Lecture Notes in Mathematics

Nonabelian Jacobian of Projective Surfaces

Geometry and Representation Theory

Authors: Reider, Igor

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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)

Table of contents (12 chapters)

  • Introduction

    Reider, Igor

    Pages 1-15

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

    Reider, Igor

    Pages 17-32

  • Some Properties of the Filtration $$\mathbf{\tilde{H}}_{-\bullet }$$

    Reider, Igor

    Pages 33-38

  • The Sheaf of Lie Algebras $$\mathcal{G}_{\Gamma }$$

    Reider, Igor

    Pages 39-73

  • Period Maps and Torelli Problems

    Reider, Igor

    Pages 75-98

Buy this book

eBook $44.99
price for USA (gross)
  • ISBN 978-3-642-35662-9
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.95
price for USA
  • ISBN 978-3-642-35661-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the eBook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Nonabelian Jacobian of Projective Surfaces
Book Subtitle
Geometry and Representation Theory
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2072
Copyright
2013
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-35662-9
DOI
10.1007/978-3-642-35662-9
Softcover ISBN
978-3-642-35661-2
Series ISSN
0075-8434
Edition Number
1
Number of Pages
VIII, 227
Topics