Geometric invariant theory and decorated principal bundles / Alexander H.W. Schmitt.

Format:

Book

Author/Creator:

Schmitt, Alexander H. W.

Contributor:

Rosengarten Family Fund.

Series:

Zurich lectures in advanced mathematics

Language:

English

Subjects (All):

Geometry, Algebraic.

Invariants.

Moduli theory.

Physical Description:

vii, 389 pages : illustrations ; 25 cm.

Place of Publication:

Zürich : European Mathematical Society, ©2008.

Summary:

"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 and features numerous examples and exercises. It addresses students and researchers with a working knowledge of elementary algebraic geometry."--Jacket.

Contents:

Algebraic groups and their representations

Geometric invariant theory for affine varieties : a first encounter

Examples from classical invariant theory

Mumford's geometric invariant theory

Criteria for stability and semistability

The variation of GIT-quotients

The analysis of unstable points

Decorated principal bundles

Rudiments of the theory of vector bundles

Decorated vector bundles : projective fibers

Principal bundles as swamps

Decorated tuples of vector bundles : projective fibers

Principal bundles as tumps

Decorated principal bundles : projective fibers

Decorated principal bundles : affine fibers.

Notes:

Includes bibliographical references (pages 363-377) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

9783037190654

3037190655

OCLC:

261176013

Publisher Number:

99996451024