Nonabelian Jacobian of projective surfaces : geometry and representation theory
- Igor Reider.
- Heidelberg ; New York : Springer, c2013.
- Physical description
- viii, 214 p. : ill. ; 24 cm.
- Lecture notes in mathematics (Springer-Verlag) 2072.
Math & Statistics Library
QA3 .L28 V.2072
- Unknown QA3 .L28 V.2072
- Reider, Igor.
- Includes bibliographical references (p. 213-214).
- 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.
- (source: Nielsen Book Data)
- Publisher's 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.
(source: Nielsen Book Data)
- Publication date
- Lecture notes in mathematics ; 2072