Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration / by Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas.

Format:

Book

Author/Creator:

Zamora Saiz, Alfonso, author.

Zúñiga-Rojas, Ronald A., author.

Series:

SpringerBriefs in Mathematics, 2191-8201

Language:

English

Subjects (All):

Geometry, Algebraic.

Mathematical physics.

Algebraic Geometry.

Mathematical Methods in Physics.

Local Subjects:

Algebraic Geometry.

Mathematical Methods in Physics.

Physical Description:

1 online resource (xiii, 127 pages) : illustrations.

Edition:

1st ed. 2021.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2021.

Summary:

This book introduces key topics on Geometric Invariant Theory, a technique to obtaining quotients in algebraic geometry with a good set of properties, through various examples. It starts from the classical Hilbert classification of binary forms, advancing to the construction of the moduli space of semistable holomorphic vector bundles, and to Hitchin’s theory on Higgs bundles. The relationship between the notion of stability between algebraic, differential and symplectic geometry settings is also covered. Unstable objects in moduli problems -- a result of the construction of moduli spaces -- get specific attention in this work. The notion of the Harder-Narasimhan filtration as a tool to handle them, and its relationship with GIT quotients, provide instigating new calculations in several problems. Applications include a survey of research results on correspondences between Harder-Narasimhan filtrations with the GIT picture and stratifications of the moduli space of Higgs bundles. Graduate students and researchers who want to approach Geometric Invariant Theory in moduli constructions can greatly benefit from this reading, whose key prerequisites are general courses on algebraic geometry and differential geometry.

Contents:

Preface

Introduction

Preliminaries

Geometric Invariant Theory

Moduli Space of Vector Bundles

Unstability Correspondence

Stratifications on the Moduli Space of Higgs Bundles

References

Index.

Notes:

Includes bibliographical references and index.

Print version record.

ISBN:

3-030-67829-6

OCLC:

1243349702