Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces

TL;DR

This paper demonstrates that the intersection Betti numbers of moduli spaces of one-dimensional sheaves on certain surfaces stabilize and match stable Betti numbers of Hilbert schemes, revealing deep geometric and topological properties.

Contribution

It introduces a unified approach to analyze intersection Betti numbers, proving their stabilization and relation to Hilbert schemes for specific classes of surfaces.

Findings

Betti numbers stabilize in fixed degrees for minimal surfaces of Kodaira dimension 0.

Intersection Betti numbers coincide with stable Betti numbers of Hilbert schemes in certain ranges.

Refined stabilization results are obtained for Enriques surfaces, supporting Oberdieck's conjecture.

Abstract

In this paper, we develop a unified approach to study the intersection Betti numbers of moduli spaces of one-dimensional semistable sheaves on smooth projective surfaces. Assuming the irreducibility of such moduli spaces, we prove that their intersection Betti numbers in a certain range of degrees coincide with the stable Betti numbers of Hilbert schemes of points. As an application, for minimal surfaces of Kodaira dimension 0, we show that these intersection Betti numbers stabilize in each fixed degree, which fits into the broader context of stable cohomology for moduli spaces of sheaves. In the case of Enriques surfaces, we also prove a refined stabilization result related to Oberdieck's conjecture on perverse Hodge numbers.