The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles

We show that birational hyper-Kähler varieties of $K{3}^{[n]}$-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-Kähler varieties of $K{3}^{[n]}$-type with Brauer classes.