# Geometric Invariant Theory and Decorated Principal Bundles

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*Alexander H. W. Schmitt*

A publication of the European Mathematical Society

The book starts with an introduction to Geometric Invariant Theory
(GIT). The fundamental results of Hilbert and Mumford are exposed as
well as more recent topics such as the instability flag, the finiteness
of the number of quotients, and the variation of quotients.

In the second part, GIT 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 generalized Hitchin map.

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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in number theory.