Derived equivalence for the simple flop of type $G_2^{\dagger}$ via tilting bundles

TL;DR

This paper proves a derived equivalence for a specific type of algebraic flop using tilting bundles, also constructing a related noncommutative crepant resolution.

Contribution

It establishes the derived equivalence for the simple flop of type G_2^{ ext{dagger}} using tilting bundles and introduces a noncommutative crepant resolution.

Findings

Derived equivalence for the G_2^{ ext{dagger}} flop established.

Constructed a noncommutative crepant resolution.

Provides a new approach to simple flops from non-homogeneous roofs.

Abstract

The aim of this article is to prove the derived equivalence for a local model of the simple flop of type $G_2^{\dagger}$, which was found by Kanemitsu. This flop is the only known simple flop that comes from a non-homogeneous roof. The proof of the derived equivalence is done by using tilting bundles, and also produces a noncommutative crepant resolution of the singularity that is derived equivalent to both sides of the flop.