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

TL;DR

This paper reviews recent advances in the theory of vector bundle moduli on higher-dimensional manifolds, introducing new constructions and relations between different moduli spaces, including a natural morphism to Donaldson-Uhlenbeck spaces.

Contribution

It presents a new natural morphism from multi-Gieseker moduli spaces to Donaldson-Uhlenbeck moduli spaces and discusses relations between Gieseker-Maruyama moduli spaces via Thaddeus-flips.

Findings

Construction of an algebro-geometric analogue of Donaldson-Uhlenbeck compactification.

Establishment of relations between Gieseker-Maruyama moduli spaces with different polarisations.

Proof of the existence of a natural morphism from multi-Gieseker to Donaldson-Uhlenbeck moduli spaces.

Abstract

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 polarisations 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.