On the Orlov conjecture for hyper-Kähler varieties via hyperholomorphic bundles
Davesh Maulik, Junliang Shen, Qizheng Yin
TL;DR
The paper develops a Fourier-transform framework for hyper-Kähler varieties of $K3^{[n]}$-type using Markman’s projectively hyperholomorphic bundles to connect derived equivalences with motivic and, under Franchetta-type hypotheses, Chow-motive statements. It proves that derived-equivalent HK varieties of $K3^{[n]}$-type have isomorphic homological motives preserving the cup-product, and that moduli spaces $M_{S,v}$ of stable sheaves on a fixed $K3$ surface have isomorphic $h_{ ext{hom}}$ to the Hilbert schemes $S^{[n]}$, with potential lift to Chow motives in Picard rank $1$. The approach relies on a cohomology-to-motive bridge via a carefully normalized Fourier transform package, an explicit use of the extended Mukai lattice, and universality arguments across moduli spaces, together with Franchetta properties to upgrade homological results to Chow-level statements. Collectively, these results provide substantial evidence toward the multiplicative Orlov conjecture for hyper-Kähler varieties and illuminate the role of generically defined cycles in translating derived-equivalence data into motivic isomorphisms with cup-product compatibility.
Abstract
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.