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On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety PDF

211 Pages·2005·1.5 MB·English
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PUTangSp March 1, 2004 On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety PUTangSp March 1, 2004 PUTangSp March 1, 2004 On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety Mark Green and Phillip Griffiths PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD PUTangSp March 1, 2004 PUTangSp March 1, 2004 Contents Chapter 1. Introduction 3 1.1 General comments 3 Chapter 2. The Classical Case when n=1 23 Chapter 3. Differential Geometry of Symmetric Products 33 Chapter 4. Absolute differentials (I) 45 4.1 Generalities 45 4.2 Spreads 52 Chapter 5. Geometric Description of TZn(X) 57 = 5.1 The description 57 5.2 Intrinsic formulation 60 Chapter 6. Absolute Differentials (II) 65 6.1 Absolute differentials arise from purely geometric considerations 65 6.2 A non-classical case when n=1 71 6.3 The differential of the tame symbol 74 Chapter 7. The Ext-definition of TZ2(X) for X an algebraic surface 89 7.1 The definition of TZ2(X) 89 = 7.2 The map THilb2(X)→TZ2(X) 93 7.3 Relation of the Puiseaux and algebraic approaches 97 7.4 Further remarks 102 Chapter 8. Tangents to related spaces 105 8.1 The definition of TZ1(X) 105 = 8.2 AppendixB:Dualityandthedescriptionof TZ1(X)usingdiffer- = ential forms 120 8.3 Definitions of TZ1(X) for X a curve and a surface 128 = 1 8.4 Identification of the geometric and formal tangent spaces to CH2(X) for X a surface 146 8.5 Canonical filtration on TCHn(X) and its relation to the conjec- tural filtration on CHn(X) 150 PUTangSp March 1, 2004 vi CONTENTS Chapter 9. Applications and examples 155 9.1 The generalization of Abel’s differential equations (cf. [36]) 155 9.2 On the integration of Abel’s differential equations 167 9.3 Surfaces in P3 172 9.4 Example: (P2,T) 182 Chapter 10. Speculations and questions 193 10.1 Definitional issues 193 10.2 Obstructedness issues 194 10.3 Null curves 197 10.4 Arithmetic and geometric estimates 198 Bibliography 203 PUTangSp March 1, 2004 CONTENTS 1 ABSTRACT InthisworkweshallproposedefinitionsforthetangentspacesTZn(X)and TZ1(X) to the groups Zn(X) and Z1(X) of 0-cycles and divisors, respec- tively,onasmoothn-dimensionalalgebraicvariety. Althoughthedefinitions arealgebraicandformal,themotivationbehindthemisquitegeometricand much of the text is devoted to this point. It is noteworthy that both the regulardifferentialformsofalldegreesandthefieldofdefinitionentersignif- icantly into the definition. An interesting and subtle algebraic point centers around the construction of the map THilbp(X)→TZp(X). Another inter- esting algebraic/geometric point is the necessary appearance of spreads and absolute differentials in higher codimension. For an algebraic surface X we shall also define the subspace TZ2 (X)⊂ rat TZ2(X) of tangents to rational equivalences, and we shall show that there is a natural isomorphism T CH2(X)∼=TZ2(X)/TZ2 (X) f rat where the left hand side is the formal tangent space to the Chow groups defined by Bloch. This result gives a geometric existence theorem, albeit at the infinitesimal level. The “integration” of the infinitesimal results raises veryinterestinggeometricandarithmeticissuesthatarediscussedatvarious places in the text. PUTangSp March 1, 2004 PUTangSp March 1, 2004 Chapter One Introduction 1.1 GENERAL COMMENTS In this work we shall define the tangent spaces TZn(X) and TZ1(X) tothespacesof0-cyclesandofdivisorsonasmooth,n-dimensionalcomplex algebraic variety X. We think it may be possible to use similar methods to defineTZp(X)forallcodimensions,butwewerenotabletodothisbecause of one significant technical point. Although the final definitions, as given in sections7and8below,arealgebraicandformalthemotivationbehindthem is quite geometric. This is explained in the earlier sections; we have chosen to present the exposition in the monograph following the evolution of our geometric understanding of what the tangent spaces should be rather than beginning with the formal definition and then retracing the steps leading to the geometry. Briefly, for0-cyclesanarcisinZn(X)isgivenbyZ-linearcombinationof arcsinthesymmetricproductsX(d),wheresuchanarcisgivenbyasmooth algebraic curve B together with a regular map B → X(d). If t is a local uniformizingparameteronB weshallusethenotationt→x (t)+···+x (t) 1 d for the arc in X(d). Arcs in Zn(X) will be denoted by z(t). We set |z(t)|= support of z(t), and if o∈B is a reference point we denote by Zn (X) the {x} subgroup of arcs in Zn(X) with limt→0|z(t)| = x. The tangent space will then be defined to be TZn(X)={arcs in Zn(X)}/≡ 1st where ≡ is an equivalence relation. Although we think it should be pos- 1st sible to define ≡ axiomatically, as in differential geometry, we have only 1st been able to do this in special cases. Among the main points uncovered in our study we mention the following: (a) The tangent spaces to the space of algebraic cycles is quite different from — and in some ways richer than — the tangent space to Hilbert schemes. PUTangSp March 1, 2004 4 CHAPTER1 This reflects the group structure on Zp(X) and properties such as (cid:1) (z(t)+z(cid:2)(t))(cid:2) =z(cid:2)(t)+z(cid:2)(cid:2)(t) (1.1) (−z(t))(cid:2) =−z(cid:2)(t) where z(t) and z(cid:2)(t) are arcs in Zp(X) with respective tangents z(cid:2)(t) and z(cid:2)(cid:2)(t). As a simple illustration, on a surface X an irreducible curve Y with a normal vector field ν may be obstructed in Hilb1(X) — e.g., the 1st or- der variation of Y in X given by ν may not be extendable to 2nd order. However, considering Y in Z1(X) as a codimension-1 cycle the 1st order variationgivenbyν extendsto2nd order. Infact,itcanbeshownthatboth TZ1(X) and TZn(X) are smooth, in the sense that for p=1, n every map Spec(C[(cid:2)]/(cid:2)2)→Zp(X) is tangent to a geometric arc in Zp(X). Forthesecondpoint,itiswellknownthatalgebraiccyclesincodimension p (cid:1) 2 behave quite differently from the classical case p = 1 of divisors. It turns out that infinitesimally this difference is reflected in a very geometric and computable fashion. In particular, (b) The differentials Ωk for all degrees k with 1 ≤ k ≤ n necessarily X/C enter into the definition of TZn(X). Remark that a tangent to the Hilbert scheme at a smooth point is uniquely determined by evaluating 1-forms on the corresponding normal vector field to the subscheme. However, for Zn(X) the forms of all degrees are required toevaluateonatangentvector,anditisinthissensethatagainthetangent spacetothespaceof0-cycleshasaricherstructurethantheHilbertscheme. Moreover, weseein(b)thatthegeometryofhighercodimensionalalgebraic cycles is fundamentally different from that of divisors. A third point is the following: For an algebraic curve one may give the definition of TZ1(X) either complex-analytically or algebro-geometrically with equivalent end results. However, it turns out that (c) For n≥2, even if one is only interested in the complex geometry of X the field of definition of an arc z(t) in Zn(X) necessarily enters into the description of z(cid:2)(0). Thus, although one may formally define TZn(X) in the analytic category, it is only in the algebraic setting that the definition is satisfactory from a geometric perspective. One reason is the following: Any reasonable set of axiomaticpropertiesonfirstorderequivalenceofarcsinZn(X)—including (1.1) above — leads for n (cid:1) 2 to the defining relations for absolute Ka¨hler differentials (cf. section 6.2 below). However, only in the algebraic setting is it the case that the sheaf of Ka¨hler differentials over C coincides with the sheaf of sections of the cotangent bundle (essentially, one cannot differen- tiate an infinite series term by term using Ka¨hler differentials). For subtle geometric reasons, (b) and (c) turn out to be closely related. (d) A fourth significant difference between divisors and higher codimen- sional cycles is the following: For divisors it is the case that If z ≡ 0 for a squence u tending to 0, then z ≡ 0. uk rat k 0 rat

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