Cohen-Macaulay modules and the Bondal-Orlov conjecture

TL;DR

This paper introduces the CM-degree concept to construct tilting bundles on resolutions, enabling proofs of the Bondal-Orlov conjecture in specific small resolution cases by ensuring derived category equivalences.

Contribution

It develops a new approach using CM-degree to produce tilting bundles compatible with flops, advancing the understanding of derived categories in algebraic geometry.

Findings

Constructed tilting generators using CM-degree

Proved the Bondal-Orlov conjecture in certain small resolutions

CM degree is preserved under strict transform

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.