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On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety. (AM-157) : Annals of Mathematics Studies - Mark Green

On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety. (AM-157)

Annals of Mathematics Studies

Paperback Published: 9th January 2005
ISBN: 9780691120447
Number Of Pages: 208

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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 Angeniol 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.

Introductionp. 3
The classical case when n = 1p. 22
Differential geometry of symmetric productsp. 31
Absolute differentials (I)p. 42
Geometric description of [actual symbol not reproducible]Z[superscript n](X)p. 54
Absolute differentials (II)p. 61
The Ext-definition of TZ[superscript 2](X) for X an algebraic surfacep. 84
Tangents to related spacesp. 100
Applications and examplesp. 150
Speculations and questionsp. 186
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691120447
ISBN-10: 0691120447
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 208
Published: 9th January 2005
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.91
Weight (kg): 0.03

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