Nonabelian Jacobian of Projective Surfaces
Subtitle: Geometry and Representation Theory
By: by Igor Reider
Contributors: SpringerLink (Online service)
Series: Lecture Notes in Mathematics -- 2072
Publisher: Springer Berlin Heidelberg; Imprint: Springer
Published: 2013
Place of publication: Berlin, Heidelberg
Language: eng
Format: [electronic resource] :
Number of pages: 227
ISBN-13: 9783642356629
Subjects: Matrix Theory Linear and Multilinear Algebras; Mathematics; Algebraic Geometry; Matrix theory
Description: 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.
Table of contents: title: 1 Introduction; title: 2 Nonabelian Jacobian J(X; L; d): main properties; title: 3 Some properties of the filtration H; title: 4 The sheaf of Lie algebras G; title: 5 Period maps and Torelli problems; title: 6 sl2-structures on F; title: 7 sl2-structures on G; title: 8 Involution on G; title: 9 Stratification of T; title: 10 Configurations and theirs equations; title: 11 Representation theoretic constructions; title: 12 J(X; L; d) and the Langlands Duality.