Complex algebraic compactifications of the moduli space of Hermitian-Yang-Mills connections on a projective manifold

TL;DR

This paper explores three different ways to compactify the moduli space of Hermitian-Yang-Mills connections on a projective manifold, establishing relationships and continuity among these compactifications.

Contribution

It introduces a gauge theoretic compactification for higher dimensions and compares it with existing sheaf-theoretic compactifications, proving their continuity and complex structure.

Findings

Established a gauge theoretic compactification in arbitrary dimensions.

Proved continuity of comparison maps between different compactifications.

Endowed the gauge theoretic compactification with a complex analytic structure.

Abstract

In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves. In the present article, following fundamental work of Tian generalising the analysis of Uhlenbeck and Donaldson in complex dimension two, we define a gauge theoretic compactification by adding certain ideal connections at the boundary. Extending work of Jun Li in the case of bundles on algebraic surfaces, we exhibit comparison maps from the sheaf theoretic compactifications and prove their continuity. The continuity, together with a delicate analysis of the fibres of the map from the moduli space of slope semistable sheaves allows us to endow the gauge theoretic compactification with the structure of a complex analytic space.