The D-equivalence conjecture for hyper-K\"ahler varieties via hyperholomorphic bundles

TL;DR

This paper proves that birational hyper-K"ahler varieties of $K3^{[n]}$-type are derived equivalent, using hyperholomorphic bundles to construct Fourier-Mukai kernels, thus confirming the D-equivalence conjecture in these cases.

Contribution

It establishes the D-equivalence conjecture for $K3^{[n]}$-type hyper-K"ahler varieties and introduces a method using hyperholomorphic bundles for constructing derived equivalences.

Findings

Birational hyper-K"ahler varieties of $K3^{[n]}$-type are derived equivalent.

Constructs Fourier-Mukai kernels from hyperholomorphic bundles.

Proves a stronger version of the D-equivalence conjecture with Brauer classes.

Abstract

We show that birational hyper-K\"ahler varieties of $K3^{[n]}$-type are derived equivalent, establishing the D-equivalence conjecture in these cases. The Fourier-Mukai kernels of our derived equivalences are constructed from projectively hyperholomorphic bundles, following ideas of Markman. Our method also proves a stronger version of the D-equivalence conjecture for hyper-K\"ahler varieties of $K3^{[n]}$-type with Brauer classes.