Principal bundles on projective varieties and the Donaldson-Uhlenbeck compactification

Let $H$ be a semisimple algebraic group. We prove the semistable reduction theorem for $\mu$–semistable principal $H$–bundles over a {\it smooth projective variety $X$} defined over the field $\text{bc}$. When $X$ is a {\it smooth projective surface} and $H$ is simple, we construct the algebro–geometric Donaldson–Uhlenbeck compactification of the moduli space of $\mu$–semistable principal $H$–bundles with fixed characteristic classes and describe its points. For large characteristic classes we show that the moduli space of $\mu$–stable principal $H$–bundles is non–empty.