Nonabelian Jacobian of projective surfaces: geometry and representation theory / Igor Reider

Auteur: Reider, Igor - AuteurType de document: MonographieCollection: Lecture notes in mathematics ; 2072Langue: anglaisPays: AllemagneÉditeur: Berlin : Springer, cop. 2013Description: 1 vol. (VIII-214 p.) ; 24 cm ISBN: 9783642356612 ; br. Résumé: 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. (Source : Springer).Bibliographie: Bibliogr. p. 213-214. Sujets MSC: 14C05 Algebraic geometry -- Cycles and subschemes -- Parametrization (Chow and Hilbert schemes)
14D07 Algebraic geometry -- Families, fibrations -- Variation of Hodge structures
14C34 Algebraic geometry -- Cycles and subschemes -- Torelli problem
14J10 Algebraic geometry -- Surfaces and higher-dimensional varieties -- Families, moduli, classification: algebraic theory
14J60 Algebraic geometry -- Surfaces and higher-dimensional varieties -- Vector bundles on surfaces and higher-dimensional varieties, and their moduli
32S60 Several complex variables and analytic spaces -- Singularities -- Stratifications; constructible sheaves; intersection cohomology
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Bibliogr. p. 213-214

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. (Source : Springer)

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