Local polar varieties in the geometric study of singularities

TL;DR

This paper explores the role of local polar varieties in understanding the structure and invariants of complex analytic singularities, unifying geometric, topological, and algebraic perspectives through Whitney conditions.

Contribution

It introduces a framework linking local polar varieties with Whitney stratifications and invariants, providing new insights into the geometry of singularities and their dual varieties.

Findings

Local polar varieties analyze limit directions of tangent hyperplanes.

Whitney conditions are shown to have a Lagrangian nature.

A Plücker-type formula relates polar multiplicities to dual variety degrees.

Abstract

This text presents several aspects of the theory of equisingularity of complex analytic spaces from the standpoint of Whitney conditions. The goal is to describe from the geometrical, topological, and algebraic viewpoints a canonical locally finite partition of a reduced complex analytic space $X$ into nonsingular strata with the property that the local geometry of $X$ is constant on each stratum. Local polar varieties appear in the title because they play a central role in the unification of viewpoints. The geometrical viewpoint leads to the study of spaces of limit directions at a given point of $X\subset \C^n$ of hyperplanes of $\C^n$ tangent to $X$ at nonsingular points, which in turn leads to the realization that the Whitney conditions, which are used to define the stratification, are in fact of a Lagrangian nature. The local polar varieties are used to analyze the structure of the set of limit directions of tangent hyperplanes. This structure helps in particular to understand how a singularity differs from its tangent cone, assumed to be reduced. The multiplicities of local polar varieties are related to local topological invariants, local vanishing Euler-Poincar\'e characteristics, by a formula which turns out to contain, in the special case where the singularity is the vertex of the cone over a reduced projective variety, a Pl\"ucker-type formula for the degree of the dual of a projective variety. The degree of the dual of a projective variety is expressed in terms of Euler-Poincar\'e characteristics attached to the minimal Whitney stratification of the variety.