Blowups and gauge fields

The relationship between stable holomorphic vector bundles on a compact com plex surface and the same such objects on a blowup of the surface is invest igated, where stability is with respect to a Gauduchon metric on the surfac e and naturally derived such metrics on the blowup. The main results are: descriptions of holomorphic vector bundles on a blowu p; conditions relating ( semi)-stability of these to that of their direct i mages on the surface; sheaf-theoretic constructions for stabilizing unstabl e bundles and desingularising moduli of stable bundles; an analysis of the behavior of Hermitian-Einstein connections on bundles over blowups as the u nderlying Gauduchon metric degenerates; the definition of a topology on equ ivalence classes of stable bundles on blowups over a surface and a proof th at this topology is compact in many cases.