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

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. Contents Chapter 1. Introduction 1.1 General comments Chapter 2. The Classical Case When n = 1 Chapter 3. Differential Geometry of Symmetric Products Chapter 4. Absolute Differentials (I) 4.1 Generalities 4.2 Spreads Chapter 5. Geometric Description of T̳Zn(X) 5.1 The description 5.2 Intrinsic formulation Chapter 6. Absolute Differentials (II) 6.1 Absolute differentials arise from purely geometric considerations 6.2 A nonclassical case when n = 1 6.3 The differential of the tame symbol Chapter 7. The Ext-definition of TZ2(X) for X an Algebraic Surface 7.1 The definition of T̳Z2(X) 7.2 The map T Hilb2(X) → TZ2(X) 7.3 Relation of the Puiseaux and algebraic approaches 7.4 Further remarks Chapter 8. Tangents to Related Spaces 8.1 The definition of T̳Z1(X) for a surface X 8.2 Duality and the description of T̳Z1(X) using differential forms 8.3 Definitions of T̳Z 1,1(X) for X a curve and a surface 8.4 Identification of the geometric and formal tangent spaces to CH2(X) for X a surface 8.5 Canonical filtration on TCHn(X) and its relation to the conjectural filtration on CHn(X) Chapter 9. Applications and Examples 9.1 The generalization of Abel’s differential equations 9.2 On the integration of Abel’s differential equations 9.3 Surfaces in P3 9.4 Example: (P2, T ) Chapter 10. Speculations and Questions 10.1 Definitional issues 10.2 Obstructedness issues 10.3 Null curves 10.4 Arithmetic and geometric estimates Bibliography Index