Derived equivalence for the simple flop of type $D_4$ via tilting bundles

TL;DR

This paper establishes a derived equivalence for the simple flop of type D_4 using tilting bundles, extending the understanding of derived categories in algebraic geometry and relating to K3 surfaces.

Contribution

It constructs tilting bundles for the D_4 flop and proves their derived equivalence, advancing the study of flops and their impact on K3 surface classifications.

Findings

Derived equivalence for the D_4 flop established

Construction of tilting bundles on both sides of the flop

Derived equivalences between certain K3 surfaces deduced

Abstract

The aim of this article is to discuss the derived equivalence problem for a local model of the simple flop of type $D_4$, which was found by Kanemitsu. First, tilting bundles on both sides of the flop are constructed, and then those tilting bundles are applied to prove the derived equivalence. This derived equivalence for the flop deduces derived equivalences between general K3 surfaces of degree $12$. The study of this example of a flop is very similar to the author's previous work for the simple of flop of type $G_2^{\dagger}$, but the construction and the analysis of tilting bundles become harder.