  The relationship between stable holomorphic vector bundles on a compact
complex surface and the same such objects on a blowup of the surface is
investigated, where "stability" is with respect to a Gauduchon metric on the
surface and naturally derived such metrics on the blowup.
  The main results are: descriptions of holomorphic vector bundles on a blowup;
conditions relating (semi)-stability of these to that of their direct images on
the surface; sheaf-theoretic constructions for "stabilising" unstable bundles
and desingularising moduli of stable bundles; an analysis of the behaviour of
Hermitian-Einstein connections on bundles over blowups as the underlying
Gauduchon metric degenerates; the definition of a topology on equivalence
classes of stable bundles on blowups over a surface and a proof (using results
of a concurrent preprint) that this topology is compact in many cases---i.e., a
new compactification for moduli of stable bundles on the original surface.
