Nonabelian Jacobian of Projective Surfaces

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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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
• matrix theory

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)

• Angers, France

  Igor Reider

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