Quiver representations in toric geometry

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.

55 pages. To Miles Reid on his 60th birthday

References in corpus (12)

• La correspondance de McKay
• McKay equivalence for symplectic resolutions of singularities
• McKay correspondence
• Log Crepant Birational Maps and Derived Categories
• Syzygies, multigraded regularity and toric varieties
• How to calculate A-Hilb C^3
• Moduli spaces for Bondal quivers
• Derived McKay correspondence via pure-sheaf transforms
• Superpotentials and Higher Order Derivations
• Noncommutative resolutions of ADE fibered Calabi-Yau threefolds
• The Special McKay correspondence as an equivalence of derived categories
• On the conjecture of King for smooth toric Deligne-Mumford stacks

Cited by in corpus (9)

• Instantons, Quivers and Noncommutative Donaldson-Thomas Theory
• Lectures on Noncommutative Resolutions
• Character Varieties and the Moduli of Quiver Representations
• Some Open Questions in Quiver Gauge Theory
• Tilting objects on twisted forms of some relative flag varieties
• The Special McKay correspondence as an equivalence of derived categories
• On the neighborliness of dual flow polytopes of quivers
• Torsion pairs and filtrations in abelian categories with tilting objects
• Tilting theoretical approach to moduli spaces over preprojective algebras