Communications in Analysis and Geometry

Volume 21 (2013)

Number 3

Characterization of isolated complete intersection singularities with $\mathbb{C}^*$-action of dimension $n \geq 2$ by means of geometric genus and irregularity

Pages: 509 – 526

DOI: http://dx.doi.org/10.4310/CAG.2013.v21.n3.a2

Authors

Stephen Yau (Department of Mathematical Science, Tsinghua University, Bejing, China)

Huaiqing Zuo (Department of Mathematical Science, Tsinghua University, Bejing, China)

Abstract

Dedicated to Professor Michael Artin on the occasion of his 79th birthday It is well known that geometric genus $p_g$ and irregularity $q$ are two important invariants for isolated singularities. In this paper, we give a formula relating $p_g$ and $q$ for isolated singularities with $\mathbb{C}^*$-action in any dimension. We also give a simple characterization of the quasi-homogeneous isolated complete intersection singularities using $p_g$ and $q$ . As a corollary, we prove that q is an invariant of topological type for two-dimensional weighted homogeneous hypersurface singularities.

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