Zariski-van Kampen theorems for singular varieties --- an approach via the relative monodromy variation

TL;DR

This paper extends the Zariski-van Kampen theorem to singular varieties using relative monodromy variation operators, unifying fundamental and higher homotopy group computations for a broader class of algebraic varieties.

Contribution

It introduces a new generalization employing relative monodromy variation operators, enabling analysis of more complex singular varieties and unifying fundamental and higher homotopy group results.

Findings

Provides a new theorem for singular varieties using relative monodromy operators.

Unifies the computation of fundamental and higher homotopy groups.

Confirms a conjecture for non-singular varieties.

Abstract

The classical Zariski-van Kampen theorem gives a presentation of the fundamental group of the complement of a complex algebraic curve in $\mathbb{P}^2$. The first generalization of this theorem to singular (quasi-projective) varieties was given by the first author. In both cases, the relations are generated by the standard monodromy variation operators associated with the special members of a generic pencil of hyperplane sections. In the present paper, we give a new generalization in which the relations are generated by the relative monodromy variation operators introduced by D. Ch\'eniot and the first author. The advantage of using the relative operators is not only to cover a larger class of varieties but also to unify the Zariski-van Kampen type theorems for the fundamental group and for higher homotopy groups. In the special case of non-singular varieties, the main result of this paper was conjectured by D. Ch\'eniot and the first author.