On the weak Lefschetz principle in birational geometry

TL;DR

This paper explores the weak Lefschetz principle in birational geometry, tracing its origins from algebraic topology to recent developments, highlighting unexpected scenarios where similar theorems hold and their mathematical significance.

Contribution

It provides an expository overview of the evolution and applications of the weak Lefschetz principle in birational geometry, emphasizing new contexts where it applies.

Findings

Various scenarios where Lefschetz-type theorems hold in geometry

Impact of these theorems on understanding birational properties

Historical development from topology to modern algebraic geometry

Abstract

This is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry. In doing so, the main theme of the article is the following: there are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. These scenarios are somewhat unexpected and have had a profound impact in mathematics.