Compact moduli spaces for slope-semistable sheaves

We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional complex projective manifolds. This is achieved by considering slopesemistability 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 Le Potier and Li. Furthermore, we describe the geometry of the newly constructed moduli spaces by relating them to moduli spaces of simple sheaves and to Gieseker-Maruyama moduli spaces.