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Mathematics - Algebra | Nonabelian Jacobian of Projective Surfaces - Geometry and Representation Theory

Nonabelian Jacobian of Projective Surfaces

Geometry and Representation Theory

Series: Lecture Notes in Mathematics, Vol. 2072

Reider, Igor

2013, VIII, 227 p.

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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.

Content Level » Research

Keywords » 14J60,14C05,16G30 - Lie algebra - surfaces - vector bundles - zero-cycles

Related subjects » Algebra

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