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

TL;DR

This paper investigates the Orlov conjecture for hyper-K"ahler varieties, demonstrating that derived equivalent varieties and certain moduli spaces share isomorphic homological motives, thus supporting the conjecture.

Contribution

It proves that derived equivalent hyper-K"ahler varieties and moduli spaces of sheaves on K3 surfaces have isomorphic homological motives, advancing the understanding of the Orlov conjecture.

Findings

Derived equivalent hyper-K"ahler varieties have isomorphic homological motives.

Moduli spaces of stable sheaves on a K3 surface share the same homological motives.

Results support the Orlov conjecture and Fu-Vial's conjecture.

Abstract

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.