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

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.