  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.

Differential Geometry;Algebraic Geometry;Complex Variables