Geometric Invariant Theory and Decorated Principal Bundles

The book starts with an introduction to Geometric Invariant Theory (GIT). The fundamental results of Hilbert and Mumford are exposed as well as some more recent topics such as the work of Kempf and others on the instability flag, the finiteness of the number of different GIT quotients by Bialynicki--Birula and Dolgachev/Hu, and the variation of GIT quotients by Dolgachev/Hu and Thaddeus. In the second part, GIT is applied to solve some classification problem of holomorphic principal bundles. The algebro-geometric version of the notion of semistability coming from gauge theory is introduced and the moduli spaces for the semistable objects are constructed as quasi-projective varieties which are equipped with a projective Hitchin map to an affine variety. Via the universal Kobayashi--Hitchin correspondence, these moduli spaces are related to moduli spaces of solutions of certain vortex type equations. Potential applications include the study of representation spaces of the fundamental group of compact Riemann surfaces. The book concludes with a brief discussion of generalizations of these findings to higher dimensional base varieties, positive characteristic, and parabolic bundles. The text is fairly self-contained (e.g, the necessary background from the theory of principal bundles is included) and features numerous examples and exercises. It addresses students and researchers with a working knowledge of elementary algebraic geometry.