© 2013

Nonabelian Jacobian of Projective Surfaces

Geometry and Representation Theory


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


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.


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

Authors and affiliations

  1. 1.AngersFrance

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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)