Moduli of vector bundles on higher-dimensional base manifolds — Construction and variation

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson–Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker–Maruyama moduli spaces with respect to two different chosen polarizations are related via Thaddeus-flips through other “multi-Gieseker”-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson–Uhlenbeck moduli space.