Ulrich Sheaves on the Hilbert Square of K3 and Abelian Surfaces
Anindya Mukherjee, Pabitra Barik
TL;DR
This work establishes the existence of Ulrich sheaves on the Hilbert square $\mathrm{Hilb}^2(S)$ for polarized K3 or abelian surfaces $S$ admitting an Ulrich bundle. The authors build Ulrich objects by descending from a surface Ulrich bundle to the symmetric square $\mathrm{Sym}^2(S)$ and lifting through the crepant Hilbert–Chow resolution $\rho: \mathrm{Hilb}^2(S)\to \mathrm{Sym}^2(S)$, carefully comparing cohomology under polarisation adjustments. They introduce and bound Ulrich complexity, proving $\mathrm{ucomp}(Hilb^2(S)) \le 2r^2$ with $r=\mathrm{ucomp}(S)$ and showing no Ulrich line bundles exist, hence $2\le \mathrm{ucomp}(Hilb^2(S)) \le 8$ for general polarizations. The results provide the first explicit Ulrich constructions on irreducible holomorphic symplectic fourfolds and suggest the method may extend to higher Hilbert schemes and other hyperkähler varieties obtained from crepant resolutions.
Abstract
We prove the existence of Ulrich sheaves on the Hilbert scheme of two points on a polarized K3 surface or an abelian surface. The construction proceeds by descending Ulrich bundles on the surface to the symmetric square and lifting them to the Hilbert square via the crepant Hilbert Chow resolution. Finally, we estimate a bound for Ulrich complexity of the Hilbert square.