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 “stabilizing” unstable bundles
and desingularising moduli of stable bundles; an analysis of the behavior 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 that this topology is compact in
many cases.
