On the Orlov conjecture for hyper-Kähler varieties via hyperholomorphic bundles

We study Fourier transforms induced by Markman’s projectively hyperholomorphic bundles on products of hyper-Kähler varieties of $K3^{[n]}$-type. As applications, we prove the following. (a) Derived equivalent hyper-Kähler 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.

💡 Research Summary

The paper tackles the long‑standing Orlov conjecture, which predicts that derived‑equivalent smooth projective varieties should have isomorphic Chow motives, and its multiplicative refinement proposed by Fu–Vial, requiring the isomorphism to respect the cup‑product. While the conjecture is known for K3 surfaces, very little is proved in higher dimensions. The authors focus on hyper‑Kähler varieties of K3^