Compact moduli spaces for slope-semistable sheaves

TL;DR

This paper introduces a new approach to construct compact moduli spaces of slope-semistable sheaves on higher-dimensional manifolds by using movable curves, resolving wall-crossing issues and generalizing classical compactifications.

Contribution

It develops a modular compactification of vector bundle moduli spaces using slope stability with respect to specific curves, extending the Donaldson-Uhlenbeck construction to higher dimensions.

Findings

Resolved wall-crossing phenomena for sheaf moduli spaces.

Constructed a modular compactification for slope-stable bundles.

Linked new moduli spaces to existing sheaf and Gieseker-Maruyama moduli spaces.

Abstract

We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a projective n-fold and a curve C that arises as the complete intersection of n-1 very ample divisors, we construct a modular compactification of the moduli space of vector bundles that are slope-stable with respect to C. Our construction generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification by Joseph Le Potier and Jun Li. Furthermore, we describe the geometry of the newly construced moduli spaces by relating them to moduli spaces of simple sheaves and to Gieseker-Maruyama moduli spaces.