Survey on some aspects of Lefschetz theorems in algebraic geometry

TL;DR

This survey reviews classical Lefschetz theorems in algebraic geometry, focusing on fundamental groups and their connection to Deligne's Weil II program, based on lectures from 2015-2016.

Contribution

It provides a comprehensive overview of Lefschetz theorems and explores their relation to Deligne's Weil II, clarifying their role in modern algebraic geometry.

Findings

Clarifies the connection between Lefschetz theorems and Weil II

Summarizes classical results on fundamental groups

Highlights open questions in the area

Abstract

We survey classical material around Lefschetz theorems for fundamental groups, and show the relation to parts of Deligne's program in Weil II. The notes are based on some aspects of the material for the Santal\'o lectures at the Universidad Complutense de Madrid (October 2015) and the Rademacher lectures at the University of Pennsylvania (February 2016). They appear in Revista Matem\'atica Complutense.