On the Orlov conjecture for hyper-K\"ahler varieties via hyperholomorphic bundles

We study Fourier transforms induced by Markman's projectively hyperholomorphic bundles on products of hyper-K\"ahler varieties of $K3^{[n]}$-type. As applications, we prove the following. (a) Derived equivalent hyper-K\"ahler varieties of $K3^{[n]}$-type have isomorphic homological motives preserving the cup-product. (b) All smooth projective moduli spaces of stable sheaves on a given $K3$ surface have isomorphic homological motives preserving the cup-product. (c) Assuming the Franchetta properties for the self-products of polarized $K3$ surfaces, the isomorphisms in (b) can be lifted to Chow motives for $K3$ surfaces of Picard rank 1. These results provide evidence for the Orlov conjecture and a conjecture of Fu-Vial.