Nonabelian Jacobian of projective surfaces : geometry and representation theory
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Bibliographic Information
Nonabelian Jacobian of projective surfaces : geometry and representation theory
（Lecture notes in mathematics, 2072）
Springer, c2013
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
L/N||LNM||2072200026166507
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Note
Includes bibliographical references (p. 213-214)
Description and Table of Contents
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
- 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.
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