Around the tangent cone theorem

TL;DR

This paper revisits the Tangent Cone theorem, exploring its algebraic and topological aspects and illustrating its applications across various geometric and topological contexts.

Contribution

It provides a comprehensive analysis of the Tangent Cone theorem in both algebraic and topological frameworks, with diverse examples from geometry and topology.

Findings

The theorem applies to smooth quasi-projective varieties.

It relates characteristic and resonance varieties near the origin.

Examples include hyperplane arrangements and configuration spaces.

Abstract

A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several classes of examples from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their Milnor fibers, configuration spaces, and elliptic arrangements.