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Nonabelian Jacobian of Projective Surfaces

Geometry and Representation Theory

  • Book
  • © 2013

Overview

Authors:
  1. Igor Reider

    1. Angers, France

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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
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  2. Introduction

    • Igor Reider
    Pages 1-15

  3. Nonabelian Jacobian J(X; L, d): Main Properties

    • Igor Reider
    Pages 17-32

  4. Some Properties of the Filtration \(\mathbf{\tilde{H}}_{-\bullet }\)

    • Igor Reider
    Pages 33-38

  5. The Sheaf of Lie Algebras \(\mathcal{G}_{\Gamma }\)

    • Igor Reider
    Pages 39-73

  6. Period Maps and Torelli Problems

    • Igor Reider
    Pages 75-98

  7. sl2-Structures on \({\mathcal{F}}^{{\prime}}\)

    • Igor Reider
    Pages 99-111

  8. sl2-Structures on \({\mathcal{G}}_{\Gamma }\)

    • Igor Reider
    Pages 113-122

  9. Involution on \({\mathcal{G}}_{\Gamma }\)

    • Igor Reider
    Pages 123-132

  10. Stratification of T π

    • Igor Reider
    Pages 133-144

  11. Configurations and Theirs Equations

    • Igor Reider
    Pages 145-173

  12. Representation Theoretic Constructions

    • Igor Reider
    Pages 175-196

  13. J(X; L, d) and the Langlands Duality

    • Igor Reider
    Pages 197-212

  14. Back Matter

    Pages 213-216
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Keywords

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.

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

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