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Nonabelian Jacobian of Projective Surfaces

Geometry and Representation Theory

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

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-viii
  2. Introduction

    • Igor Reider
    Pages 1-15
  3. Period Maps and Torelli Problems

    • Igor Reider
    Pages 75-98
  4. Involution on \({\mathcal{G}}_{\Gamma }\)

    • Igor Reider
    Pages 123-132
  5. Stratification of T π

    • Igor Reider
    Pages 133-144
  6. Configurations and Theirs Equations

    • Igor Reider
    Pages 145-173
  7. Representation Theoretic Constructions

    • Igor Reider
    Pages 175-196
  8. J(X; L, d) and the Langlands Duality

    • Igor Reider
    Pages 197-212
  9. Back Matter

    Pages 213-216

About this book

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
  • matrix theory

Reviews

From the reviews:

“The book is well written, listing the main ideas in sections, and giving the successive results as they appear. The idea of a Jacobian on surfaces is new and important, and this book is the initiation of the study of this interesting object.” (Arvid Siqveland, Mathematical Reviews, November, 2013)

Authors and Affiliations

  • Angers, France

    Igor Reider

Bibliographic Information

  • Book Title: Nonabelian Jacobian of Projective Surfaces

  • Book Subtitle: Geometry and Representation Theory

  • Authors: Igor Reider

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-35662-9

  • Publisher: Springer Berlin, Heidelberg

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

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2013

  • Softcover ISBN: 978-3-642-35661-2Published: 15 March 2013

  • eBook ISBN: 978-3-642-35662-9Published: 02 March 2013

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: VIII, 227

  • Topics: Algebraic Geometry, Linear Algebra

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions