Buch, Englisch, 389 Seiten, PB, Format (B × H): 170 mm x 240 mm
Buch, Englisch, 389 Seiten, PB, Format (B × H): 170 mm x 240 mm
Reihe: Zurich Lectures in Advanced Mathematics
ISBN: 978-3-03719-065-4
Verlag: EMS Press
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.
Zielgruppe
The text addresses students and researchers with a working knowledge of elementary algebraic geometry.
