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The classical Zariski-van Kampen theorem gives a presentation of thefundamental 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, therelations are generated by the standard monodromy variation operatorsassociated with the special members of a generic pencil of hyperplane sections.In the present paper, we give a new generalization in which the relations aregenerated by the relative monodromy variation operators introduced by D.Ch\'eniot and the first author. The advantage of using the relative operatorsis not only to cover a larger class of varieties but also to unify theZariski-van Kampen type theorems for the fundamental group and for higherhomotopy groups. In the special case of non-singular varieties, the main resultof this paper was conjectured by D. Ch\'eniot and the first author.

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