Lefschetz properties through a topological lens

TL;DR

This paper explores the algebraic Lefschetz properties through a topological perspective, connecting classical theorems in algebraic geometry with recent advances in commutative algebra and topology.

Contribution

It provides a comprehensive overview of the Lefschetz properties, emphasizing their topological aspects and recent developments in algebraic and geometric contexts.

Findings

Connections between the Hard Lefschetz Theorem and algebraic structures

Topological interpretations of Lefschetz properties in commutative algebra

Recent advances in artinian Gorenstein rings related to Lefschetz properties

Abstract

These lecture notes were prepared for the Lefschetz Preparatory School, a graduate summer course held in Krakow, May 6-10, 2024. They present the story of the algebraic Lefschetz properties from their origin in algebraic geometry to some recent developments in commutative algebra. The common thread of the notes is a bias towards topics surrounding the algebraic Lefschetz properties that have a topological flavor. These range from the Hard Lefschetz Theorem for cohomology rings to commutative algebraic analogues of these rings, namely artinian Gorenstein rings, and topologically motivated operations among such rings.