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Geometric Invariant Theory and Decorated Principal Bundles
Zurich Lectures in Advanced Mathematics

Alexander H.W. Schmitt (Freie Universität Berlin, Germany)

Geometric Invariant Theory and Decorated Principal Bundles

ISBN print 978-3-03719-065-4, ISBN online 978-3-03719-565-9
DOI 10.4171/065
July 2008, 396 pages, softcover, 17.0 x 24.0 cm.
48.00 Euro

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.

Keywords: Geometric invariant theory, reductive group, representation, group action, categorical, good, geometric quotient, principal bundle, decorated bundle, moduli space

Further Information

Color picture on page 138

Color picture on page 256

Review in Zentralblatt MATH 1159.14001

Review in MR 2437660 (2010c:14038)