The first chapter is an introduction to geometric invariant theory, as developed by D. Mumford. Fundamental results by Hilbert and Mumford are explained here, together with more recent topics, such as the instability flag, the finiteness of the number of quotients and the variation of quotients. In the second chapter, geometric invariant theory is applied to solve the classification problem of decorated principal bundles on a compact Riemann surface. The solution is a quasi-projective moduli scheme, which parameterizes those objects that satisfy a semistability condition originating from gauge theory. The moduli space is equipped with a generalised Hitchin map. Via the universal Hitchin-Kobayashi correspondence, these moduli spaces are related to moduli spaces of solutions of certain vortex type equations. Possible applications include a study of representation spaces of the fundamental group of compact Riemann surfaces. The book concludes with a brief discussion of generalisations of these results to higher dimensional base varieties, positive characteristics and parabolic bundles.

Reviewer:

lkř