Year: 2020
Type: preprint
Abstract: One version of the classical Lefschetz hyperplane theorem states that for $U \subset \mathbb P^n$ a smooth quasi-projective variety of dimension at least $2$, and $H \cap U$ a general hyperplane section, the resulting map on étale fundamental groups $π_1(H \cap U) \rightarrow π_1(U)$ is surjective. We prove a generalization, replacing the hyperplane by a general $\operatorname{PGL}_{n+1}$-translate of an arbitrary projective variety: If $U \subset \mathbb P^n$ is a normal quasi-projective variety, $X$ is a geometrically irreducible projective variety of dimension at least $n + 1 - \dim U$, and $Y$ is a general $\operatorname{PGL}_{n+1}$-translate of $X$, then the map $π_1(Y \cap U) \rightarrow π_1(U)$ is surjective.
Source: arXiv (Cornell University)
Author: Landesman, Aaron
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3027830690