Let $H$ be a semisimple algebraic group. We prove the semistable reduction theorem for $\mu$-semistable principal $H$-bundles over a smooth projective variety $X$ defined over the field C. When $X$ is a 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.

In this article our main goal is to establish an $L^2$ version of the wisted invariance of plurigenera. We equally simplify the original "two towers" approach of Yum-Tong Siu.