Nonabelian Jacobian of Projective Surfaces : Geometry and Representation Theory / by Igor Reider.

Format:

Book

Author/Creator:

Reider, Igor, author.

Contributor:

SpringerLink (Online service)

Series:

Lecture Notes in Mathematics, 0075-8434 ; 2072.

Lecture Notes in Mathematics, 0075-8434 ; 2072

Language:

English

Subjects (All):

Geometry, Algebraic.

Matrices.

Algebraic Geometry.

Linear and Multilinear Algebras, Matrix Theory.

Local Subjects:

Algebraic Geometry.

Linear and Multilinear Algebras, Matrix Theory.

Physical Description:

1 online resource (VIII, 227 pages).

Contained In:

Springer eBooks

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2013.

System Details:

text file PDF

Summary:

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.

Contents:

1 Introduction

2 Nonabelian Jacobian J(X; L; d): main properties

3 Some properties of the filtration H

4 The sheaf of Lie algebras G

5 Period maps and Torelli problems

6 sl2-structures on F

7 sl2-structures on G

8 Involution on G

9 Stratification of T

10 Configurations and theirs equations

11 Representation theoretic constructions

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

Other Format:

Printed edition:

ISBN:

9783642356629

Access Restriction:

Restricted for use by site license.