## Kodai Mathematical Journal

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

#### Abstract

The classical Zariski-van Kampen theorem gives a presentation of the fundamental group of the complement of a complex algebraic curve in $\mathbf{P}^2$. 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. Chéniot 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. Chéniot and the first author.

#### Article information

Source
Kodai Math. J., Volume 42, Number 1 (2019), 75-98.

Dates
First available in Project Euclid: 19 March 2019

https://projecteuclid.org/euclid.kmj/1552982507

Digital Object Identifier
doi:10.2996/kmj/1552982507

Mathematical Reviews number (MathSciNet)
MR3934614

Zentralblatt MATH identifier
07081614

#### Citation

Eyral, Christophe; Petrov, Peter. Zariski-van Kampen theorems for singular varieties—an approach via the relative monodromy variation. Kodai Math. J. 42 (2019), no. 1, 75--98. doi:10.2996/kmj/1552982507. https://projecteuclid.org/euclid.kmj/1552982507