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Non-commutative crepant resolutions, an overview

Michel Van den Bergh

TL;DR

Non-commutative crepant resolutions (NCCRs) are endomorphism algebras $\Lambda=\operatorname{End}_Y(\mathcal{T})$ of tilting bundles on (potential) crepant resolutions $Y$ of Gorenstein varieties $X$, providing derived equivalences $\mathcal{D}(Y)\simeq \mathcal{D}(\Lambda)$. The paper surveys construction methods—including quotient singularities, tilting on resolutions, toric and GIT quotients with modules of covariants—and mutation techniques, illustrating how NCCRs arise across diverse geometric settings. It explains how NCCRs relate to crepant categorical resolutions, SKMS/schobers, and tilting theory, highlighting Bridgeland’s 3-fold flop results and the interplay between commutative and noncommutative crepant pictures. The discussion emphasizes both successes (e.g., BKR-type results in low dimension and toric cases) and limitations (e.g., instances where polynomiality of stringy $E$-functions fails or where crepant resolutions do not exist), outlining a landscape where NCCRs serve as noncommutative bridges for birational geometry and derived categories.

Abstract

Non-commutative crepant resolutions (NCCRs) are non-commutative analogues of the usual crepant resolutions that appear in algebraic geometry. In this paper we survey some results around NCCRs.

Non-commutative crepant resolutions, an overview

TL;DR

Non-commutative crepant resolutions (NCCRs) are endomorphism algebras of tilting bundles on (potential) crepant resolutions of Gorenstein varieties , providing derived equivalences . The paper surveys construction methods—including quotient singularities, tilting on resolutions, toric and GIT quotients with modules of covariants—and mutation techniques, illustrating how NCCRs arise across diverse geometric settings. It explains how NCCRs relate to crepant categorical resolutions, SKMS/schobers, and tilting theory, highlighting Bridgeland’s 3-fold flop results and the interplay between commutative and noncommutative crepant pictures. The discussion emphasizes both successes (e.g., BKR-type results in low dimension and toric cases) and limitations (e.g., instances where polynomiality of stringy -functions fails or where crepant resolutions do not exist), outlining a landscape where NCCRs serve as noncommutative bridges for birational geometry and derived categories.

Abstract

Non-commutative crepant resolutions (NCCRs) are non-commutative analogues of the usual crepant resolutions that appear in algebraic geometry. In this paper we survey some results around NCCRs.
Paper Structure (21 sections, 32 theorems, 44 equations)

This paper contains 21 sections, 32 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Crepant resolutions and derived equivalences
  4. Non-commutative rings
  5. Bridgeland's result
  6. Flops
  7. Maps with fibers of dimension $\le 1$
  8. Non-commutative (crepant) resolutions
  9. Generalities
  10. Relation with crepant categorical resolutions
  11. Constructions of non-commutative crepant resolutions
  12. Quotient singularities
  13. Crepant resolutions with tilting complexes
  14. Resolutions with partial tilting complexes
  15. Three-dimensional affine toric varieties
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that $X$ has canonical Gorenstein singularities. Then the Hodge numbers of $Y$ for a crepant resolution $Y\rightarrow X$ are independent of $Y$.

Theorems & Definitions (93)

  • Theorem 1.1: Kon95Batyrev1, see also §\ref{['sec:stringy']} below
  • Conjecture 1.2: BonOrl, KawDK
  • Remark 1.3
  • Example 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7: KellerKrause
  • Remark 1.8
  • Example 1.9
  • Theorem 1.10
  • ...and 83 more