Methods and Applications of Analysis

Volume 24 (2017)

Number 2

Special issue dedicated to Henry B. Laufer on the occasion of his 70th birthday: Part 2

Guest Editors: Stephen S.-T. Yau (Tsinghua University, China); Gert-Martin Greuel (University of Kaiserslautern, Germany); Jonathan Wahl (University of North Carolina, USA); Rong Du (East China Normal University, China); Yun Gao (Shanghai Jiao Tong University, China); and Huaiqing Zuo (Tsinghua University, China)

Equisingular and equinormalizable deformations of isolated non-normal singularities

Pages: 215 – 276

DOI: http://dx.doi.org/10.4310/MAA.2017.v24.n2.a3

Author

Gert-Martin Greuel (Department of Mathematics, University of Kaiserslautern, Germany)

Abstract

We present new results on equisingularity and equinormalizability of families with isolated non–normal singularities (INNS) of arbitrary dimension. We define a $\delta$-invariant and a $\mu$-invariant for an INNS and prove necessary and sufficient numerical conditions for equinormalizability and weak equinormalizability using $\delta$ and $\mu$. For families of generically reduced curves, we investigate the topological behavior of the Milnor fibre and characterize topological triviality of such families. Finally we state some open problems and conjectures. In addition we give a survey of classical results about equisingularity and equinormalizability so that the article may be useful as a reference source.

Keywords

equisingularity, Milnor number, $\delta$-invariant, isolated non-normal singularities, simultaneous normalization, topological triviality

2010 Mathematics Subject Classification

14B05, 14B07, 14B10, 14H20, 32B10, 32C20, 32S05, 32S15, 32S25, 32S30

Full Text (PDF format)

Received 24 January 2017

Accepted 1 July 2017

Published 3 January 2018