211 Pages·2005·1.5 MB·English

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

Upgrade Premium

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.