Window categories for a simple $9$-fold flop of Grassmannian type

Abstract

The local simple $9$-fold flop of Grassmannian type is a birational transformation between total spaces of vector bundles on the Grassmannians $\mathrm{Gr}(2, 5)$ and $\mathrm{Gr}(3, 5)$. We produce four different derived equivalences which commute with the pushforward functors for the flopping contractions. These equivalences are realized by identifying four different window categories inside the derived category of coherent sheaves on an Artin stack. As an application, our approach provides a new proof of derived equivalence for a pair of non-birational Calabi-Yau threefolds realized as zero loci of sections of homogeneous vector bundles in Grassmannians.