Homotopical variations and high-dimensional Zariski-van Kampen theorems

In 1933, van Kampen described the fundamental groups of the complements of plane complex projective algebraic curves. Recently, Ch'eniot-Libgober proved an analogue of this result for higher homotopy groups of the complements of complex projective hypersurfaces with isolated singularities. Their description is in terms of some “homotopical variation operators”. We generalize here the notion of “homotopical variation” to (singular) quasi-projective varieties. This is a first step for further generalizations of van Kampen’s theorem. A conjecture, with a first approach, is stated in the special case of non-singular quasi-projective varieties.