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Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration

  • Book
  • © 2021

Overview

Authors:
  1. Alfonso Zamora Saiz

    1. Applied Mathematics, Technical University of Madrid, Madrid, Spain

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    You can also search for this author in PubMed   Google Scholar

  2. Ronald A. Zúñiga-Rojas

    1. School of Mathematics, University of Costa Rica, San José, Costa Rica

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    You can also search for this author in PubMed   Google Scholar

  • Introduces key topics on Geometric Invariant Theory through examples and applications
  • Covers Hilbert classification of binary forms and Hitchin's theory on Higgs bundles
  • Takes particular note of unstable objects in module problems

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction

    • Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas
    Pages 1-5

  3. Preliminaries

    • Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas
    Pages 7-30

  4. Geometric Invariant Theory

    • Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas
    Pages 31-58

  5. Moduli Space of Vector Bundles

    • Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas
    Pages 59-80

  6. Unstability Correspondence

    • Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas
    Pages 81-99

  7. Stratifications on the Moduli Space of Higgs Bundles

    • Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas
    Pages 101-119

  8. Back Matter

    Pages 121-127
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Keywords

About this book

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.



Reviews

“The book is well written and introduces various explicit examples. It is very suitable for those who want to have a basic idea and global scope of this theory and apply the idea to study other classification problems in algebraic geometry.” (‪Pengfei Huang, Mathematical Reviews, September, 2022)

Authors and Affiliations

  • Applied Mathematics, Technical University of Madrid, Madrid, Spain

    Alfonso Zamora Saiz

  • School of Mathematics, University of Costa Rica, San José, Costa Rica

    Ronald A. Zúñiga-Rojas

About the authors

Alfonso Zamora Saiz is a Professor at the School of Computer Science Engineering, Technical University of Madrid, Spain. Holding a PhD in algebraic geometry from the Complutense University of Madrid (2013), he has been a visiting PhD student at Cambridge University and Columbia University, a postdoc at the IST in Lisbon, Lecturer at the California State University Channel Islands and a Professor at the CEU San Pablo University in Madrid. His research interests include algebra, geometry and topology in pure mathematics, as well as data analytical applications and mathematics education.

Ronald A. Zúñiga-Rojas is a Professor at the School of Mathematics, University of Costa Rica (UCR), and is currently a member of both Center of Mathematical and Meta-Mathematical Research (CIMM-UCR) and the Center of Pure and Applied Mathematics Research (CIMPA-UCR). He completed the Doctor’s Degree in Mathematics at University of Porto, Portugal, in 2015, in a PhD Programin association with the University of Coimbra in Portugal. His research interests lay on pure mathematics, focused on algebraic geometry, algebraic topology, and differential geometry.


Bibliographic Information

  • Book Title: Geometric Invariant Theory, Holomorphic Vector Bundles and the Harder-Narasimhan Filtration

  • Authors: Alfonso Zamora Saiz, Ronald A. Zúñiga-Rojas

  • Series Title: SpringerBriefs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-67829-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Softcover ISBN: 978-3-030-67828-9Published: 25 March 2021

  • eBook ISBN: 978-3-030-67829-6Published: 24 March 2021

  • Series ISSN: 2191-8198

  • Series E-ISSN: 2191-8201

  • Edition Number: 1

  • Number of Pages: XIII, 127

  • Number of Illustrations: 4 b/w illustrations, 12 illustrations in colour

  • Topics: Algebraic Geometry, Mathematical Methods in Physics

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