Geometric invariant theory, holomorphic vector bundles and the Harder-Narasimhan filtration / Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas.

Format:

Book

Author/Creator:

Zamora Saiz, Alfonso, author.

Zúñiga-Rojas, Ronald A., author.

Series:

SpringerBriefs in mathematics.

SpringerBriefs in Mathematics

Language:

English

Subjects (All):

Geometry, Algebraic.

Moduli theory.

Invariants.

Physical Description:

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

Edition:

1st ed.

Place of Publication:

Cham, Switzerland : 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.

Contents:

Intro

Preface

Contents

List of Symbols

1 Introduction

2 Preliminaries

2.1 Algebraic Varieties and Groups

2.1.1 Algebraic Varieties

2.1.2 Group Actions

2.2 Sheaf Theory and Schemes

2.2.1 Sheaves and Cohomology

2.2.2 Schemes

2.3 Holomorphic Vector Bundles

2.3.1 Vector Bundles

2.3.2 Line Bundles

2.3.3 Divisors

3 Geometric Invariant Theory

3.1 Quotients and the Notion of Stability

3.2 Hilbert-Mumford Criterion

3.3 Symplectic Stability

3.4 Examples

3.5 Maximal Unstability

4 Moduli Space of Vector Bundles

4.1 GIT Construction of the Moduli Space

4.2 Harder-Narasimhan Filtration

4.3 Other Constructions of the Moduli Space of Vector Bundles

4.3.1 Analytical Construction of the Moduli Space of Vector Bundles

4.3.2 Moduli Space of Representations of the Fundamental Group

4.4 Moduli Space of Higgs Bundles

4.4.1 Hitchin's Construction

4.4.2 Higher Rank and Dimensional Higgs Bundles

5 Unstability Correspondence

5.1 Correspondence for Vector Bundles

5.1.1 Main Correspondence: Holomorphic Vector Bundles

5.1.2 Other Correspondences for Augmented Bundles

5.2 Quiver Representations

5.3 (G,h)-Constellations

6 Stratifications on the Moduli Space of Higgs Bundles

6.1 Shatz Stratification

6.2 C-Action and Białynicki-Birula Stratification

6.3 Stratifications in Rank Three

6.3.1 Sketch of the Proof of Theorem 6.1

6.3.2 Relationship Between Shatz and Biłynicki-Birula Stratifications for Rank Three Higgs Bundles

6.4 Homotopy Groups

References

Index.

Notes:

Includes bibliographical references and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

3-030-67829-6

OCLC:

1243349702