Kodai Mathematical Journal
Online ISSN : 1881-5472
Print ISSN : 0386-5991
ISSN-L : 0386-5991
Zariski-van Kampen theorems for singular varieties—an approach via the relative monodromy variation
Christophe EyralPeter Petrov
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2019 年 42 巻 1 号 p. 75-98

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The classical Zariski-van Kampen theorem gives a presentation of the fundamental group of the complement of a complex algebraic curve in P2. 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.

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© 2019 Department of Mathematics, Tokyo Institute of Technology
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