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Nonabelian Jacobian of Projective Surfaces

Geometry and Representation Theory

  • Igor Reider

Part of the Lecture Notes in Mathematics book series (LNM, volume 2072)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Igor Reider
    Pages 1-15
  3. Igor Reider
    Pages 75-98
  4. Igor Reider
    Pages 123-132
  5. Igor Reider
    Pages 133-144
  6. Igor Reider
    Pages 145-173
  7. Igor Reider
    Pages 175-196
  8. Igor Reider
    Pages 197-212
  9. Back Matter
    Pages 213-216

About this book

Introduction

The Jacobian of a smooth projective curve is undoubtedly one of the most remarkable and beautiful objects in algebraic geometry. This work is an attempt to develop an analogous theory for smooth projective surfaces - a theory of the nonabelian Jacobian of smooth projective surfaces.
Just like its classical counterpart, our nonabelian Jacobian relates to vector bundles (of rank 2) on a surface as well as its Hilbert scheme of points. But it also comes equipped with the variation of Hodge-like structures, which produces a sheaf of reductive Lie algebras naturally attached to our Jacobian. This constitutes a nonabelian analogue of the (abelian) Lie algebra structure of the classical Jacobian. This feature naturally relates geometry of surfaces with the representation theory of reductive Lie algebras/groups.
This work’s main focus is on providing an in-depth study of various aspects of this relation. It presents a substantial body of evidence that the sheaf of Lie algebras on the nonabelian Jacobian is an efficient tool for using the representation theory to systematically address various algebro-geometric problems. It also shows how to construct new invariants of representation theoretic origin on smooth projective surfaces.

Keywords

14J60,14C05,16G30 Lie algebra surfaces vector bundles zero-cycles

Authors and affiliations

  • Igor Reider
    • 1
  1. 1.AngersFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-35662-9
  • Copyright Information Springer-Verlag Berlin Heidelberg 2013
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-35661-2
  • Online ISBN 978-3-642-35662-9
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site