Helmut Hamm: On Lefschetz-Zariski-van Kampen theorems (Research Seminar on Geometry, Algebra and Topology: Moduli Spaces of Complex Curves)
Wednesday, 30.11.2022 16:15 im Raum M3
The classical Lefschetz theorem on hyperplane sections helps to calculate the homology groups of smooth complex projective varieties. The original tool was a Lefschetz pencil, later on this method was replaced by Morse theory which made it easier to pass to homotopy groups. On the other hand, van Kampen developed a method to calculate the fundamental group of the complement of a smooth plane projective curve, and Zariski showed that the computation of the fundamental group of the complement of an arbitrary smooth projective hypersurface can be reduced to this case. All this can be subsumed in the study of the homology and homotopy groups of a smooth quasi-projective variety using a general hyperplane section.
There are corresponding local theorems. These are connected with the preceding ones by taking the cone over a (quasi)projective variety.
The case of homology is meanwhile well understood but passing from homology to homotopy groups involves problems because homotopy excision is only allowed under heavy restrictions.