On the tangent space to the space of algebraic cycles on a smooth algebraic variety

Mark C. Green, Phillip Griffiths - Collection Annals of Mathematic Studies

200 pages, parution le 21/02/2005

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Résumé

In recent years, considerable progress has been made in studying algebraic cycles using infinitesimal methods. These methods have usually been applied to Hodge-theoretic constructions such as the cycle class and the Abel-Jacobi map. Substantial advances have also occurred in the infinitesimal theory for subvarieties of a given smooth variety, centered around the normal bundle and the obstructions coming from the normal bundle's first cohomology group. Here, Mark Green and Phillip Griffiths set forth the initial stages of an infinitesimal theory for algebraic cycles.

The book aims in part to understand the geometric basis and the limitations of Spencer Bloch's beautiful formula for the tangent space to Chow groups. Bloch's formula is motivated by algebraic K-theory and involves differentials over Q. The theory developed here is characterized by the appearance of arithmetic considerations even in the local infinitesimal theory of algebraic cycles. The map from the tangent space to the Hilbert scheme to the tangent space to algebraic cycles passes through a variant of an interesting construction in commutative algebra due to Angéniol and Lejeune-Jalabert. The link between the theory given here and Bloch's formula arises from an interpretation of the Cousin flasque resolution of differentials over Q as the tangent sequence to the Gersten resolution in algebraic K-theory. The case of 0-cycles on a surface is used for illustrative purposes to avoid undue technical complications.

L'auteur - Mark C. Green

Mark Green is Professor of Mathematics and Director of the Institute for Pure and Applied Mathematics at the University of California, Los Angeles.

L'auteur - Phillip Griffiths

Phillip Griffiths is Professor in the School of Mathematics at the Institute of Advanced Study.

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Sommaire

Introduction
The classical case when n = 1
Differential geometry of symmetric products
Absolute differentials (I)
Geometric description of T Zⁿ(X)
Absolute differentials (II)
The Ext-definition of T Z² (X) for X an algebraic surface
Tangents to related spaces
Applications and examples
Speculations and questions
Bibliography
Index

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Caractéristiques techniques

	PAPIER
Éditeur(s)	Princeton University Press
Auteur(s)	Mark C. Green, Phillip Griffiths
Collection	Annals of Mathematic Studies
Parution	21/02/2005
Nb. de pages	200
Format	15,5 x 23,5
Couverture	Broché
Poids	303g
Intérieur	Noir et Blanc
EAN13	9780691120447
ISBN13	978-0-691-12044-7