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In this paper we study the relationship between three compactifications ofthe moduli space of Hermitian-Yang-Mills connections on a fixed Hermitianvector bundle over a projective algebraic manifold of arbitrary dimension. Viathe Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic tothe moduli space of stable holomorphic vector bundles, and as such it admits analgebraic compactification by Gieseker-Maruyama semistable torsion-freesheaves. A recent construction due to the first and third authors gives anothercompactification as a moduli space of slope semistable sheaves. In the presentarticle, following fundamental work of Tian generalising the analysis ofUhlenbeck and Donaldson in complex dimension two, we define a gauge theoreticcompactification by adding certain ideal connections at the boundary. Extendingwork of Jun Li in the case of bundles on algebraic surfaces, we exhibitcomparison maps from the sheaf theoretic compactifications and prove theircontinuity. The continuity, together with a delicate analysis of the fibres ofthe map from the moduli space of slope semistable sheaves allows us to endowthe gauge theoretic compactification with the structure of a complex analyticspace.

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