Moduli Spaces and Vector Bundles

doi:10.1017/CBO9781139107037

Vector bundles and their associated moduli spaces are of fundamental importance in algebraic geometry. In recent decades this subject has been greatly enhanced by its relationships with other areas of mathematics, including differential geometry, topology and even theoretical physics, specifically gauge theory, quantum field theory and string theory. Peter E. Newstead has been a leading figure in this field almost from its inception and has made many seminal contributions to our understanding of moduli spaces of stable bundles. This volume has been assembled in tribute to Professor Newstead and his contribution to algebraic geometry. Some of the subject's leading experts cover foundational material, while the survey and research papers focus on topics at the forefront of the field. This volume is suitable for both graduate students and more experienced researchers.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Preface
pp vii-viii
- By L. Brambila-Paz, S. B. Bradlow, O. García-Prada, S. Ramanan
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Acknowledgments
pp ix-x
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Part I - Lecture Notes
pp 1-1
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1 - Lectures on Principal Bundles
pp 2-28
- By V. Balaji, SIPCOT Information Technology Park
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2 - Brill-Noether Theory for Stable Vector Bundles
pp 29-50
- By Ivona Grzegorczyk, University Channel Islands, Montserrat Teixidor i Bigas, Tufts University
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3 - Introduction to Fourier-Mukai and Nahm Transforms with an Application to Coherent Systems on Elliptic Curves
pp 51-98
- By U. Bruzzo, Istituto Nazionale di Fisica Nucleare, D. Hernández Ruipérez, Universidad de Salamanca, C. Tejero Prieto, Universidad de Salamanca
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4 - Geometric Invariant Theory
pp 99-127
- By P. E. Newstead, University of Liverpool
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5 - Deformation Theory for Vector Bundles
pp 128-164
- By N. Nitsure, Tata Institute of Fundamental Research
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6 - The Theory of Vector Bundles on Algebraic Curves with Some Applications
pp 165-210
- By S. Ramanan, SIPCOT Information Technology Park
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Part II - Survey Articles
pp 211-211
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7 - Moduli of Sheaves from Moduli of Kronecker Modules
pp 212-228
- By L. Álvarez-Cónsul, Instituto de Ciencias Matemáticas, A. King, University of Bath
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8 - Coherent Systems: a Brief Survey
pp 229-264
- By S. B. Bradlow, University of Illinois
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9 - Higgs Bundles Surface and Group Representations
pp 265-310
- By Oscar García-Prada, Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM
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10 - Quotients by Non-Reductive Algebraic Group Actions
pp 311-366
- By Frances Kirwan, Mathematical Institute
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11 - Dualities on T*SUX (2, OX)
pp 367-387
- By E. Previato, Boston University
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12 - Moduli Spaces for Principal Bundles
pp 388-424
- By A.H.W. Schmitt, Freie Universität Berlin, Institut für Mathematik
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Part III - Research Articles
pp 425-425
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13 - Beilinson Type Spectral Sequences on Scrolls
pp 426-436
- By M. Aprodu, Institute of Mathematics “Simion Stoilow”, V. Brînzănescu, University of Piteşti
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14 - Coherent Systems on a Nodal Curve
pp 437-455
- By Usha N. Bhosle, School of Mathematics, Tata Institute of Fundamental Research
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15 - Brill-Noether Bundles and Coherent Systems on Special Curves
pp 456-472
- By L. Brambila-Paz, Angela Ortega, Universidad Michoacana
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16 - Higgs Bundles in the Vector Representation
pp 473-483
- By N. Hitchin, Mathematical Institute
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17 - Moduli Spaces of Torsion Free Sheaves on Nodal Curves and Generalisations – I
pp 484-505
- By C.S. Seshadri, Chennai Mathematical Institute
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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Acknowledgments
pp ix-x

Part I - Lecture Notes
pp 1-1

1 - Lectures on Principal Bundles
pp 2-28

2 - Brill-Noether Theory for Stable Vector Bundles
pp 29-50

3 - Introduction to Fourier-Mukai and Nahm Transforms with an Application to Coherent Systems on Elliptic Curves
pp 51-98

4 - Geometric Invariant Theory
pp 99-127

5 - Deformation Theory for Vector Bundles
pp 128-164

6 - The Theory of Vector Bundles on Algebraic Curves with Some Applications
pp 165-210

Part II - Survey Articles
pp 211-211

7 - Moduli of Sheaves from Moduli of Kronecker Modules
pp 212-228

8 - Coherent Systems: a Brief Survey
pp 229-264

9 - Higgs Bundles Surface and Group Representations
pp 265-310

10 - Quotients by Non-Reductive Algebraic Group Actions
pp 311-366

11 - Dualities on T*SUX (2, OX)
pp 367-387

12 - Moduli Spaces for Principal Bundles
pp 388-424

Part III - Research Articles
pp 425-425

13 - Beilinson Type Spectral Sequences on Scrolls
pp 426-436

14 - Coherent Systems on a Nodal Curve
pp 437-455

15 - Brill-Noether Bundles and Coherent Systems on Special Curves
pp 456-472

16 - Higgs Bundles in the Vector Representation
pp 473-483

17 - Moduli Spaces of Torsion Free Sheaves on Nodal Curves and Generalisations – I
pp 484-505

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