A simpler kernel for stratified Mukai flops

TL;DR

This work addresses the problem of identifying the Fourier–Mukai kernel for the derived equivalence associated with stratified Mukai flops, focusing on Grassmannians of planes in the $k=2$ case. By recasting the problem in the matrix-factorization setting via Knörrer periodicity and magic-window theory, the authors construct and analyze a natural kernel on the higher-dimensional spaces, proving that the kernel is the structure sheaf of the fiber product $\overline{\Delta}=E_+\times_{E_0}E_-$, interpreted as a matrix-factorization kernel of $W'-W$. They compute the $\mathrm{GL}(S)$-weights to verify the kernel lies in the appropriate window and provide an explicit free resolution, showing how the CKL kernel is recovered in this MF framework. The results reveal that the apparent complexity of stratified Mukai flops is an artifact of passing to derived categories via Knörrer periodicity, and that in the MF category the kernel admits a simple geometric description for $k=2$. While the $k>2$ generalization remains open, the paper clarifies the link between window equivalences, matrix factorizations, and known geometric kernels, offering a concrete and implementable description in this setting.

Abstract

We reinvestigate the problem of describing the Fourier-Mukai kernel for the derived equivalence associated to a stratified Mukai flop. For the case of Grassmannians of planes we give a very simple geometric construction of the kernel, using the framework of matrix factorizations.

Theorems & Definitions (19)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Lemma 3.2
• proof
• Remark 3.4
• Lemma 4.1
• proof