maximal ideal cycles and the fundamental cycles are defined on the exceptional sets of resolution spaces of normal complex surface singularities. The former (resp. later) is determined by the analytic (resp. topological) structure of the singularities. We study such cycles for normal surface singularities with ${\Bbb C}^*$-action. Assuming the existence of a reduced homogeneous function of the minimal degree, we prove that these two cycles coincide if the coefficients on the central curve of the exceptional set of the minimal good resolution coincide.