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Cohen-Macaulay modules and the Bondal-Orlov conjecture

Ananyo Dan, Yirui Xiong

TL;DR

The paper advances the Bondal-Orlov conjecture for crepant resolutions by introducing a CM-degree framework to construct tilting generators on small resolutions. It proves that CM-degree is preserved under strict transform and shows that when a relative very ample line bundle has CM-degree equal to the exceptional locus dimension, existing tilting generators remain tilting after the flop, yielding derived equivalences in new small resolution cases. The work develops a robust theory of almost-full sheaves and tilting factors, provides vanishing and tensor-product results, and connects these to both tilting bundles and maximal Cohen-Macaulay modules. A key contribution is the explicit link between CM-degree and tilting behavior, enabling concrete verifications in Mukai-type flops and providing a practical route to non-commutative crepant resolutions in higher dimensions. Overall, the results broaden the scope of Bondal-Orlov type equivalences by supplying verifiable criteria that are preserved through flops and by clarifying the relationship between geometric tilting objects and CM-module theory.

Abstract

Most of the known examples of derived categories of small resolutions arise as the derived category of the endormorphism algebra of tilting bundles or complexes. Given two resolutions connected by a flop, if the strict transform of a tilting bundle is again tilting, then the derived categories of the two resolutions are equivalent, thereby proving the Bondal-Orlov conjecture in this setup. Unfortunately, it is difficult to produce tilting bundles that are compatible with flops. In this article, we introduce the notion of CM-degree of locally-free sheaves on resolutions and use them to construct tilting generators. In particular, we show that if there exists a relative very ample line bundle on the resolution with CM degree equal to the dimension of the exceptional locus, then the generator bundles constructed by Van den Bergh and Toda-Uehara are also tilting bundles. The advantage of our approach is that the CM degree is preserved under strict transform. As a consequence we prove the Bondal-Orlov conjecture in certain cases of small resolutions.

Cohen-Macaulay modules and the Bondal-Orlov conjecture

TL;DR

The paper advances the Bondal-Orlov conjecture for crepant resolutions by introducing a CM-degree framework to construct tilting generators on small resolutions. It proves that CM-degree is preserved under strict transform and shows that when a relative very ample line bundle has CM-degree equal to the exceptional locus dimension, existing tilting generators remain tilting after the flop, yielding derived equivalences in new small resolution cases. The work develops a robust theory of almost-full sheaves and tilting factors, provides vanishing and tensor-product results, and connects these to both tilting bundles and maximal Cohen-Macaulay modules. A key contribution is the explicit link between CM-degree and tilting behavior, enabling concrete verifications in Mukai-type flops and providing a practical route to non-commutative crepant resolutions in higher dimensions. Overall, the results broaden the scope of Bondal-Orlov type equivalences by supplying verifiable criteria that are preserved through flops and by clarifying the relationship between geometric tilting objects and CM-module theory.

Abstract

Most of the known examples of derived categories of small resolutions arise as the derived category of the endormorphism algebra of tilting bundles or complexes. Given two resolutions connected by a flop, if the strict transform of a tilting bundle is again tilting, then the derived categories of the two resolutions are equivalent, thereby proving the Bondal-Orlov conjecture in this setup. Unfortunately, it is difficult to produce tilting bundles that are compatible with flops. In this article, we introduce the notion of CM-degree of locally-free sheaves on resolutions and use them to construct tilting generators. In particular, we show that if there exists a relative very ample line bundle on the resolution with CM degree equal to the dimension of the exceptional locus, then the generator bundles constructed by Van den Bergh and Toda-Uehara are also tilting bundles. The advantage of our approach is that the CM degree is preserved under strict transform. As a consequence we prove the Bondal-Orlov conjecture in certain cases of small resolutions.
Paper Structure (17 sections, 19 theorems, 93 equations)

This paper contains 17 sections, 19 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local cohomology
  4. Duality theorem
  5. Tilting factorization
  6. Setup
  7. Example (Tilting factors of $\mathop{\mathrm{\mathcal{O}}}\nolimits_X$):
  8. Tilting bundle
  9. Properties of almost full sheaves
  10. Criterion for almost full sheaves
  11. Rank one almost full sheaves
  12. Tensor product of almost full sheaves
  13. Vanishing result
  14. Application to the Bondal-Orlov conjecture
  15. Setup
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(X,x)$ be an isolated Cohen-Macaulay singularity of dimension at least $3$ and $\widetilde{X}$ be a small resolution of $X$. If $\mathcal{L}$ is an invertible sheaf on $\widetilde{X}$ with CM-degree $m$, then $H^i((\mathcal{L}^{\vee})^{\otimes j})=0$ for all $i>0$ and $1 \le j \le m$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 3.1
  • Definition 3.2
  • Theorem 3.3
  • ...and 33 more