Igor Reider

Nonabelian Jacobian of Projective Surfaces

Geometry and Representation Theory

• Broschiertes Buch

Produktbeschreibung

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.

Produktdetails

• Lecture Notes in Mathematics 2072
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-642-35661-2
• 2013
• Seitenzahl: 224
• Erscheinungstermin: 15. März 2013
• Englisch
• Abmessung: 235mm x 155mm x 13mm
• Gewicht: 354g
• ISBN-13: 9783642356612
• ISBN-10: 3642356613
• Artikelnr.: 36768622

Inhaltsangabe

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.

Rezensionen

From the reviews:
"The book is well written, listing the main ideas in sections, and giving the successive results as they appear. The idea of a Jacobian on surfaces is new and important, and this book is the initiation of the study of this interesting object." (Arvid Siqveland, Mathematical Reviews, November, 2013)