Singularities and Characteristic Classes for Differentiable Maps

TL;DR

This paper introduces a new branch of Thom polynomial theory for holomorphic map singularities, replacing point counting with weighted Euler characteristics, and applies it to study the topology of weighted homogeneous map-germs.

Contribution

It develops a novel approach to singularity theory by integrating weighted Euler characteristics into Thom polynomial computations for holomorphic maps.

Findings

New framework for Thom polynomial theory for holomorphic singularities

Application to the topology of weighted homogeneous map-germs

Extension of singularity analysis without corank restrictions

Abstract

This is a note on my mini-course in the International Workshop on Real and Complex Singularities held at ICMC-USP (Sao Carlos, Brazil) in July 2012. Here we introduce a new branch of the Thom polynomial theory for singularities of holomorphic maps, in which we replace counting singular points by computing weighted Euler characteristics. The main purpose is to apply this theory to the study on the vanishing topology of weighted homogeneous map-germs of finite A-codimension without any corank condition.