Quiver representations in toric geometry

TL;DR

This paper introduces explicit methods for studying moduli spaces of quiver representations and derived categories in toric geometry, emphasizing noncommutative approaches and derived equivalences.

Contribution

It presents a noncommutative geometric approach to semiprojective toric varieties using quivers and explores derived category techniques, including tilting bundles and the McKay correspondence.

Findings

Constructed semiprojective toric varieties via geometric invariant theory.

Described derived equivalences and tilting bundles in toric geometry.

Extended McKay correspondence beyond G-Hilbert schemes.

Abstract

This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of moduli spaces of quiver representations and derived categories arising in toric geometry. The first main goal is to present the noncommutative geometric approach to semiprojective toric varieties via quivers. To achieve this, we use geometric invariant theory to construct both semiprojective toric varieties and moduli spaces of quiver representations. The second main goal builds on the first by presenting an introduction to explicit methods in derived categories of coherent sheaves in toric geometry. We recall the notion of tilting bundles with examples, and describe the McKay correspondence as a derived equivalence in some detail following Bridgeland, King and Reid. We also describe extensions of their result beyond the $G$-Hilbert scheme to other fine moduli spaces of bound quiver representations.