Periods and Algebraic derham Cohomology

Transcription

1 Universität Leipzig Fakultät für Mathematik und Informatik Mathematisches Institut Periods and Algebraic derham Cohomology Diplomarbeit im Studiengang Diplom-Mathematik Leipzig, vorgelegt von Benjamin Friedrich, geboren am 22 Juli 1979

2 Abstract It is known that the algebraic derham cohomology group H i dr (X 0/Q) of a nonsingular variety X 0 /Q has the same rank as the rational singular cohomology group H i sing (Xan ; Q) of the complex manifold X an associated to the base change X 0 Q C However, we do not have a natural isomorphism H i dr (X 0/Q) = H i sing (Xan ; Q) Any choice of such an isomorphism produces certain integrals, so called periods, which reveal valuable information about X 0 The aim of this thesis is to explain these classical facts in detail Based on an approach of Kontsevich [K, pp 62 64], different definitions of a period are compared and their properties discussed Finally, the theory is applied to some examples These examples include a representation of ζ(2) as a period and a variation of mixed Hodge structures used by Goncharov [G1]

3 Contents 1 Introduction 1 2 The Associated Complex Analytic Space 4 21 The Definition of the Associated Complex Analytic Space 4 22 Algebraic and Analytic Coherent Sheaves 5 3 Algebraic derham Theory Classical derham Cohomology Algebraic derham Cohomology Basic Lemmas The Long Exact Sequence in Algebraic derham Cohomology Behaviour Under Base Change Some Spectral Sequences 18 4 Comparison Isomorphisms Situation Analytic derham Cohomology The Long Exact Sequence in Analytic derham Cohomology Some Spectral Sequences for Analytic derham Cohomology Comparison of Algebraic and Analytic derham Cohomology Complex Cohomology Comparison of Analytic derham Cohomology and Complex Cohomology Singular Cohomology Comparison of Complex Cohomology and Singular Cohomology Comparison Theorem An Alternative Description of the Comparison Isomorphism A Motivation for the Theory of Periods 41 5 Definitions of Periods First Definition of a Period: Pairing Periods Second Definition of a Period: Abstract Periods Third Definition of a Period: Naïve Periods 46 6 Triangulation of Algebraic Varieties Semi-algebraic Sets Semi-algebraic Singular Chains 53 7 Periods Revisited Comparison of Definitions of Periods Torsors Fourth Definition of a Period: Effective Periods 63 8 Examples First Example: A 1 C \ {0} Second Example: Quadratic Forms Third Example: Elliptic Curves Fourth Example: A ζ-value Fifth Example: The Double Logarithm Variation of Mixed Hodge Structures 77 A Sign Conventions and Related Material 94

4 1 Introduction The prehistory of Algebraic Topology dates back to Euler, Riemann and Betti, who started the idea of attaching various invariants to a topological space With his simplicial (co)homology theory, Poincaré was the first to give an instance of what in modern terms we would call a contravariant functor H from the category of (sufficiently nice) topological spaces to the category of cyclic complexes of abelian groups Many of such functors have been found so far; the most common examples are the standard cohomology theories (ie those satisfying the Eilenberg-Steenrod axioms), which measure quite different phenomena relating to diverse branches of mathematics It is a beautiful basic fact that all these standard cohomology theories agree (when restricted to an appropriate subcategory) This does not imply that we cannot hope for more If the topological space in question enjoys additional structure, one defines more elaborate invariants which take values in an abelian category of higher complexity For example, Hodge theory gives us a functor from the category of compact Kähler manifolds to the category of cyclic complexes of pure Hodge structures In this thesis, we will concentrate on spaces originating from Algebraic Geometry; these may be regarded as spaces carrying an algebraic structure Generalizing the concept of derham theory to nice schemes over Q gives us algebraic derham cohomology groups where each is nothing but a full Q-lattice inside the C-vector space of the corresponding classical derham cohomology group So, after tensoring with C, algebraic derham cohomology agrees with all the standard cohomology theories with complex coefficients However, a natural isomorphism between the original Q-vector space and a standard cohomology group with rational coefficients cannot exist We will illustrate this phenomenon in the following example (see Example 4121 for details) Let X an := C be the complex plane with the point 0 deleted and let t be the standard coordinate on X an Then the first singular cohomology group of X an is generated by the dual σ of the unit circle σ := S 1 H 1 sing(x an ; Q) = Q σ and H 1 sing(x an ; C) = C σ ; while for the first classical derham cohomology group, we have Under the comparison isomorphism H 1 dr (Xan ; C) = C dt t H 1 sing(x an ; C) = H 1 dr (Xan ; C) the generator σ of H 1 sing (Xan ; C) is mapped to ( dt S 1 t ) 1 dt t = 1 dt 2πi t If we view X an as the complex manifold associated to the base change to C of the algebraic variety X 0 := Spec Q[t, t 1 ] over Q, we can also compute the algebraic de Rham cohomology group H 1 dr (X 0/Q) of X 0 and embed it into H 1 dr (Xan ; C) H 1 dr (X 0/Q) = Q dt t H1 dr (Xan ; C) = C dt t 1

5 Thus we get two Q-lattices inside H 1 dr (Xan ; C), H 1 sing (Xan ; Q) and H 1 dr (X 0/Q), which do not coincide In fact, they differ by the factor 2πi our first example of what we will call a period Other examples will produce period numbers like π, ln 2, elliptic integrals, or ζ(2), which are interesting also from a number theoretical point of view (cf page 47) There is some ambiguity about the precise definition of a period; actually we will give four definitions in total: (i) pairing periods (cf Definition 511 on page 43) (ii) abstract periods (cf Definition 521 on page 45) (iii) naïve periods (cf Definition 532 on page 46) (iv) effective periods (cf Definition 731 on page 63) For X 0 a nonsingular variety over Q, we have a natural pairing between the i th algebraic derham cohomology of X 0 and the i th singular homology group of the complex manifold X an associated to the base change X 0 Q C H sing i (X an ; Q) H i dr (X 0/Q) C The numbers which can appear in the image of this pairing (or its version for relative cohomology) are called pairing periods; this is the most traditional way to define a period In [K, p 62], Kontsevich gives the alternative definition of effective periods which does not need algebraic derham cohomology and, at least conjecturally, gives the set of all periods some extra algebraic structure We present his ideas in Subsection 73 Abstract periods describe just a variant of Kontsevich s definition In fact, we have a surjection from the set of effective periods to the set of abstract ones (cf page 65), which is conjectured to be an isomorphism Naïve periods are defined in an elementary way and are used to provide a connection between pairing periods and abstract periods In Kontsevich s paper [K, p 63], it is used that the notion of pairing and abstract period coincide The aim of this thesis is to show that the following implications hold true (cf Theorem 711) abstract period naïve period pairing period The thesis is organized as follows The discussion of the various definitions of a period makes up the principal part of the work filling sections five to seven Section two gives an introduction to complex analytic spaces Additionally, we provide the connection to Algebraic Geometry by defining the associated complex analytic space of a variety In Section three, we define algebraic derham cohomology for pairs consisting of a variety and a divisor on it We also give some working tools for this cohomology The aim of Section four is to give a comparison theorem (Theorem 4101) which states that algebraic derham and singular cohomology agree In Section five, we present the definition of pairing, abstract, and naïve periods and prove some of their properties Section six provides some facts about the triangulation of algebraic varieties 2

6 Section seven contains the main result (Theorem 711) about the implications between the various definitions of a period mentioned above Furthermore, we give the definition of effective periods which motivated the definition of abstract periods In the last section, Section eight, we consider five examples to give an application of the general theory Among them is a representation of ζ(2) as an abstract period and the famous double logarithm variation of mixed Hodge structures used by Goncharov [G1] whose geometric origin is emphasized Conventions By a variety, we will always mean a reduced, quasi-projective scheme We will often deal with a variety X 0 defined over some algebraic extension of Q As a rule, skipping the subscript 0 will always mean base change to C X := X 0 Q C = X 0 Spec Q Spec C (An exception is section three, where arbitrary base fields are used) The complex analytic space associated to X will be denoted by X an (cf Subsection 21) The sign conventions used throughout this thesis are listed in the appendix Acknowledgments I am greatly indebted to my supervisor Prof A Huber- Klawitter for her guidance and her encouragement I very much appreciated the informal style of our discussions in which she vividly pointed out to me the central ideas of the mathematics involved I would also like to thank my fellow students R Munck, M Witte and K Zehmisch who read the manuscript and gave numerous comments which helped to clarify the exposition 3

7 2 The Associated Complex Analytic Space Let X be a variety over C The set X of closed points of X inherits the Zariski topology However, we can also equip this set with the standard topology: For smooth X this gives a complex manifold; in general we get a complex analytic space X an The main reference for this section is [Ha, B1] 21 The Definition of the Associated Complex Analytic Space We consider an example before giving the general definition of a complex analytic space Example 211 Let D n C n be the polycylinder D n := {z C n z i < 1, i = 1,, n} and O D n the sheaf of holomorphic functions on D n For a set of holomorphic functions f 1,, f m Γ(D n, O D n) we define X D n := {z D n f 1 (z) = = f m (z) = 0} O XD n := O D n/(f 1,, f m ) (1) The locally ringed space (X D n, O XD n ) from this example is a complex analytic space In general, complex analytic spaces are obtained by glueing spaces of the form (1) Definition 212 (Complex analytic space, [Ha, B1, p 438]) A locally ringed space (X, O X ) is called complex analytic if it is locally (as a locally ringed space) isomorphic to one of the form (1) A morphism of complex analytic spaces is a morphism of locally ringed spaces For any scheme (X, O X ) of finite type over C we have an associated complex analytic space (X an, O X an) Definition 213 (Associated complex analytic space, [Ha, B1, p 439]) Assume first that X is affine We fix an isomorphism X = Spec C[x 1,, x n ]/(f 1,, f m ) and then consider the f i as holomorphic functions on C n in order to set X an := {z C n f 1 (z) = = f m (z) = 0} O X an := O C n/(f 1,, f m ), where O C n denotes the sheaf of holomorphic functions on C n For an arbitrary scheme X of finite type over C, we take a covering of X by open affine subsets U i The scheme X is obtained by glueing the open sets U i, so we can use the same glueing data to glue the complex analytic space (U i ) an into an analytic space X an This is the associated complex analytic space of X This construction is natural and we obtain a functor an from the category of schemes of finite type over C to the category of complex analytic spaces Note that its restriction to the subcategory of smooth schemes maps into the category of complex manifolds as a consequence of the inverse function theorem (cf [Gun, Thm 6, p 20]) 4

8 Example 214 The complex analytic space associated to complex projective space is again complex projective space, but considered as a complex manifold To avoid confusion in the subsequent sections, the notation CP n will be reserved for complex projective space in the category of schemes, whereas we write CP n an for complex projective space in the category of complex analytic spaces For any scheme X of finite type over C, we have a natural map of locally ringed spaces φ : X an X (2) which induces the identity on the set of closed points X of X Note that φ O X = O X an 22 Algebraic and Analytic Coherent Sheaves Let us consider sheaves of O X -modules The equality of functors Γ(X,?) = Γ(X an, φ 1?) gives an equality of their right derived functors H i (X;?) = R i Γ(X;?) = R i Γ(X an ; φ 1?) Since φ 1 is an exact functor, the spectral sequence for the composition of the functors φ 1 and Γ(X an ;?) degenerates and we obtain H i (X an ; φ 1?) = R i Γ(X an ;?) φ 1 = R i Γ(X an ; φ 1?) Thus the natural map for F a sheaf of O X -modules φ 1 F φ F gives a natural map of cohomology groups H i (X; F) = H i (X an ; φ 1 F) H i (X an ; φ F) (3) Sheaf cohomology behaves particularly nice for coherent sheaves, this notion being defined as follows Definition 221 (Coherent sheaf) We define a coherent sheaf F on X (resp X an ) to be a sheaf of O X -modules (resp O X an-modules) that is Zariski-locally (resp locally in the standard topology) isomorphic to the cokernel of a morphism of free O X -modules (resp O X an-modules) of finite rank where r, s N O r U O s U F U 0, for U X (resp U X an ) open, (4) For sheaves on X this agrees with the definition given in [Ha, II5, p 111] For this alternate definition, we need some notation: If U = Spec A is an affine variety and M an A-module, we denote by M the sheaf on U associated to M (ie the sheaf associated to the presheaf V Γ(V ; O V ) A M for V U open, see [Ha, II5, p 110]) 5

9 Lemma 222 (cf [Ha, II5 Exercise 54, p 124]) A sheaf F of O X -modules is coherent if and only if X can be covered by open affine subsets U i = Spec A i such that F Ui = Mi for some finitely generated A i -modules M i Proof if : The A i s are Noetherian rings Therefore any finitely generated A i - module M i will be finitely presented A r i A s i M i 0 Since localization is an exact functor, we get O r U i O s U i M i 0, which proves the if -part only if : Wlog we may assume that the open subsets U X in (4) are affine U = Spec A Then Γ(U; O U ) = A and O U = à Now the A-module M := coker ( Γ(U; OU) r Γ(U; OU) s ) is clearly finitely generated Since = coker(a r A s ) A r A s M 0 gives we conclude à r Ãs M 0, F U = coker(o r U O s U) = coker(ãr Ãs ) = M As an immediate consequence of the definition of a coherent sheaf F on X, we see that the sheaf F an := φ F will be coherent as well: If is exact, so is O r U O s U F U 0 O r U an Os U an φ F U an 0, since φ 1 is exact and tensoring is a right exact functor There is a famous theorem by Serre usually referred to as GAGA, since it is contained in his paper Géométrie algébrique et géométrie analytique [Se:gaga] Theorem 223 (Serre, [Ha, B21, p 440]) Let X be a projective scheme over C Then the map F F an induces an equivalence between the category of coherent sheaves on X and the category of coherent sheaves on X an Furthermore, the natural map (3) is an isomorphism for all i H i (X; F) H i (X an ; F an ) 6

10 Let us state a corollary of Theorem 223, which is not included in [Ha] Corollary 224 In the situation of Theorem 223, we also have a natural isomorphism for hypercohomology for all i H i (X; F ) = H i (X an ; F an), where F is a bounded complex of coherent sheaves on X Here we only require the boundary morphisms of F to be morphisms of sheaves of abelian groups They do not need to be O X -linear Before we begin proving Corollary 224, we need some homological algebra Lemma 225 Let A be an abelian category and F [0] G,, (5) a morphism of double complexes of A-objects (cf the appendix), where F [0] is a double complex concentrated in the zeroth row with F being a complex vanishing below degree zero, ie F n = 0 for n < 0, and G, is a double complex living only in non-negative degrees If for all q Z 0 F q G 0,q G 1,q is a resolution of F q, then the map of total complexes induced by (5) is a quasi-isomorphism F tot G, Proof From (5) we obtain a morphism of spectral sequences h p II hq I F [0] h n F h p II hq I G, h n tot G, Both spectral sequences degenerate because of { h p h p F if q = 0, II hq I F [0] = 0 else and h p II hq I G, = { h p F if q = 0, 0 else and so we get an isomorphism on the initial terms Hence we also have an isomorphism on the limit terms and our assertion follows Remark 226 A similar statement holds with F [0] considered as a double complex concentrated in the zeroth column 7

11 Godement resolutions As we also need Γ-acyclic resolutions that behave functorial in the proof of Corollary 224, we now describe the concept of Godement resolutions For any sheaf F on X (or X an ) define g(f) := x X i F x, where i : x X denotes the closed immersion of the point x The sheaf g(f) is flabby and we have a natural inclusion F g(f) Setting G i+1 := g ( coker(g i 1 G i ) ) with G 1 := F and G 0 := g(f) gives an exact sequence 0 F G 0 G 1 We define the Godement resolution of F to be G F := 0 G 0 G 1 It is Γ-acyclic and functorial in F Extending this definition to a bounded complex F G,q F := G F q G F (6) := tot G, F yields a map of double complexes and a quasi-isomorphism by Lemma 225 F [0] G, F G F Let x X be a closed point of X The map φ from (2) gives a commutative square {x} an φ {x} i X an i X and a natural morphism (cf [Ha, II5, p 110]) φ F x φ φ F x, where we consider the stalk F x as a sheaf on {x} This map induces another map i F x i φ φ F x = (i φ) φ F x = (φ i) φ F x = φ i φ F x, and yet a third map (cf loc cit) ε : φ i F x i φ F x, (7) 8

12 which is an isomorphism, as can be seen on the stalks Consequently, ε x : φ F x φ F x, ε y : 0 0, for y x φ g(f) = φ x i F x = x φ i F x (7) = x i φ F x = x i (φ F) x = g(φ F x ) Thus we get first then φ G F p = G φ F p, φ G F = G φ F This gives us a natural map of hypercohomology groups H i (X; F ) = h i Γ(X; G F ) = hi Γ(X an ; φ 1 G F ) h i Γ(X an ; φ G F ) = hi Γ(X an ; G φ F ) = hi Γ(X an ; G F an ) = Hi (X an ; F an) (8) Proof of Corollary 224 We claim that the natural map (8) is an isomorphism for all i if X is projective If F has has length one, Theorem 223 tells us that this is indeed true So let us assume that (8) is an isomorphism for all complexes of length n and let F be a complex of coherent sheaves on X of length n + 1 Wlog F n+1 0 but F p = 0 for p > n + 1 We write σ n F for the complex F cut off above degree n The short exact sequence 0 F n+1 [ n 1] F σ n F 0 remains exact if we take the inverse image along φ 0 F n+1 an [ n 1] F an σ n F an 0 Using the naturality of (8), we obtain the following ladder with commuting squares H n 1+i (X; F n+1 ) H i (X; F ) H i (X; σ n F ) H n 1+i (X an ; F n+1 an ) H i (X an ; F an) H i (X an ; σ n F an) and the induction step follows from the 5-lemma 9

13 3 Algebraic derham Theory In this section, we define the algebraic derham cohomology H dr (X, D/k) of a smooth variety X over a field k and a normal-crossings-divisor D on X (cf definitions 323, 324, and 326) We also give some working tools for this cohomology: a base change theorem (Proposition 351) and two spectral sequences (Corollary 363 and Proposition 364) 31 Classical derham Cohomology This subsection only serves as a motivation for the following giving an overview of classical derham theory For a complex manifold, analytic derham, complex, and singular cohomology are defined and shown to be equal to classical derham cohomology All material presented here will be generalized to a relative setup later on Let M be a complex manifold We recall two standard exact sequences, (i) the analytic derham complex of holomorphic differential forms on M 0 C M Ω 0 M Ω 1 M, and (ii) the classical derham complex of smooth C-valued differential forms on M 0 C M E 0 M d E 1 M d, where C M is the constant sheaf with fibre C on M In both cases exactness is a consequence of the respective Poincaré lemmas [W, 418, p 155] and [GH, p 25] Now consider the commutative diagram C M [0] = C M [0] Ω M E M We can rephrase the exactness of the sequences (i) and (ii) by saying that the vertical maps are quasi-isomorphisms We indicate quasi-isomorphisms by a tilde Hence the natural inclusion Ω M E M is a quasi-isomorphism as well Therefore the hypercohomology of the two complexes coincides H (M; Ω M) = H (M; E M) The sheaves E p M are fine, since they admit a partition of unity In particular they are acyclic for the global section functor Γ(M,?) and we obtain H (M; E M) = h Γ(M; E M) The right-hand-side is usually called the classical derham cohomology of M, denoted H dr (M; C) 10

14 The equalities above give H dr (M; C) = H (M; Ω M) We refer to the right-hand-side H (M; Ω M ) as analytic derham cohomology, for which we want to use the same symbol H dr (M; C) The hypercohomology H (M; Ω M ) turns out to be a good candidate for generalizing derham theory to algebraic varieties Both variants of derham cohomology agree with complex cohomology H (M; C) := H (M; C M ) = H (M; C M [0]) We have yet a third resolution of the constant sheaf C M given by the complex of singular cochains: For any open set U M, we write C p sing (U; C) for the vector space of singular p-cochains on U with coefficients in C The sheaf of singular p- cochains is now defined as C p sing (M; C) : U Cp sing (U; C) for U M open with the obvious restriction maps The sheaves C p sing (M; C) are flabby: The restriction maps are the duals of injections between vector spaces of singular p-chains and the functor Hom C (?, C) is exact In particular these sheaves are acyclic for the global section functor Γ(M;?) and we obtain H (M; C sing(m; C)) = h Γ(M; C sing(m; C)) = h C sing(m; C) = H sing(m; C) By the following lemma, we conclude H (M; C) = H sing (M; C) Lemma 311 For any locally contractible, locally path-connected topological space M the sequence is exact 0 C M C 0 sing(m; C) C 1 sing(m; C) Proof Note first that C M = h 0 C sing (M; C), since M is locally path-connected For the higher cohomology sheaves h p C sing (M; C), p > 0, we observe that any element s x of the stalk h p C sing (M; C) x at x M not only lifts to a section s of C p sing (U; C) for some contractible open subset x U M, but that we can assume s to be a cocycle by eventually shrinking U Now this s is also a coboundary because of h p C sing(u; C) = H p sing (U; C) = 0 for all p > 0 We summarize this subsection in the following proposition Proposition 312 Let M be a complex manifold Then we have a chain of natural isomorphisms between the various cohomology groups defined in this subsection H dr (M; C) = H (M; Ω M) = H (M; C) = H sing(m; C) 11

15 32 Algebraic derham Cohomology Let X be a smooth variety defined over a field k and D a divisor with normal crossings on X; where having normal crossings means, that locally D looks like a collection of coordinate hypersurfaces, or more precisely: Definition 321 (Divisor with normal crossings, [Ha, p 391]) A divisor D X is said to have normal crossings, if each irreducible component of D is nonsingular and whenever s irreducible components D 1,, D s meet at a closed point P, then the local equations f 1,, f s of the D i form part of a regular system of parameters f 1,, f d at P It is proved in [Ma, 12, p 78] that in this case the f 1,, f d are linearly independent modulo m 2 P, where m P is the maximal ideal of the local ring O X,P at P By the inverse function theorem for holomorphic functions [Gun, Thm 6, p 20], we find in a neighbourhood of any P X an a holomorphic chart z 1,, z d such that D an is given as the zero-set {z 1 z s = 0} We are now going to define algebraic derham cohomology groups H dr (X/k), H dr (D/k) and H dr (X, D/k) Remark 322 In [Ha:dR], algebraic derham cohomology is defined for varieties with arbitrary singularities However, the relative version of algebraic derham cohomology discussed in [Ha:dR] deals with morphisms of varieties, not pairs of them 321 The Smooth Case Definition 323 (Algebraic derham cohomology for a smooth variety) We set H dr (X/k) := H (X; Ω X/k ), where Ω X/k is the complex of algebraic differential forms on the smooth variety X over k (cf [Ha, II8, p 175]) 322 The Case of a Divisor with Normal Crossings We write D = r i=1 D i as a sum of its irreducible components and use the short-hand notation (cf Figure 1) D I := D i0 i p := p D ik for I = {i 0,, i k } k=0 Associated to the decomposition D = D i is a simplicial scheme D := D a Dab Dabc, this notion being defined as follows Figure 1: Divisor with normal crossings 12

16 Simplicial sets and simplicial schemes We consider a category Simplex with objects [m] := {0,, m}, m N 0, and <-order-preserving maps as morphisms (Thus the existence of a map f : [m] [n] implies m n) This category can be thought of as the prototype of a simplicial complex: Let K be a simplicial complex and assign to [m] the set of m-simplices of K Any morphism f : [m] [n] maps each n-simplex to its m-dimensional f-face f (e 0,, e n ) = (e f(0),, e f(m) ) Thus K can be described by a contravariant functor to the category Sets of sets Simplex Sets A contravariant functor from Simplex to the category of sets, schemes, is called a simplicial set, simplicial scheme and so forth [GM, Ch 1, 22, p 9] (There, as a slight generalization, the maps f : [m] [n] are only required to be non-decreasing instead of being <-order-preserving, thus allowing for degenerate simplices ) In the above situation we define a simplicial scheme D : Simplex Schemes by the assignment [m] (f : [m] [n]) 1 i 0 < <i m r here D (f) is the sum of the natural inclusions D i0 i m (D (f) : D i0 i n D i0 i m ) ; D i0 i n D if(0) i f(m) If δl m : [m] [m + 1] denotes the unique <-order-preserving map, whose image does not contain l, we can represent D by a diagram r a=1 D (δ0 0) D a D (δ1 0) 1 a<b r D ab D (δ 1 0 ) D (δ 1 1 ) D (δ 1 2 ) 1 a<b<c r D abc (D ) Now the differentials come into play again We consider the contravariant functor i Ω?/k defined for subschemes Z of Supp(D) i Ω?/k : Z i Ω Z/k where i stands for the natural inclusion Z Supp(D) We enlarge the scope of i Ω?/k to schemes of the form a Z a, Z a D by making the convention ( ) ( ) i Ω?/k Z a := a a i i Ω Z a/k Composing D with the contravariant functor i Ω?/k yields a diagram r i Ω D a/k a=1 d 0 0 d 0 1 i Ω D ab /k 1 a<b r d 1 0 d 1 1 d a<b<c r i Ω D abc /k, 13

17 where we have written d m l for (i Ω?/k D )(δl m ) Summing up these maps d m l with alternating signs m d m := ( 1) l d m l gives a new diagram l=0 r i Ω D a/k a=1 d 0 1 a<b r i Ω D ab /k d 1 1 a<b<c r i Ω D abc /k, which turns out to be a double complex (cf shows the appendix) as a little calculation m+1 d m+1 d m = = = 0 l=0 k=0 m ( 1) l+k d m+1 d m k 0 k<l m+1 l ( 1) l+k d m+1 l d m k + 0 l k m ( 1) l+k d m+1 d m k }{{} = ( 1) l+k d m+1 = 0 l k m 0 k <l m+1 l k+1 dm l ( 1) k +l 1 d m+1 l d m k We denote this double complex by Ω, D /k and its total complex (cf the appendix) by Ω D/k := tot Ω, D /k Definition 324 (Algebraic derham cohomology for a divisor) If D is a divisor with normal crossings on a smooth variety X over k, we define its algebraic derham cohomology by H dr (D/k) := H (D; Ω D/k ) Remark 325 Actually, we do not need that D X has codimension 1 for this definition; but D having normal crossings is essential 323 The Relative Case The natural restriction maps Ω X/k (D i X) Ω D i /k sum up to a natural map of double complexes Ω X/k [0] (Supp(D) X) Ω, D /k, (9) 14

18 where we view Ω X/k [0] as a double complex concentrated in the zeroth column Somewhat sloppy, we will often write i F instead of (Supp(D) X) F Taking the total complex in (9) yields a natural map f : Ω X/k i Ω D/k, for whose mapping cone (cf the appendix) ( ) M f = i Ω D/k [ 1] Ω X/k with differential dd f 0 d X we write Ω X,D/k Definition 326 (Relative algebraic derham cohomology) For X a smooth variety over k and D a divisor with normal crossings on X, we define the relative algebraic derham cohomology of the pair (X, D) by H dr (X, D/k) := H (X; Ω X,D/k ) These definitions may seem a bit technical at first glance However, they will hopefully become more transparent in the sequel, when the existence of a long exact sequence in algebraic derham cohomology and various comparison isomorphisms are proved 33 Basic Lemmas In this subsection we gather two basic lemmas for the ease of reference Lemma 331 If i : Z X is a closed immersion and I an injective sheaf of abelian groups on Z, then i I is also injective Proof Let A B be an injective morphism of sheaves of abelian groups on X Since i 1 is an exact functor, the induced map i 1 A i 1 B will be injective as well Now the adjoint property of i 1 provides us with the following commutative square Hom X (B, i I) = Hom Z (i 1 B, I) Hom X (A, i I) = Hom Z (i 1 A, I) Because I is injective, the map on the right-hand-side is surjective Hence so is the left-hand map, which in turn implies the injectivity of i I, since A and B were arbitrary Lemma 332 If i : Z X is a closed immersion and F a complex of sheaves on Z, which is bounded below, then there is a natural isomorphism H (X; i F ) = H (Z; F ) 15

19 Proof The point is that i is an exact functor for closed immersions, which can be easily checked on the stalks Thus if F I is a quasi-isomorphism between F and a complex I consisting of injective sheaves, the induced map i F i I will be a quasi-isomorphism as well But the sheaves i I p are injective by Lemma 331, hence we can use i I to compute the hypercohomology of i F H (X; i F ) = h Γ(X; i I ) Now the claim follows from the identity of functors Γ(X, i?) = Γ(Z;?) H (X; i F ) = h Γ(X; i I ) = h Γ(Z; I ) = H (Z; F ) 34 The Long Exact Sequence in Algebraic derham Cohomology Proposition 341 We have a natural long exact sequence in algebraic derham cohomology (as defined in Definition 323, 324 and 326) H p 1 dr (D/k) H p dr (X, D/k) Hp dr (X/k) Hp dr (D/k) H p+1 dr (X, D/k), where X is a smooth variety over k and D a normal-crossings-divisor on X Proof The short exact sequence of the mapping cone 0 i Ω D/k [ 1] Ω X,D/k Ω X/k 0 gives us a long exact sequence in hypercohomology H p (X; i Ω D/k [ 1]) H p (X; Ω X,D/k ) Hp (X; Ω X/k ) Hp+1 (X; i Ω D/k [ 1]) H p+1 (X; Ω X,D/k ), which we can rewrite as By Lemma 332, we see and our assertion follows H p 1 (X; i Ω D/k ) H p dr (X, D/k) Hp dr (X/k) Hp (X; i Ω D/k ) H p+1 dr (X, D/k) H (X; i Ω D/k ) = H (D; Ω D/k ) = H dr (D/k) 16

20 35 Behaviour Under Base Change Let X 0 be a smooth variety defined over a field k 0 and D 0 a divisor with normal crossings on X 0 For the base change to an extension field k of k 0 we write Then we have the following proposition X := X 0 k0 k and D := D 0 k0 k Proposition 351 In the above situation, there are natural isomorphisms between algebraic derham cohomology groups (as defined in definitions 323, 324, 326) H dr (X/k) = H dr (X 0/k 0 ) k0 k, H dr (D/k) = H dr (D 0/k 0 ) k0 k and H dr (X, D/k) = H dr (X 0, D 0 /k 0 ) k0 k Proof Denote by π : X X 0 the natural projection By [Ha, Prop II810, p 175], we have Ω X/k = π Ω X 0 /k 0 Now the natural map factors through yielding a natural map π 1 Ω X 0 /k 0 π Ω X 0 /k 0 π 1 Ω X 0 /k 0 π 1 Ω X 0 /k 0 k0 k π 1 Ω X 0 /k 0 k0 k π Ω X 0 /k 0 = Ω X/k, which provides us with a morphism of spectral sequences h p H q (X; π 1 Ω X 0 /k 0 k0 k) H n (X; π 1 Ω X 0 /k 0 k0 k) h p H q (X; π Ω X 0 /k 0 ) H n (X; π Ω X 0 /k 0 ) We can rewrite the first initial term using the exactness of the functor? k0 k H q (X; π 1 Ω p X 0 /k 0 k0 k) = H q (X; π 1 Ω p X 0 /k 0 ) k0 k = H q (X 0 ; Ω p X 0 /k 0 ) k0 k Since cohomology commutes with flat base extension for quasi-coherent sheaves [Ha, Prop III93, p 255], the map H q (X 0 ; Ω p X 0 /k 0 ) k0 k H q (X; π Ω p X 0 /k 0 ) is an isomorphism Hence we also have an isomorphism on the limit terms, for which we obtain H n (X; π 1 Ω X 0 /k 0 k0 k) = H n (X; π 1 Ω X 0 /k 0 ) k0 k = H n (X 0 ; Ω X 0 /k 0 ) k0 k = H n dr (X 0/k 0 ) k0 k 17

21 and H n (X; π Ω X 0 /k 0 ) = H n (X; Ω X/k ) = H n dr (X/k) The other two statements can be proved analogously (Observe that the sheaves Ω p D 0 /k 0 and Ω p X 0,D 0 /k 0 are quasi-coherent by [Ha, Prop II58(c), p 115]) 36 Some Spectral Sequences Before we proceed in the discussion of algebraic derham cohomology, we have to make a digression and provide some spectral sequences which we will need later on 361 Čech Cohomology We adopt the notation from [Ha, III4]: Let X be a topological space, U = (U j ) j J an open covering of X and F a sheaf of abelian groups on X We assume J to be well-ordered As usual, we use the short-hand notation U I := U i0 i m := m U ik for I = {i 0,, i m } k=0 Recall that in this situation we have a Čech functor defined on the category Ab(X) of sheaves of abelian groups on X (cf [Ha, III4, p 220]) Č p (U,?) : Ab(X) Ab(X) F i F UI, I =p+1 I J where i stands for the respective inclusions U I X, and a differential d : Čp (U; F) Čp+1 (U; F), which makes Č (U, F) into a complex We explain this d below We also need Čech groups Č p (U; F) := Γ(X; Čp (U; F)) and of course Čech cohomologyȟ p (U; F) := h p Č (U; F) Clearly, we can consider the Čech functor also for closed coverings; but then Čech cohomology will not behave particularly well We can give the following fancy definition of the Čech complex Č (U; F) The assignment U : Simplex Schemes [m] i 0 < <i m U i0 i m (f : [m] [n]) (U (f) : U i0 i n ) U i0 i m 18

22 provides us with a simplicial scheme U (cf page 13), and thus we have a diagram Denote by a U (δ0 0) U a U (δ1 0) ε : a<b U ab I =m+1 U (δ 1 0 ) U (δ 1 1 ) U (δ 1 2 ) U I X a<b<c the natural morphism induced by U I X Then ε ε 1 F = i F UI, I =m+1 U abc where i : U I X Composing U with the contravariant functor ε ε 1 F : U I i F UI yields a diagram with maps a d 0 0 i F xua d 0 1 I =m+1 i F Uab a<b I =m+1 d 1 0 d 1 1 d 1 2 d m l := ( ε ε 1 F U ) (δ m l ) Summing up these maps with alternating signs a<b<c i F Uabc d m := m ( 1) l d m l l=0 gives us the Čech complex Č (U; F) We have the following lemma Lemma 361 ([Ha, III42, p 220]) Let X be a topological space, F a sheaf of abelian groups on X, and U = (U j ) j J an open covering of X Then the Čech complex Č (U; F), as defined above, is a resolution for F, ie the sequence 0 F Č0 (U; F) d0 Č1 (U; F) d1 is exact In particular, we have a natural isomorphism H (X; F) = H (X; Č (U; F)) 19

23 362 A Spectral Sequence for Hypercohomology of an Open Covering We generalize the well-known spectral sequence of an open covering U of a space X (where H q (F) is the presheaf V H q (V ; F V )) Ȟ p (U; H q (F)) H n (X; F) to the case of a complex F of abelian sheaves and its hypercohomology Let X be a topological space, U = (U j ) j J an open covering of X and F a complex of sheaves of abelian groups on X, which is bounded below We define a presheaf H p (F ) : V H p (V ; F V ) for V X open as a sheafified version of hypercohomology Proposition 362 In the situation above, the following spectral sequence converges E p,q 2 := Ȟp (U; H q (F )) E n := H n (X; F ) Proof We choose a quasi-isomorphism F I with I being a complex of flabby sheaves Note that the double complex Č I := Č (U; I ) consists of flabby sheaves, as well Now we consider the two spectral sequences of a double complex (cf [GM, Ch 1, 35, p 20]) for the double complex Č I := Č (U; I ) = Γ(X; Č I ) The first spectral sequence (10) Lemma 361), we have Therefore h p I hq II Č I h n tot Č I and (10) h p II hq I Č I h n tot Č I (11) h q IIČ I p = h q Γ(X; Č I p ) = H q (X; I p ) = Since Č I p is a flabby resolution of I p (by { Γ(X; I p ) if q = 0, 0 else { h p I hq IIČ I h p Γ(X; I ) if q = 0, = 0 else { H p (X; F ) if q = 0, = 0 else Thus the first spectral sequence degenerates and we obtain a natural isomorphism H (X; F ) = h tot Č I The second spectral sequence (11) From Č p I q = Γ ( X; i I q ) U I = I =p+1 = I =p+1 I =p+1 Γ ( X; i I q U I ) Γ ( U I ; I q U I ), 20

24 we see h q I Čp I = h q Γ ( U I ; I U ) I I =p+1 = H q( U I ; F U ) I I =p+1 = Γ ( U I ; H q (F ) ) UI I =p+1 = Γ ( X; i H q (F ) ) UI I =p+1 = Γ ( X; i H q (F ) ) UI I =p+1 = Γ ( X; Čp( U; H q (F ) )) = Čp( U; H q (F ) ) This gives and concludes the proof h p II hq I Č I = h p Č (U; H q (F )) = Ȟp (U; H q (F )) 363 A Spectral Sequence for Algebraic derham Cohomology of a Smooth Variety We are especially interested in Proposition 362 for F = Ω X/k Corollary 363 Let X be a smooth variety over a field k and U an open covering of X Then we have a convergent spectral sequence for algebraic derham cohomology of smooth varieties (as defined in Definition 323) E p,q 2 := h p I = +1 H q dr (U I/k) E n := H n dr (X/k) Proof Let us compute the initial terms of the spectral sequence Ȟ(U; H q (F )) H n (X; F ) of Proposition 362 for F = Ω X/k Ȟ p (U; H q (Ω X/k )) = hp Γ ( X; Č ( U; H q (Ω X/k ( ))) ) = h p Γ X; i H q (Ω X/k ) U I = h p I = +1 I = +1 Γ ( X; i H q (Ω X/k ) U I ) = h p Γ ( U I ; H q (Ω X/k ) ) U I I = +1 = h p H q (U I ; Ω X/k U I ) I = +1 21

25 For the limit term we get immediately and the corollary is proved = h p I = +1 = h p I = +1 H q (U I ; Ω U I /k ) H q dr (U I/k) H (X; Ω X/k ) = H dr (X/k) 364 A Spectral Sequence for Algebraic derham Cohomology of a Divisor with Normal Crossings For a divisor D with normal crossings, we have a spectral sequence expressing H dr (D/k) in terms of H dr (D i 0 i m /k) Recall that (cf page 14) Ω D/k = tot Ω, D /k = tot I i Ω D I /k Since we are interested in hypercohomology, we replace the Ω by their Godement resolutions (cf page 8), where we shall use the following abbreviations and G p D I /k := Gp Ω D I /k, G p,q D /k := Gp Ω,q D/k = I =q+1 Gp i Ω D I /k ( ) = I =q+1 i G p D I /k, (we have equality at ( ) since i is exact for closed immersions i) G D/k := G e Ω D/k = G tot Ω, D /k = tot G, D /k Furthermore, we write G p D I /k := Γ( D I ; G p D I /k ), G p,q D /k := Γ( D; G p,q ) D /k and G p D/k := Γ( D; G p ) D/k for the groups of global sections of these sheaves We consider the second spectral sequence for the double complex G p,q D /k For the limit terms, we obtain h p II hq I G, D /k hn tot G, D /k h n tot G, D /k = hn G D/k = hn Γ(D; G D/k ) = Hn (D; Ω D/k ) = Hn dr (D/k) 22

26 Let us compute the initial terms h q G,p D /k = hq Γ(D; = h q = I =p+1 I =p+1 = I =p+1 = I =p+1 = I =p+1 I =p+1 Thus we have proved the following proposition i G D I /k ) Γ(D; i G D I /k ) h q Γ(D; i G D I /k ) h q Γ(D I ; G D I /k ) H q (D I ; Ω D I /k ) H q dr (D I/k) Proposition 364 Let X be a smooth variety over a field k and D a divisor with normal crossings on X Then we have a convergent spectral sequence for algebraic de Rham cohomology of a divisor (as defined in Definition 324) E p,q 2 := h p I = +1 H q dr (D I/k) E n := H n dr (D/k) 23

27 4 Comparison Isomorphisms If we consider a smooth variety X defined over C and a divisor D with normal crossings on X, then their algebraic derham cohomology as defined in Subsection 32 turns out to be naturally isomorphic to the singular cohomology of the associated complex analytic spaces X an and D an Our proof proceeds in three steps: (1) First we mimic the definition of algebraic derham cohomology in the complex analytic setting using holomorphic differential forms instead of algebraic ones, thus defining analytic derham cohomology groups H dr (Xan ; C), H dr (Dan ; C) and H dr (Xan, D an ; C) These are then shown to be isomorphic to their algebraic counterparts using an application of Serre s GAGA-type results by Grothendieck (2) In the second step, analytic derham cohomology is proved to coincide with complex cohomology, (3) which in turn is isomorphic to singular cohomology, as we show in the third step (In the smooth case this has already been shown in Proposition 312) 41 Situation Throughout this section, X will be a smooth variety over C and D a divisor with normal crossings on X As usual, we denote the complex analytic spaces associated to X and D by X an and D an, respectively (cf Definition 213) 42 Analytic derham Cohomology If we replace in Subsection 32 every occurrence of the complex of algebraic differential forms Ω Y/C on a complex variety Y by the complex of holomorphic differential forms Ω Y an on the associated complex analytic space Y an, we obtain complexes Ω X an, and thus are able to define: Ω, D an, Ω D an and Ω X an,d an Definition 421 (Analytic derham cohomology) For X an and D an as in Subsection 41, we define analytic derham cohomology by H dr (Xan ; C) := H (X an ; Ω X an), H dr (Dan ; C) := H (D an ; Ω D an) and H dr (Xan, D an ; C) := H (X an ; Ω X an,d an) ( ) Here ( ) is the definition of analytic derham cohomology for complex manifolds already considered on page 11, which we listed again for completeness Note that (Ω Y/C ) an = Ω Y an, hence (Ω X/C ) an = Ω X an, (Ω, D /C ) an = Ω, D an, ( Ω D/C ) an = Ω D an and (12) ( Ω X,D/C ) an = Ω X an,d an 24

28 Remark 422 We could easily generalize Definition 421 to arbitrary complex manifolds (and normal-crossings-divisors on them) not necessarily associated to something algebraic, but we will not need this 43 The Long Exact Sequence in Analytic derham Cohomology The results of Subsection 34 carry over one-to-one to the complex analytic case Proposition 431 We have naturally a long exact sequence in analytic derham cohomology (as defined in Definition 421) H p 1 dr (Dan ; C) H p dr (Xan, D an ; C) H p dr (Xan ; C) H p dr (Dan ; C) H p+1 dr (Xan, D an ; C), where X an and D an are as in Subsection 41 Moreover, by Equation (8) on page 9 and Equation (12), we have a natural morphism from the long exact sequence in algebraic derham cohomology to the one in analytic derham cohomology H p dr (X, D/C) Hp dr (X/C) Hp dr (D/C) H p dr (Xan, D an ; C) H p dr (Xan ; C) H p dr (Dan ; C) (13) We want to show that all the vertical maps are isomorphisms 44 Some Spectral Sequences for Analytic derham Cohomology The proof of Proposition 363 applies in the analytic case as well and we obtain Proposition 441 We have a convergent spectral sequence for analytic derham cohomology (as defined in Definition 421) E p,q 2 := h p H q an dr (UI I = +1 ; C) E n := H n dr (Xan ; C), where X an is associated to a smooth variety X over C and U = (U j ) j J is an open covering of X Furthermore, the natural map of hypercohomology groups (cf (8) and (12)) H (U; Ω U/C ) H (U an ; Ω U an) for U X open, provides us with a natural map of spectral sequences Ȟ p (U; H q (Ω X/C )) Hn (X; Ω X/C ) Ȟ p (U an ; H q (Ω X an)) Hn (X an ; Ω X an), 25

29 which we can rewrite using Corollary 363 and Proposition 441 h p H q dr (U I/C) H n dr (X/C) I = +1 an (U ; C) H n dr (Xan ; C) h p I = +1 H q dr Likewise, we can transfer Proposition 364 to the analytic case Proposition 442 We have a convergent spectral sequence E p,q 2 := h p H q dr (Dan I ; C) E n := H n dr (Dan ; C), I = +1 I where D an is associated to a normal-crossings-divisor D on a smooth variety X defined over C as in Subsection 41 Again, we have a natural map of spectral sequences We write (cf page 22) ( ) G p,q D := G p,q an D /C and G p,q D := Γ(D an ; G p,q an D ) an Now the natural map of zeroth homology (cf Equation (3) for i = 0) G, D /C G, D an gives us a natural map of spectral sequences an h p I hq II G, D /C hn tot G, D /C h p I hq II G, D h n tot G, an D an With propositions 364 and 442, this map takes the form h p H q dr (D I/C) H n dr (D/C) I = +1 h p I = +1 H q dr (Dan I ; C) H n dr (Dan ; C) 45 Comparison of Algebraic and Analytic derham Cohomology The proof of H dr (X/C) = H dr (Xan ; C), ie H (X; Ω X/C ) = H (X an ; Ω Xan) is easy in the projective case, because of (12) and Corollary 224 The affine case is contained in a theorem of Grothendieck Theorem 451 (Grothendieck, [Gro, Thm 1, p 95]) Let X be a smooth affine variety over C Then the natural map between algebraic and analytic derham cohomology (as defined in definitions 323 and 421) is an isomorphism H dr (X/C) H dr (X an ; C) 26 (14) (15)

30 We give only some comments on the proof Let j : X X be the projective closure of X and let Z := X \ X Furthermore, we set Ω ( Z) := lim Ω X/C (nz) and X/C n Ω X an( Zan ) := lim n Ω X an(nzan ), where F(nZ ) denotes the sheaf F twisted by the n-fold of the divisor Z In his paper [Gro], Grothendieck considers the following chain of isomorphisms: H dr (X/C) = H (X; Ω X/C ) Ω p X/C are Γ(X;?)-acyclic [Ha, Thm III35, p 215] h Γ(X; Ω X/C ) h Γ(X; j Ω X/C ) h Γ(X; Ω X/C ( Z)) [Ha, Prop III29, p 209] lim n h Γ(X; Ω X/C (nz)) Thm 223 (GAGA) lim h Γ(X an ; Ω X an(nzan )) n [AH, Lemma 6] h Γ(X an ; Ω X an( Zan )) main step h Γ(X an ; j Ω X an) h Γ(X an ; Ω X an) Ω p X are Γ(X an ;?)-acyclic since X an is Stein an [GR, IV11, Def 1, p 103; V1, Bem, p 130] H dr (Xan ; C) = H (X an ; Ω X an) In the case that Z has normal crossings, the main step is due to Atiyah and Hodge [AH, Lemma 17] and is proved by explicit calculations Grothendieck reduces the general case to this one by using the resolution of singularities according to Hironaka [Hi1] The smooth case Choose an open affine covering U = (U j ) j J of X Since X is separated, all the U I are affine Hence by Theorem 451 we get an isomorphism on the initial terms in (14), and therefore also on the limit terms H dr (X/C) = H dr (Xan ; C) 27

31 The case of a normal-crossings-divisor Since D is a divisor with normal crossings, all the D I are smooth Thus we see from the discussion of the smooth case, that we have an isomorphism on the initial terms in (15), hence also on the limit terms H dr (D/C) = H dr (Dan ; C) The relative case By the 5-lemma, this case follows immediately from the two cases considered above using diagram (13) We summarize our results so far Proposition 452 For X a smooth variety over C and D a normal-crossings-divisor on X, we have a natural isomorphism between algebraic derham cohomology of X and D (see definitions 323, 324, 326) and analytic derham cohomology (see Definition 421) of the associated complex analytic spaces X an and D an H p dr (X, D/C) Hp dr (X/C) Hp dr (D/C) H p dr (Xan, D an ; C) H p dr (Xan ; C) H p dr (Dan ; C) 46 Complex Cohomology Definition 461 (Complex cohomology) Let X an and D an be as in Subsection 41 In the absolute case, complex cohomology is simply sheaf cohomology of the constant sheaf with fibre C H (X an ; C) := H (X an ; C X an) and H (D an ; C) := H (D an ; C D an) We define a relative version by where U := X \ D and j : U X H (X an, D an ; C) := H (X an ; j! C U an), Remark 462 Taking Q instead of C in the above definition would give us rational cohomology groups H (X an ; Q), H (D an ; Q) and H (X an, D an ; Q) The short exact sequence 0 j! C U an α C X an i C D an 0 (16) with j : U an X an and i : D an X an, yields a long exact sequence in complex cohomology H p 1 (D an ; C) H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) H p+1 (X an, D an ; C) The mapping cone M C (cf the appendix) of the natural restriction map C X an i C D an gives us another short exact sequence 0 i C D an[ 1] M C C X an[0] 0 (17) 28

32 Consider the map γ := (0, α) : j! C U an[0] M C = i C D an[ 1] C X an[0], where α : j! C U an[0] C X an[0] was defined in (16) and 0 : j! C U an i C D an[ 1] denotes the zero map In order to show that α is a morphism of complexes, we only have to check the compatibility of the differentials, but this follows from the fact that (16) is a complex The exactness of the sequence in (16) translates into γ being a quasi-isomorphism This gives us the following diagram H p 1 (D an ; C) H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) H p (X an ; i C D an[ 1]) H p (X an ; M C ) H p (X an ; C X an[0]) H p+1 (X an ; i C D an[ 1]) Unfortunately, this diagram does not commute (there is a sign mismatch), so we consider the natural transformation and write down a new diagram ε p = ( 1) p id for p Z H p 1 (D an ; C) H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) ε p 1 ε p ε p ε p H p (X an ; i C D an[ 1]) H p (X an ; M C ) H p (X an ; C X an[0]) H p+1 (X an ; i C D an[ 1]) This diagram commutes as we see by replacing the sheaves involved in (16) by injective resolutions A, B and C and applying Lemma 463 below to the complexes A, B and C of their global sections Lemma 463 Let 0 A α B β C 0 be a short exact sequence of complexes of abelian groups and 0 C [ 1] ι M π B 0 the mapping cone of β : B C Then we have a quasi-isomorphism and a commutative diagram γ := (0, α) : A M = C [ 1] B h p 1 C δ h p A α h p B β h p C ε p 1 1 ε p γ ε p 2 ε p h p 1 C ι h p M π h p B δ h p C Proof The assertion about the quasi-isomorphism γ : A M follows from the exactness of 0 A B C 0 The middle square commutes because of π γ = α So we are left to show the commutativity of the squares 1 and 2 29

33 Ad 2 ) From the diagram C p ι M p+1 d M M p π B p and the definition of the connecting morphism δ we see for b B p δ [b] = [β(b)] = β [b], hence (δ ε p )[b] = (ε p β )[b] Ad 1 ) Pick an element c C p 1 and choose a preimage b B p 1 With A p α B p d B B p 1 β C p 1 we get Applying γ gives δ[c] = [α 1 d B b] h p A (γ δ)[c] = (0, α) [α 1 d B b] = [0, d B b] h p M = h p (C [ 1] B ) or after subtracting the coboundary d M (0, b) = (β(b), d B b) = (c, d B b) (γ δ)[c] = [ c, 0] = ι [c], hence (ε p γ δ)[c] = (ι ε p 1 )[c] 47 Comparison of Analytic derham Cohomology and Complex Cohomology We apply the Čech functor (cf Subsection 361) for the closed covering D := (Di an ) r i=1 to the constant sheaf C D an, and thus get Čq (D; C) := Čq (D; C D an) Now the natural inclusions C D an I Ω 0 DI an sum up to a natural morphism of sheaves on D an where i : D an I Č q (D; C) = Ω 0,q D an = I =q+1 I =q+1 D an Since the complexes Ω D an I i C D an I i Ω 0 D an I, and i and are exact functors, this shows that Ω,q D an (18) are resolutions of the sheaves C D an I is a resolution of Čq (D; C) 30

34 If we consider Č (D; C)[0] as a double complex (cf the appendix) concentrated in the zeroth row, we can rewrite (18) as or, after taking the total complex, as Č (D; C)[0] Ω, D an Č (D; C) Ω D an This morphism is a quasi-isomorphism by Lemma 225 (and Remark 226) and fits into the following commutative diagram C X an[0] i C D an[0] α C X an[0] i Č (D; C) Ω X an i Ω D an The map α is a quasi-isomorphism as a consequence of the following proposition Proposition 471 We have an exact sequence 0 C D an Č0 (D; C) Č1 (D; C) Proof (cf [Ha, Lemma III42, p 220]) This can be checked on the stalks and is therefore a purely combinatorial problem Let x D an and assume wlog that the irreducible components D1 an,, Dan s but no other component of D an are passing through x For the stalks at x, we write Č p := Čp (D; C) x = C D an i0 ip,x 1 i 0 < <i p s Observe that exactness at the zeroth and first step is obvious Using the coordinate functions for a set of indices j 0,, j p Č p = C D an C i0 ip,x Dj an 0 jp,x we define a homotopy operator 1 i 0 < <i p s α i0 i p α j0 j p k p : Čp Čp 1 such that for any α Čp the coordinates of its image k p (α) satisfy { k p 0 if j 0 = 1 (α) j0 j p 1 = else α 1 j0 j p 1 Now an elementary calculation shows that for p 1 d p 1 k p + k p+1 d p = idčp Hence k is a homotopy between the identity map and the zero map on Č, and we conclude that h p Č = 0 for p 1 31

35 Thus we have a commutative diagram C X an[0] i C D an[0] Ω X an i Ω D an, where the vertical maps are quasi-isomorphisms As a consequence, we also have a quasi-isomorphism between the respective mapping cones 0 i C D an[ 1] M C C X an[0] 0 0 i Ω D an[ 1] Ω X an,d an Ω X an 0 (19) Taking hypercohomology proves the following proposition Proposition 472 We have a natural isomorphism between the long exact sequence in analytic derham cohomology (cf Definition 421) and the sequence in complex cohomology (cf Definition 461) H p 1 dr (Xan ; C) H p 1 dr (Dan ; C) H p dr (Xan, D an ; C) H p 1 (X an ; C) H p 1 (D an ; C) H p (X an, D an ; C), where X an and D an are as usual (see Subsection 41) 48 Singular Cohomology We extend the definition of singular cohomology given on page 11 H sing(x an ; C) := h Γ ( X an ; C sing(x an ; C) ) and H sing(d an ; C) := h Γ ( D an ; C sing(d an ; C) ) by a relative version Definition 481 (Relative singular cohomology) As usual, let X an and D an be as in Subsection 41 We set C sing(x an, D an ; C) := ker(c sing(x an ; C) i C sing(d an ; C)) and define H sing(x an, D an ; C) := h Γ(X an ; C sing(x an, D an ; C)) Remark 482 Similarly, one defines singular cohomology H sing (Xan, D an ; k) with coefficients in a field k different from C We will need H sing (Xan, D an ; Q) occasionly Since? Q C is an exact functor, we have H sing(x an, D an ; Q) Q C = H sing(x an, D an ; C) 32

36 The sheaf C sing (Xan, D an ; C) is Γ(X an ;?)-acyclic because of the surjectivity of C sing(x an ; C) C sing(d an ; C) = Γ(X an ; i C sing(d an ; C)) Thus the short exact sequence 0 C sing(x an, D an ; C) C sing(x an ; C) i C sing(d an ; C) 0 gives rise to a long exact sequence in singular cohomology H p 1 sing (Dan ; C) H p sing (Xan, D an ; C) H p sing (Xan ; C) H p sing (Dan ; C) H p+1 sing (Xan, D an ; C) 49 Comparison of Complex Cohomology and Singular Cohomology We have a commutative diagram 0 j! C U an[0] C X an[0] i C D an[0] 0 α 0 C sing (Xan, D an ; C) C sing (Xan ; C) i C sing (Dan ; C) 0, (20) where β and γ are quasi-isomorphisms, since the singular cochain complex is a resolution of the constant sheaf (see Proposition 311) and i is exact for closed immersions Therefore α is also a quasi-isomorphism Taking hypercohomology gives a natural isomorphism between the long exact sequence in complex cohomology and the sequence in singular cohomology Thus we have finally proved the following theorem 410 Comparison Theorem Theorem 4101 (Comparison theorem) Let X be a smooth variety defined over C and D a divisor on X with normal crossings (cf Definition 321) The associated complex analytic spaces (cf Subsection 21) are denoted by X an and D an, respectively Then we have natural isomorphisms between the long exact sequences in algebraic derham cohomology (cf definitions 323, 324, 326), β analytic derham cohomology (cf Definition 421), complex cohomology (cf Definition 461), and singular cohomology (cf Definition 481) γ 33

37 as shown in the commutative diagram below H p dr (X, D/C) Hp dr (X/C) H p dr (D/C) algebraic derham cohomology H p dr (Xan, D an ; C) H p dr (Xan ; C) H p dr (Dan ; C) analytic derham cohomology H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) complex cohomology H p sing (Xan, D an ; C) H p sing (Xan ; C) H p sing (Dan ; C) singular cohomology Proof Combine propositions 452 and 472 with the result from Subsection An Alternative Description of the Comparison Isomorphism For computational purposes, it will be useful to describe the isomorphism from the Comparison Theorem 4101 H dr (Xan ; C) = H sing(x an ; C) more explicitely We will formulate our result in Proposition 4115 and explain its significance in Subsection 4115 We start by defining yet another cohomology theory 4111 Smooth Singular Cohomology Let M be a complex manifold We write std p for the standard p-simplex spanned by the basis {(0,, 0, 1, 0,, 0) i = 1,, p + 1} of R p+1 A singular p-simplex i σ : std p M is said to be smooth, if σ extends to a smooth map defined on a neighbourhood of the standard p-simplex std p in its p-plane We denote by C p (M; C) the C-vector space generated by all smooth p-simplices of M, thus getting a subcomplex C (M; C) of the complex C sing (M; C) of all singular simplices on M We can dualize this subcomplex and define a complex of sheaves C (M; C) C sing (M; C) C (M; C) := C (M; C), C (M; C) : V C (V ; C) for V M open The sheaves C p (M; Q) are flabby and define smooth singular cohomology of M Definition 4111 (Smooth singular cohomology of a manifold) Let M be a complex manifold We define smooth singular cohomology groups H (M; C) := h C (M; C) = H ( M; C (M; C) ) 34

38 Since smooth singular cohomology coincides with singular cohomology for smooth manifolds (cf [Br, p 291]), we can show that C (M; C) is a resolution of the constant sheaf C M exactly as we did for C sing (M; C) in Lemma 311 Thus we have the following lemma Lemma 4112 For M a complex manifold, the following sequence is exact 0 C M C 0 (M; C) C 1 (M; C) We are now going to extend the definition of smooth singular cohomology to the case of a divisor and to the relative case Let X an and D an be as in Subsection 41 First, recall how Ω D an was defined (cf pages 14 and 24): We composed the simplicial scheme D with the functor i Ω?, then obtained a double complex Ω, an D by an summing up certain maps and finally denoted the total complex of Ω, D by Ω an D an Similarly, we can use the functor i C (? an ; C), compose it to the simplicial scheme D and denote the resulting double complex by C, (D an ; C) For the corresponding total complex we write C (D an ; C) We define C (X an, D an ; C) similar to Ω X an,d an (cf pages 14 and 24): The natural restriction maps C (X an ; C) i C (D j an ; C) sum up to a natural map of double complexes C (X an ; C)[0] i C, (D an ; C), (21) where we view C (X an ; C)[0] as a double complex concentrated in the zeroth column Taking the total complex in (21) yields a natural map C (X an ; C) i C (D an ; C), for whose mapping cone (cf the appendix) we write C (X an, D an ; C) Thus we are able to formulate: Definition 4113 (Smooth singular cohomology for the case of a divisor and for the relative case) We define smooth singular cohomology groups H (D an ; C) := h C (D an ; C) = H (D an ; C (D an ; C)) and H (X an, D an ; C) := h C (X an, D an ; C) = H (X an ; C (X an, D an ; C)), where X an and D an are as in Subsection 41 Remark 4114 Similarly one defines smooth singular cohomology with coefficients in Q All statements made below about smooth singular cohomology remain valid for rational coefficients except those involving holomorphic differentials As usual, we have a long exact sequence in smooth singular cohomology H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) 35

39 4112 Comparison with Analytic derham Cohomology Our motivation for considering smooth singular cohomology instead of ordinary singular cohomology comes from the following fact: We can integrate differential-p-forms over smooth p-simplices γ and thus have a natural morphism As a consequence of Stoke s theorem we get a well-defined map of complexes Ω p X C (X p an ; C) an Cp (X an ; C) C ω γ ω γ ω = γ γ dω, Ω X an C (X an ; C) This map is functorial, hence gives a natural transformation of functors With the definitions of the complexes Ω? an C (? an ; C) Ω D an, C (D an ; C) and Ω X an,d an, C (X an, D an ; C) (cf pages 24 and 35), we get a commutative diagram 0 i Ω D an[ 1] Ω X an,d an Ω Xan 0 0 i C (D an ; C)[ 1] C (X an, D an ; C) C (X an ; C) 0 (22) Recall the short exact sequence (17) of the mapping cone M C from Subsection i C D an[ 1] M C C X an[0] 0 and Diagram (19) relating this short exact sequence with the first line of (22) Combining (19) and (22) gives a big diagram 0 i C D an[ 1] M C C X an[0] 0 0 i Ω D an[ 1] Ω X an,d an Ω Xan 0 α β γ 0 i C (D an ; C)[ 1] C (X an, D an ; C) C (X an ; C) 0 (23) with α, β, and γ being just the composition of the two vertical maps We want to show that α, β, and γ are quasi-isomorphisms We already know that γ is a quasiisomorphism by Lemma 4112 If α is also a quasi-isomorphism, then it will follow 36

40 from the 5-lemma that β is a quasi-isomorphism as well We split α as a composition of a quasi-isomorphism (cf Proposition 471) and a map α α : C D an[0] Č eα (D; C) C (D an ; C) The map α is a map of total complexes induced by the morphism of double complexes Č (D; C)[0] C, (D an ; C), where we consider Č (D; C)[0] as a double complex concentrated in the zeroth row By Lemma 4112 the sequence 0 C D an I C (D 0 I an ; C) C (D 1 I an ; C) is exact Hence the sequence 0 Čq (D; C) C 0,q (D an ; C) C 1,q (D an ; C) is also exact, since and i are exact functors Applying Lemma 225 shows that α is a quasi-isomorphism Thus all vertical maps in diagram (23) are indeed quasiisomorphisms Taking hypercohomology provides us with isomorphisms between long exact cohomology sequences generalizing the Comparison Theorem 4101 H p (X an ; M C ) H p (X an ; C) H p (D an ; C) H p dr (Xan, D an ; C) H p dr (Xan ; C) H p dr (Dan ; C) H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) (24) 4113 Comparison with Singular Cohomology We will compare smooth singular cohomology with ordinary singular cohomology First we need morphisms between the defining complexes We already have a natural map C sing(x an ; C) C (X an ; C) induced by the duals of the standard inclusions C (V ; C) C sing (V ; C) for V X an open Next we discuss the case of a divisor: By restricting singular cochains from D an to an irreducible component Dj an of D an, we get a map where i : D an j C sing(d an ; C) i C sing(d j an ; C), D an is the natural inclusion By summing up the composition maps C sing(d an ; C) i C sing(d an j we obtain a morphism of complexes ; C) i C (D j an ; C), C sing(d an ; C) C,0 (D an ; C), 37

41 which gives a morphism of double complexes C sing(d an ; C)[0] C, (D an ; C), where we consider C sing (Dan ; C)[0] as a double complex concentrated in the zeroth column Now we turn to the relative case of a pair (X an ; D an ): Let M sing denote the mapping cone of C sing(x an ; C) i C sing(d an ; C) By the functoriality of the mapping cone construction, we get a commutative diagram 0 i C sing (Dan ; C)[ 1] M sing C sing (Xan ; C) 0 α β γ 0 i C (D an ; C)[ 1] C (X an, D an ; C) C (X an ; C) 0 It is not hard to show directly that α, β, γ are quasi-isomorphisms However, we can use a shorter indirect argument: We have the following commutative diagram (cf diagrams (20), (23) and the one above) 0 i C D an[ 1] α M C β C X an[0] γ 0 0 i C (D an ; C)[ 1] C (X an, D an ; C) C (X an ; C) 0 α α β β γ γ 0 i C sing (Dan ; C)[ 1] M sing C sing (Xan ; C) 0 In Subsection 4112, we have already seen that α, β, γ are quasi-isomorphisms Furthermore α and γ are quasi-isomorphisms (cf Subsection 49), hence so is β (by the 5-lemma) Therefore α, β, γ have to be quasi-isomorphisms as well Taking hypercohomology in this diagram gives us H p (X an ; M C ) H p (X an ; C) β H p (D an ; C) γ H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) H p (X an ; M sing ) Hp sing (Xan ; C) H p sing (Dan ; C) (25) In order to get rid of the mapping cones M C and M sing, we apply Lemma 463 two each of the two short exact sequences of complexes below (where G? denotes the Godement resolution see page 8) α 0 Γ(X an ; G j! C ) U an Γ(Xan ; G C ) X an Γ(Xan ; G i ) C D an 0 0 Γ(X an ; G C sing (Xan,D an ;C) ) Γ(Xan ; G C sing (Xan ;C) ) Γ(Xan ; G i C sing (Dan ;C) ) 0 38

42 and obtain a commutative diagram (cf definitions 461 and 481) H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) ε p ε p H p (D an ; C) H p (X an ; C) H p (X an ; MC) α γ β H p sing (Dan ; C) H p sing (Xan ; C) H p sing (Xan, D an ; C) H p sing (Dan ; C) ε p H p sing (Xan ; C) ε p H p (X an ; M sing ) (26) Thus we have the following supplement to the Comparison Theorem

43 4114 Supplement to the Comparison Theorem 4101 Proposition 4115 With the notation of Theorem 4101 and definitions 4111, 4113, we have the following commutative diagram, all whose vertical maps are isomorphisms H p (X an, D an ; C) H p dr (Xan, D an ; C) H p dr (Xan ; C) H p dr (Dan ; C) H p (X an, D an ; C) H p (X an ; C) H p (D an ; C) H p (X an ; C) H p (D an ; C) H p sing (Xan, D an ; C) H p sing (Xan ; C) H p sing (Dan ; C) Proof Combine diagrams (24), (25) and (26) 4115 Application of the Supplement to the Comparison Theorem With Proposition 4115, we can compute the isomorphism from the Comparison Theorem 4101 alternatively as analytic derham cohomology analytic derham cohomology α complex cohomology smooth singular cohomology β singular cohomology singular cohomology Why should this be useful? We show that we can easily find the image of a derham cohomology class under the second isomorphism β 1 α by computing a few integrals First, we define smooth singular homology H as the dual of H (cf definitions 4111, 4113), or equivalently: Definition 4116 (Smooth singular homology) For X an, D an as usual (see Subsection 41), we define smooth singular homology groups H (X an ; C) := h C (X an ; C) H (D an ; C) := h tot I = +1 i C (D an I ; C) H (X an, D an ; C) := h mapping cone ( tot I = +1 i C (D an I ; C) C (X an ; C) ) This gives immediately an isomorphism between singular homology and its smooth version Now if [ω] is a derham cohomology class in the image of h Γ(X an ; Ω X an) H dr (Xan ; C), represented by a differential form ω Γ(X an ; Ω X an), we can find the image (β 1 α)[ω] of [ω] in H sing (Xan ; C) as follows: Let {γ 1,, γ t } be a basis of H (X an ; C) and 40

44 {γ 1,, γ t } its dual Then {β 1 γ 1,, β 1 γ t } is a basis of H sing (Xan ; C) and we can express (β 1 α)[ω] in terms of this basis (β 1 α)[ω] = = = t βγ i, (β 1 α)[ω] β 1 γi i=1 t γ i, α[ω] β 1 γi i=1 t ( ω i=1 γ i ) β 1 γi, where, denotes the homology-cohomology-pairing for singular and smooth singular (co)homology Thus, if [ω] H dr (Xan ; C) is a cohomology class represented by a differential form ω Γ(X an ; Ω X an), we can find its image (β 1 α)[ω] in H sing (Xan ; C) by computing the integrals γ i ω for i = 1,, t For an application, see Example 4121 The case of a divisor and the relative case can be dealt with analogously The only difference is that a cohomology class [ω] in the image of or h Γ(D an ; Ω D an) H dr (Dan ; C) h Γ(X an ; Ω X an ;D an) H dr (Xan, D an ; C) is represented by a formal sum I ω I of differential forms ω I living on either X an or one of the various DI an Similarly, a homology class γ H (D an ; C) or γ H (X an, D an ; C) is represented by a formal sum I Γ I of smooth simplicial chains Γ I living on X an and the DI an s In computing the pairing γ, [ω], a summand γ J, [ω I ] can only give a non-zero contribution if the domains of definition coincide DI an = DJ an (Here ω I lives on DI an with the convention that ω corresponds to X an ) 412 A Motivation for the Theory of Periods Assume we are dealing with a smooth variety X 0 defined over Q and a normalcrossings-divisor D 0 on X 0 We denote the base change to C by X and D, respectively It seems a natural question to ask whether the isomorphism H dr (X 0, D 0 /Q) Q C =H dr (X, D/C) H sing(x an, D an ; Q) Q C =H sing(x an, D an ; C) is induced by a natural isomorphism of the form H dr (X 0, D 0 /Q) H sing(x an, D an ; Q) The following example shows that the answer to this question is no and this negative result may be regarded as the starting point of the theory of periods (This example uses absolute (co)homology, but taking D 0 := we can reformulate it for relative (co)homology) 41

45 Example 4121 Let X 0 := Spec Q[t, t 1 ] = A 1 Q \ {0} be the affine line with point 0 deleted Then the singular homology group H sing 1 (X an ; Q) of X an = C is generated by the unit circle σ := S 1 Hence H 1 sing (Xan ; Q) is generated by its dual σ For the algebraic derham cohomology group H 1 dr (X 0/Q) of X 0 we get H 1 dr (X 0/Q) = H 1 (X 0 ; Ω X 0 /Q ) ( ) = h 1 Γ(X 0 ; Ω X 0 /Q ) = coker(d : Q[t, t 1 ] Q[t, t 1 ]dt) = Q dt t Here we used at ( ) that the sheaves Ω p X 0 /Q are quasi-coherent, hence acyclic for the global section functor Γ(X 0 ;?) by [Ha, Thm III35, p 215], since X 0 is affine Under the isomorphism H 1 dr (X/C) H 1 sing (X; C) the generator dt t is mapped to 2πi σ because of dt σ t = 2πi But 2πi is not a rational number, hence the isomorphism H 1 dr (X/C) H1 sing(x; C) is not induced by a map H 1 dr (X 0/Q) H 1 sing(x; Q) The complex number 2πi is our first example of a period 42

46 5 Definitions of Periods A period is to be thought of as an integral that occurs in a geometric context In their papers [K] and [KZ], Kontsevich and Zagier list various ways of how to define a period It is stated in their papers without reference, that all these variants give the same definition We give a partial proof of this statement in the Period Theorem First Definition of a Period: Pairing Periods Let X 0 be a smooth variety defined over Q and D 0 a divisor with normal crossings on X 0 We denote by X an and D an the complex analytic spaces (cf Subsection 21) associated to the base change to C of X 0 and D 0 From the discussion of derham cohomology in the previous sections (especially Proposition 351 and Theorem 4101), we see that inside the C-vector space H p dr (Xan, D an ; C) there sits a Q-lattice H p dr (X 0, D 0 /Q) of full rank Since H dr (Xan, D an ; C) (4101) = H sing(x an, D an ; C) = H sing(x an, D an ; Q) Q C, the classical perfect pairing, : H sing (X an, D an ; Q) H sing(x an, D an ; Q) Q gives us a new non-degenerated pairing which is natural in (X 0, D 0 ), : H sing (X an, D an ; Q) H dr (X 0, D 0 /Q) C, (27) Definition 511 (Pairing period, short: p-period) The complex numbers γ, ω 0 C for ω 0 H dr (X 0, D 0 /Q), γ H sing (X an, D an ; Q), which appear in the image of the natural pairing (27), are called p-periods The set of all p-periods for all pairs (X 0, D 0 ) will be denoted by P p We call a p-period γ, ω 0 special or a p -period, if ω 0 is contained in the image of h Γ(X 0 ; Ω X 0,D 0 /Q ) H dr (X 0, D 0 /Q) and write P p for the set of all special pairing periods for all X 0, D 0, ω 0, γ Running through bases {ω 1,, ω t } of H dr (X 0, D 0 /Q) and {γ 1,, γ t } of H sing (X an, D an ; Q) yields the so-called period matrix P := ( γ i, ω j ) t i,j=1 of (X 0, D 0 ) Remark 512 As a consequence of the Comparison Theorem 4101, the period matrix must be a square matrix (cf [K, p 63]) In loc cit also a statement about its determinant is made, being a square root of a rational number times a power of 2πi determinant = ± a (2πi) n for a Q, n N 0 An indication of a proof is not given there 43

47 52 Second Definition of a Period: Abstract Periods There exists another definition of a period, which also emphasizes their geometric origin but does not involve algebraic derham cohomology For this definition the following data is needed, X 0 a smooth algebraic variety defined over Q of dimension d, D 0 a divisor on X 0 with normal crossings, ω 0 Γ(X 0 ; Ω d X 0 /Q) an algebraic differential form of top degree, γ H sing d (X an, D an ; Q) a homology class of singular chains on the complex manifold X an with boundary on the divisor D an As usual, we denote by X an and D an the complex analytic spaces (cf Subsection 21) associated to the base change to C of X 0 and D 0 Thus, we are dealing with objects X an, D an and γ of real dimension 2d, 2d 2 and d, respectively The differential form ω 0 on X 0 gives rise to a differential form ω := π ω 0 on the base change X = X 0 Q C, where π : X X 0 is the natural projection Now ω 0 is closed for dimension reasons, hence ω is closed as well We choose a representative Γ α 1 γ of the preimage α 1 γ of γ under the isomorphism (cf Proposition 4115) and define α : H d (Xan, D an ; Q) γ H sing d (X an, D an ; Q) ω := ω, Γ d where the chain Γ d consists of the d-simplices of Γ (that is we ignore those d q 1- simplices of Γ that live on one of the DI an for I = q + 1, q 0 see page 12 for the definition of D I ) Observe that this integral γ ω is well-defined The restriction of ω to some irreducible component Dj an of D an is a holomorphic d-form on a complex manifold of dimension d 1, hence zero Therefore the integral ω evaluates to zero for smooth singular simplices that are supported on D an Now if Γ, Γ are two representatives for α 1 γ, we have Γ d Γ d (Γ d+1) modulo simplices living on some D an I for a smooth singular chain Γ of dimension d+1 Using Stoke s theorem we get ω since ω is closed Γ d Γ C d+1 (Xan, D an ; Q) Γ d ω = (Γ d+1 ) ω = dω = 0, Γ d+1 44

48 Definition 521 (Abstract period, short: a-period) We will call the complex number γ ω the a-period of the quadruple (X 0, D 0, ω 0, γ) and denote the set of all a-periods for all (X 0, D 0, ω 0, γ) by P a Remark 522 This definition was motivated by Kontsevich s discussion of effective periods [K, def 20, p 62], which we will partly quote in subsection 73 Modulo Conjecture 734, abstract and effective periods are essentially the same We can ask whether a different definition of the tuples (X 0, D 0, ω 0, γ) yields more period values A partial answer is given by the following remarks Remark 523 (cf [K, p 62]) For example, we could have considered algebraic varieties defined over Q However, doing so does not give us more period values Proof Any variety X 0 defined over Q is already defined over a finite extension Q of Q, ie there exists a variety X 0 such that X 0 = X 0 Q Q Via we may consider X 0 X 0 Spec Q Spec Q as a variety defined over Q An analogous argument applies to a normal-crossings-divisor D 0 on X 0 By possibly replacing Q by a finite extension field, we can therefore assume wlog that D 0 is the base extension of a normal-crossings-divisor D 0 on X 0 D 0 = D 0 Q Q Also any differential d-form ω 0 Γ(X 0 ; Ω d ) can be considered as a X 0 /Q differential form ω 0 on X 0 by eventually replacing Q by a finite extension Finally, we discuss singular homology classes First observe that Spec C Q Q = Hom C (C Q Q, C) = Hom Q (Q, C) is a finite discrete set of points Thus X is a finite union of disjoint copies of X X = X 0 Q C = X 0 Q Q Q C C C = (X 0 Q C) C ( Q Q C) ) = X Hom Q (Q, C) = X σ : Q C 45

49 Similarly, D = σ D Therefore we have H sing d (X an, D an ; Q) = σ H sing d (X an, D an ; Q) If γ H sing d (X an, D an ; Q) is some singular homology class, we can pick any σ : Q C and find γ := {σ } γ H sing d (X an, D an ; Q) = σ H sing d (X an, D an ; Q) Now which finishes the proof ω = γ γ ω Remark 524 (cf [K, p 62]) The requirement of X to be smooth was also made only for convenience and is not a real restriction We postpone the proof of Remark 524 until the Period Theorem 711 is proved 53 Third Definition of a Period: Naïve Periods For the last definition of a period, we need the notion of semi-algebraic sets Let K be any field contained in R Definition 531 (K-semi-algebraic sets, [Hi2, Def 11, p 166]) A subset of R n is said to be K-semi-algebraic, if it is of the form {x R n f(x) 0} for some polynomial f K[x 1,, x n ] or can be obtained from sets of this form in a finite number of steps, where each step consists of one of the following basic operations: (i) complementary set, (ii) finite intersection, (iii) finite union Denote the integral closure of Q in R by Q Note that Q is a field define Now we can Definition 532 (Naïve periods, short: n-periods) Let G R n be an oriented compact Q-semi-algebraic set which is equidimensional of dimension d, and ω 0 a rational differential d-form on R n having coefficients in Q, which does not have poles on G Then we call the complex number G ω 0 a n-period and denote the set of all n-periods for all G and ω 0 by P n This set P n enjoys additional structure 46

50 Proposition 533 The set P n is a Q-algebra Proof Additive structure: Let G 1 ω 1 and G 2 ω 2 P n be periods with domains of integration G 1 R n 1 and G 2 R n 2 Using the inclusions i 1 : R n 1 = R n 1 {1/2} {0} R n 1 R R n 2 i 2 : R n 2 = {0} { 1/2} R n 2 R n 1 R R n 2, we can write i 1 (G 1 ) i 2 (G 2 ) for the disjoint union of G 1 and G 2 With the projections p j : R n 1 R R n 2 R n j for j = 1, 2, we can lift ω j on R n j to p j ω j on R n 1 R R n 2 For q 1, q 2 Q we get q 1 ω 1 + q 2 ω 2 = q 1 (1/2 + t) p G 1ω 1 + q 2 (1/2 t) p 2ω 2 P n, 1 G 2 i 1 (G 1 ) i 2 (G 2 ) where t is the coordinate of the middle factor R of R n 1 R R n 2 This shows that P n is a Q-vector space Multiplicative structure: In order to show that P n is closed under multiplication, we write p i : R n 1 R n 2 R n i, i = 1, 2 for the natural projections and obtain ( ) ( ) ω 1 ω 2 = p 1ω 1 p 2ω 2 P n G 1 G 2 G 1 G 2 by the Fubini formula Remark 534 The Definition 532 was inspired by the one given in [KZ, p 772] and Definition A [naïve] period is a complex number whose real and imaginary part are values of absolutely convergent integrals of rational functions with rational coefficients, over domains in R n given by polynomial inequalities with rational coefficients We will not work with this definition and use the modified version 532 instead, since this gives us more flexibility in proofs From the statements made in [KZ, p 773] it would follow that both definitions of a naïve period agree Examples of naïve periods are 2 dt = ln 2, 1 t t 2 dt = π and + 1 dt 2 G s = dt = elliptic integrals, 1 t for G := {(t, s) R 2 1 t 2, 0 s, s 2 = t 3 + 1} As a problematic example, we consider the following identity 47

51 Proposition 535 (cf [K, p 62]) We have dt 1 dt 2 = ζ(2) (28) (1 t 1 ) t t 1 t 2 1 Proof This equality follows by a simple power series manipulation: For 0 t 2 < 1, we have t2 dt 1 t n 2 = log(1 t 2 ) = 1 t 1 n Let ε > 0 The power series we get 0 t 1 t 2 1 ε n=1 dt 1 dt 1 ε 2 = (1 t 1 ) t 2 n=1 t n 1 2 n converges uniformly for 0 t 2 1 ε and 0 n=1 t n 1 2n dt 2 = n=1 (1 ε) n n 2 Applying Abel s Theorem [Fi, vol 2, XII, 438, 6, p 411] at ( ), using n=1 1 < n 3 gives us 0 t 1 t 2 1 dt 1 dt 2 = lim (1 t 1 ) t 2 ε 0 (1 ε) n n=1 n 2 ( ) = n=1 1 n 2 = ζ(2) Equation (28) is not a valid representation of ζ(2) as an integral for a naïve period in t our sense, because the pole locus {t 1 = 1} {t 2 = 0} of 1 t 2 (1 t 1 ) t 2 is not disjoint with the domain of integration {0 t 1 t 2 1} But (28) gives a valid period integral according to the original definition of [KZ, p 772] see Remark 534 We will show in Example 84 how to circumvent this difficulty We will prove in the Period Theorem 711, that special pairing, abstract and naïve periods are essentially the same, ie P p = P a = P n, but have to make a digression on the triangulation of varieties first Note that in Subsection 73 we will give a fourth definition of a period 48

52 6 Triangulation of Algebraic Varieties If X 0 is a variety defined over Q we may ask whether any singular homology class γ H sing (X an ; Q) can be represented by an object described by polynomials This is indeed the case: For a precise statement we need several definitions The result will be formulated in Proposition Semi-algebraic Sets We already defined Q-semi-algebraic sets in Definition 531 Definition 611 ( Q-semi-algebraic map [Hi2, p 168]) A continuous map f between Q-semi-algebraic sets A R n and B R m is said to be Q-semi-algebraic if its graph Γ f := {( a, f(a) ) a A } A B R n+m is Q-semi-algebraic Example 612 Any polynomial map f : A B (a 1,, a n ) (f 1 (a 1,, a n ),, f m (a 1,, a n )) between Q-semi-algebraic sets A R n and B R m with f i Q[x 1,, x n ] for i = 1,, m is Q-semi-algebraic, since it is continuous and its graph Γ f R n+m is cut out from A B by the polynomials y i f i (x 1,, x n ) Q[x 1,, x n, y 1,, y m ] for i = 1,, m (29) We can even allow f to be a rational map with rational component functions f i Q(x 1,, x n ), i = 1,, m as long as none of the denominators of the f i vanish at a point of A The argument remains the same except that the expression (29) has to be multiplied by the denominator of f i Fact 613 ([Hi2, Prop II, p 167], [Sb, Thm 3, p 370]) By a result of Seidenberg-Tarski, the image (respectively preimage) of a Q-semi-algebraic set under a Q-semi-algebraic map is again Q-semi-algebraic As the name suggests any algebraic set should be in particular Q-semi-algebraic Lemma 614 Let X 0 be an algebraic variety defined over Q Then we can regard the complex analytic space X an (cf Subsection 21) associated to the base change X = X 0 eq C as a bounded Q-semi-algebraic subset X an R N (30) for some N Moreover, if f 0 : X 0 Y 0 is a morphism of varieties defined over Q, we can consider f an : X an Y an as a Q-semi-algebraic map 49

53 Proof First step X 0 = QP n : Consider CP n an with homogenous coordinates x 0,, x n, which we split as x m = a m + ib m with a m, b m R in real and imaginary part, and R N, N = 2(n + 1) 2, with coordinates {y kl, z kl } k,l=0,,n We define a map ψ : CPan n [x 0 ::x n] R N (y 00,z 00,,y nn,z nn) ( ) Re x k x l Im x [x 0 : : x n ], k x n m=0 x m }{{ 2, l n m=0 } x m 2, }{{} y kl z kl ( ) a k a l + b k b l b [a 0 + ib 0 : : a n + ib n ], k a l a k b n m=0 a2 m + b 2, l n m m=0 }{{} a2 m + b 2, m }{{} y kl z kl Rewriting the last line (with the convention 0 cos( indeterminate angle ) = 0) as ( [r 0 e iφ 0 : : r n e iφn ], r kr l cos(φ k φ l ) n, r ) kr l sin(φ k φ l ) n, m=0 r2 m m=0 r2 m (31) shows that ψ is injective: Assume ψ ( [r 0 e iφ 0 : : r n e iφn ] ) = (y 00, z 00,, y nn, z nn ) where r k 0, or equivalently y kk 0, for a fixed k We find r l ykl 2 = + z2 kl, and r k y kk arctan(z kl /y kl ) if y kl 0, π/2 if y kl = 0, z kl > 0, φ k φ l = indeterminate if y kl = z kl = 0, π/2 if y kl = 0, z kl < 0; that is the preimage of (y 00, z 00,, y nn, z nn ) is uniquely determined Therefore we can consider CP n an via ψ as a subset of R N It is bounded since it is contained in the unit sphere S N 1 R N We claim that ψ(cp n an) is also Q-semialgebraic The composition of the projection π : R 2(n+1) \ {(0,, 0)} CP n an (a 0, b 0,, a n, b n ) [a 0 + ib 0 : : a n + ib n ] with the map ψ is a polynomial map, hence Q-semi-algebraic by Example 612 Thus is Q-semi-algebraic by Fact 613 im ψ π = im ψ R N 50

54 Second step (zero set of a polynomial): We use the notation V (g) := {x CP n an g(x) = 0} for g C[x 0,, x n ] homogenous, and W (h) := {t R N h(t) = 0} for h C[y 00,, z nn ] Let X an = V (g) for some homogenous g Q[x 0,, x n ] Then ψ(x an ) R N is a Q-semi-algebraic subset, as a little calculation shows Setting for k = 0,, n g k := g(x x k ) = g(x 0 x k,, x n x k ) = g ( (a 0 a k + b 0 b k ) + i(b 0 a k a 0 b k ),, (a n a k + b n b k ) + i(b n a k a n b k ) ), where x j = a j + ib j for j = 0,, n, and we obtain h k := g(y 0k + iz 0k,, y nk + iz nk ), ψ(x an ) = ψ(v (g)) n = ψ(v (g k )) = = k=0 n ψ(cpan) n W (h k ) k=0 n ψ(cpan) n W (Re h k ) W (Im h k ) k=0 Final step: We can choose an embedding thus getting X 0 QP n, X an CP n an Since X 0 is a locally closed subvariety of QP n, X an can be expressed in terms of subvarieties of the form V (g) with g Q[x 0,, x n ], using only the following basic operations (i) complementary set, (ii) finite intersection, (iii) finite union Now Q-semi-algebraic sets are stable under these operations as well and the first assertion is proved Second assertion: The first part of the lemma provides us with Q-semi-algebraic inclusions ψ : X an φ : Y an CPan n R N, x=[x 0 ::x n] (y 00,z 00,,y nn,z nn) CPan m R M, u=[u 0 ::u m] (v 00,w 00,,v mm,w mm) 51

55 and a choice of coordinates as indicated We use the notation V (g) := {(x, u) CP n an CP m an g(x, u) = 0}, for g C[x 0,, x n, u 0,, u m ] homogenous in both x and u, W (h) := {t R N+M h(t) = 0}, for h C[y 00,, z nn, v 00,, w mm ] and Let {U i } be a finite open affine covering of X 0 such that f 0 (U i ) satisfies f 0 (U i ) does not meet the hyperplane {u j = 0} QP m for some j, and f 0 (U i ) is contained in an open affine subset V i of Y 0 This is always possible, since we can start with the open covering Y 0 {u j 0} of Y 0, take a subordinated open affine covering {V i }, and then choose a finite open affine covering {U i } subordinated to {f 1 (V i )} Now each of the maps f i := f an Ui : U an i Y an has image contained in Vi an and does not meet the hyperplane {u CPan m u j = 0} for an appropriate j f i : U an i V an i Being associated to an algebraic map between affine varieties, this map is rational [ g 0 f i : x (x) g j 1 g 0 : : (x) (x) g j 1 (x) : 1 : g j+1 (x) ] j g j+1 (x) : : g m (x) g m(x), with g k, g k Q[x 0,, x n ], k = 0,, ĵ,, m Since the graph Γ fan of f an is the finite union of the graphs Γ fi of the f i, it is sufficient to prove that (ψ φ)(γ fi ) is a Q-semi-algebraic subset of R N+M Now Γ fi = (U an i Vi an ) n k=0 k j V ( yk so all we have to deal with is g k (x) ) y j g k (x) = (U an i V (y k g k (x) y jg k (x)) Again a little calculation is necessary Setting Vi an ) g pq := u k u q g k (x x p) u j u q g k (x x p) = u k u q g k (x 0x p,, x n x p ) u j u q g k (x 0x p,, x n x p ) = ( (c k c q + d k d q ) + i(d k c q c k d q ) ) g k n k=0 k j V (y k g k (x) y jg k (x)), ( (a0 a p + b 0 b p ) + i(b 0 a p a 0 b p ),, (a n a p + b n b p ) + i(b n a p a n b p ) ) ( (c j c q + d j d q ) + i(d j c q c j d q ) ) g k( (a0 a p + b 0 b p ) + i(b 0 a p a 0 b p ),, (a n a p + b n b p ) + i(b n a p a n b p ) ), where x l = a l + ib l for l = 0,, n, u l = c l + id l for l = 0,, m, and h pq := (v kq +iw kq )g k (y 0p+iz 0p,, y np +iz np ) (v jq +iw jq )g k (y 0p+iz 0p,, y np +iz np ), 52

56 we obtain (ψ φ) ( V ( y k g k (x) y jg k (x))) = n m = (ψ φ)(v (g pq )) = = p=0 q=0 n p=0 q=0 n p=0 q=0 m (ψ φ)(u an m (ψ φ)(u an i i V an j ) W (h pq ) V an j ) W (Re h pq ) W (Im h pq ) 62 Semi-algebraic Singular Chains We need further prerequisites in order to state the announced Proposition 622 Definition 621 (Open simplex, [Hi2, p 168]) By an open simplex we mean the interior of a simplex (= the convex hull of r + 1 points in R n which span a r-dimensional subspace) For convenience, a point is considered as an open simplex as well The notation std d will be reserved for the closed standard simplex spanned by the standard basis {(0,, 0, 1, 0,, 0) i = 1,, d + 1} of R d+1 i Consider the following data ( ): X 0 a variety defined over Q, D 0 a divisor on X 0 with normal crossings, and finally γ H sing p (X an, D an ; Q), p N 0 As usual, we have denoted by X an (resp D an ) the complex analytic space associated to the base change X = X 0 eq C (resp D = D 0 eq C) By Lemma 614, we may consider both X an and D an as bounded Q-semi-algebraic subsets of R N We are now able to formulate our proposition Proposition 622 With data ( ) as above, we can find a representative of γ that is a rational linear combination of singular simplices each of which is Q-semi-algebraic The proof of this proposition relies on the following proposition due to Lojasiewicz which has been written down by Hironaka Proposition 623 (Triangulation of Q-semi-algebraic sets, [Hi2, p 170]) For {X i } a finite system of bounded Q-semi-algebraic sets in R n, there exists a simplicial decomposition R n = j j by open simplices j and a Q-semi-algebraic automorphism κ : R n R n such that each X i is a finite union of some of the κ( j) 53

57 Note 624 Although Hironaka considers R-semi-algebraic sets, it has been checked by the author, that we can safely replace R by Q in his article (including the fact he cites from [Sb]) The only problem that could possibly arise concerns a good direction lemma : Lemma 625 (Good direction lemma for R, [Hi2, p 172], [KB, Thm 5I, p 242]) Let Z be a R-semi-algebraic subset of R n, which is nowhere dense A direction v RP n 1 is called good, if any line l in R n parallel to v meets Z in a discrete (maybe empty) set of points; otherwise v is called bad Then the set B(Z) of bad directions is a Baire category set in RP n 1 This gives immediately good directions v RP n 1 \ B(Z), but not necessarily v QP n 1 \ B(Z) However, in Remark 21 of [Hi2], which follows directly after the lemma, the following statement is made: If Z is compact, then B(Z) is closed in RP n 1 In particular QP n 1 \ B(Z) will be non-empty Since we only consider bounded Q-semi-algebraic sets Z, we may take Z := Z (which is compact by Heine- Borel), and thus find a good direction v QP n 1 \ B(Z ) using B(Z ) B(Z) Lemma 626 (Good direction lemma for Q) Let Z be a bounded Q-semialgebraic subset of R n, which is nowhere dense Then the set QP n 1 \ B(Z) of good directions is non-empty Proof of Proposition 622 Applying Proposition 623 to the two-element system of Q-semi-algebraic sets X an, D an R N, we obtain a Q-semi-algebraic decomposition R N = j j of R N by open simplices j and a Q-semi-algebraic automorphism κ : R N R N We write j for the closure of j The sets K := { j κ( j) X an } and L := { j κ( j) D an } can be thought of as finite simplicial complexes, but built out of open simplices instead of closed ones We define their geometric realizations K := j and L := j j K j L Then Proposition 623 states that κ maps the pair of topological spaces ( K, L ) homeomorphically to (X an, D an ) Easy case: If X 0 is complete, so is X (by [Ha, Cor II48(c), p 102]), hence X an and D an will be compact [Ha, B1, p 439] In this situation K := { j κ( j ) X an } and L := { j κ( j ) D an } are (ordinary) simplicial complexes, whose geometric realizations coincide with those of K and L, respectively In particular H simpl (K, L; Q) = H sing ( K, L ; Q) = H sing ( K, L ; Q) = H sing (X an, D an ; Q) (32) 54

58 Here H simpl (K, L; Q) denotes simplicial homology of course We write γ simpl H simpl p (K, L; Q) and γ sing H sing p ( K, L ; Q) for the image of γ under this isomorphism Any representative Γ simpl of γ simpl is a rational linear combination Γ simpl = j a j j, a j Q of oriented closed simplices j K We can choose orientation-preserving affinelinear maps of the standard simplex std p to j These maps yield a representative σ j : std p j for j Γ simpl Γ sing := j a j σ j of γ sing Composing with κ yields Γ := κ Γ sing γ, where Γ has the desired properties In the general case, we perform a barycentric subdivision B on K twice (once is not enough) and define K and L not as the closure of K and L, but as follows (see Figure 2) K := { B 2 (K) and K }, L := { B 2 (33) (K) and L } κ 1 (X an ) j K j K j Intersection of κ 1 (X an ) with a closed 2-simplex j, where we assume that part of the boundary j does not belong to κ 1 (X an ) Open simplices of K contained in j Figure 2: Definition of K Intersection of K with j (the dashed lines show the barycentric subdivision) The point is that the pair of topological spaces ( K, L ) is a strong deformation retract of ( K, L ) Assuming this we see that in the general case with K, L defined as in (33), the isomorphism (32) still holds and we can proceed as in the easy case to prove the proposition We define the retraction map ρ : ( K [0, 1], L [0, 1]) ( K, L ) as follows: Let j K be an open simplex which is not contained in the boundary of any other simplex of K and set inner := j K, outer := j \ K 55

59 Note that inner is closed For any point p outer the ray c p from the center c of j through p leaves the set inner at a point q p, ie c p inner equals the line segment c q p ; see Figure 3 The map ρ j : j [0, 1] j { p if p inner, (p, t) q p + t (p q p ) if p outer retracts j onto inner Now these maps ρ j glue together to give the desired homotopy ρ Figure 3: Definition of q p We want to state one of the intermediate results of this proof explicitly Corollary 627 Let X 0 and D 0 be as above Then the pair of topological spaces (X an, D an ) is homotopic equivalent to a pair of (realizations of) simplicial complexes ( X simpl, D simpl ) 56

60 7 Periods Revisited 71 Comparison of Definitions of Periods We are now ready to prove the theorem announced in the introduction Main Theorem 711 (Period theorem) It holds P p = P a = P n, that is the following three definitions of a period given in Section 5 coincide special pairing periods (cf Definition 511), abstract periods (cf Definition 521), and naïve periods (cf Definition 532) Note that we will give a fourth definition of a period in Subsection 73 Proof P a P p: Let γ ω P a be an abstract period coming from (X 0, D 0, ω 0, γ) We want to understand ω 0 as an element of Γ(X 0 ; Ω d X 0,D 0 /Q ) = Γ(X 0; Ω d X 0 /Q ) Ω d 1 Γ(D 0 ; D 0 /Q ) via Γ(X 0; Ω d X 0 /Q ) Γ(X 0; Ω d X 0,D 0 /Q ), where d := dim X 0 Then ω 0 Γ(X 0 ; Ω X 0,D 0 /Q ) is a cocycle, since ω 0 is closed (for dimension reasons) and its restriction to each of the irreducible components D j of D 0 is zero (also for dimension reasons) Thus we can consider [ω 0 ] via h Γ(X 0 ; Ω X 0,D 0 /Q ) H dr (X 0, D 0 /Q) as an algebraic derham cohomology class and find (cf Subsection 4115) ω = γ, [ω 0 ] P p γ P p P n : As explained in Subsection 4115, a special pairing period is a sum of integrals γ, ω 0 P p I γ I, ω I = I γ I ω I, where ω I H dr (D I/Q) and γ I H (D I ; Q) Since P n is closed under addition by Proposition 533, it suffices to prove γ I ω I P n for all index sets I We choose an embedding D I QP n (x 0 ::x n) and equip QP n with coordinates as indicated Lemma 614 provides us with a map ψ : CP n an R N such that DI an and CPan n become Q-semi-algebraic subsets of R N Then, by Proposition 622, ψ γ I has a representative which is a rational linear combination of singular simplices Γ i, each of which is Q-semi-algebraic 57

61 We set G i := im Γ i and are left to prove ψ 1 (G i ) ω I P n This will be clear as soon as we find a rational differential form ω I on RN such that ψ ω I = ω I, since then ω I = ψ ω I = ω I P n G i ψ 1 (G i ) ψ 1 (G i ) After eventually applying a barycentric subdivision to Γ i, we may assume wlog that there exists a hyperplane in CPan, n say {x 0 = 0}, which does not meet ψ 1 (G i ) Furthermore, we may assume that ψ 1 (G i ) lies entirely in U an for U 0 an open affine subset of D I {x 0 0} (As usual, U an denotes the complex analytic space associated to the base change to C of U) The restriction of ω I to the open affine subset can be represented in the form (cf [Ha, II84A, II821, II82A]) ( ) ( ) xj1 xjd f J (x 0,, x n ) d d J =d with f J (x 1,, x n ) Q(x 0,, x n ) being homogenous of degree zero This expression defines a rational differential form on all of QP n with coefficients in Q and it does not have poles on ψ 1 (G i ) We construct the rational differential form ω I on RN with coefficients in Q(i) as follows ω I := J =d x 0 ( f J 1, y 10 + iz 10,, y ) n0 + iz n0 d y 00 + iz 00 y 00 + iz 00 ( yj1 0 + iz j1 0 y 00 + iz 00 x 0 ) d ( yjd 0 + iz jd 0 y 00 + iz 00 where we have used the notation from the proof of Lemma 614 Using the explicit form of ψ given in this proof, we obtain ( ψ f J 1, y 10 + iz 10,, y ) ( n0 + iz n0 x0 x 0 = f J y 00 + iz 00 y 00 + iz 00 x 0 2, x 1x 0 x 0 2,, x ) nx 0 x 0 2 = f J (x 0, x 1,, x n ) and ( ) ( ) ψ yj0 + iz j0 xj x 0 d = d y 00 + iz 00 x 0 2 = d This shows that ψ ω I = ω I and we are done P n P a : In this part of the proof, we will use objects over various base fields: We will use subscripts to indicate which base field is used: subscript base field 0 Q 1 Q R none R C ( xj x 0 ) ), 58

62 Furthermore, we fix an embedding Q C Let G ω R P n be a naïve period with G R n an oriented Q-semi-algebraic set, equidimensional of dimension d, and ω R a rational differential d-form on R n with coefficients in Q, which does not have poles on G The Q-semi-algebraic set G R n is given by polynomial inequalities and equalities By omitting the inequalities but keeping the equalities in the definition of G, we see that G is supported on (the set of R-valued points of) a variety Y R A n R of same dimension d This variety Y R is already defined over Q Y R = Y 0 eq R for a variety Y 0 A n e Q over Q Similarly the boundary G of G is supported on a variety E R, likewise defined over Q E R = E 0 eq R Note that E 0 is a divisor on Y 0 By eventually enlarging E 0, we may assume wlog that E 0 contains the singular locus of Y 0 In order to obtain an abstract period, we need smooth varieties The resolution of singularities according to Hironaka [Hi1] provides us with a Cartesian square where Ỹ0 is smooth and quasi-projective, π 0 is proper, surjective and birational, and Ẽ0 is a divisor with normal crossings Ẽ 0 Ỹ0 π 0 (34) E 0 Y 0 In fact, π 0 is an isomorphism away from Ẽ0 since the singular locus of Y 0 is contained in E 0 π 0 eu0 : Ũ0 U 0 (35) with Ũ0 := Ỹ0 \ Ẽ0 and U 0 := Y 0 \ E 0 We apply the functor an (associated complex analytic space cf Subsection 21) to the base change to C of the map π 0 : Ỹ0 Y 0 and obtain a projection π an : Ỹ an Y an We want to show that the strict transform of G G := π 1 an (G \ E an ) Ỹ an 59

63 can be triangulated Since CP n an is the projective closure of C n, we have C n CP n an and thus get an embedding Y an C n CP n an We also choose an embedding Ỹ an CP m an for some m N Using Lemma 614, we may consider both Y an and Ỹ an as Q-semialgebraic sets via some maps ψ : Y an CP n an R N, ψ : Ỹ an CP m an R M and In this setting, the induced projection π an : Ỹ an Y an becomes a Q-semi-algebraic map The composition of ψ with the inclusion G Y an is a Q-semi-algebraic map; hence G R N is Q-semi-algebraic by Fact 613 Since E an is also Q-semi-algebraic via ψ, we find that G \ E an is Q-semi-algebraic Again by Fact 613, πan 1 (G \ E an ) R M is Q-semi-algebraic Thus G R M, being the closure of a Q-semi-algebraic set, is Q-semi-algebraic From Proposition 622, we see that G can be triangulated G = j j, (36) where the j are (homeomorphic images of) d-dimensional simplices Our next aim is to define an algebraic differential form ω 1 replacing ω R We first make a base change in (34) from Q to Q and obtain Ẽ 1 Ỹ1 π 1 E 1 Y 1 The differential d-form ω R can be written as ω R = f J (x 1,, x n ) dx j1 dx jd, (37) J =d where x 1,, x n are coordinates of R n and f J Q(x 1,, x n ) We can use equation (37) to define a differential form ω 1 on A n Q ω R = f J (x 1,, x n ) dx j1 dx jd, J =d where now x 1,, x n denote coordinates of A n The pole locus of ω Q 1 gives us a variety Z 1 A n We set Q The restriction ω 1 X1 pullback X 1 := Y 1 \ Z 1, D 1 := E 1 \ Z 1, and X 1 := π 1 1 (X 1), D1 := π 1 1 (D 1) of ω 1 to X 1 is a (regular) algebraic differential form on X 1 ; the ω 1 := π 1(ω 1 X1 ) 60

64 is an algebraic differential form on X 1 We consider the complex analytic spaces X an, D an, Z an associated to the base change to C of X1, D1, Z 1 Since ω R has no poles on G, we have G Z an = ; hence G πan 1 (Z an ) = This shows G X = Ỹ \ π 1 an (Z an ) Since G is oriented, so is πan 1 (G\E an ), because π an is an isomorphism away from E an Every d-simplex j in (36) intersects πan 1 (G \ E an ) in a dense open subset, hence inherits an orientation As in the proof of Proposition 622, we choose orientationpreserving homeomorphisms from the standard d-simplex std d to j These maps sum up to a singular chain σ j : std d j Γ = j σ j C sing d ( X an ; Q) It might happen that the boundary of the singular chain Γ is not supported on G Nevertheless, it will always be supported on D an : The set πan 1 (G \ E an ) is oriented and therefore the boundary components of j that do not belong to G cancel if they have non-zero intersection with πan 1 (G \ E an ) Thus Γ gives rise to a singular homology class γ H sing d ( X an, D an ; Q) We denote the base change to C of ω 1 and ω 1 by ω and ω, respectively Now ω R = ω = ω G G G U an (35) = π ω = ω π 1 (G U an ) eg U ean = ω = ω = ω P a is an abstract period for the quadruple ( X 1, D 1, ω 1, γ) by Remark 523 Now Remark 524 is an easy corollary of the Period Theorem 711 Proof of Remark 524 Let X 0 be a possibly non-regular variety defined over Q of dimension d and D 0 a Cartier divisor on X 0 Furthermore, let ω 0 Ω d X 0 and γ H sing d (X an, D an ; Q) Exactly the same argumentation used for D I ω I in proving P p P n shows that γ ω P n Because of P n = P a, there must exists a smooth quadruple ( X 0, D 0, ω 0, γ) as described in the definition of an abstract period (cf Definition 521), which gives the same period value Remark 712 It might be expected that also P p = P p holds The following ideas make this conjecture probable Let γ, ω 0 be a pairing period for (X 0, D 0 ) If X 0 is affine, then the complex Ω X 0,D 0 /Q consists of quasi-coherent sheaves by [Ha, Thm III35, p 215] Hence we have a surjection H Γ(X 0 ; Ω X 0,D 0 /Q ) H dr (X 0, D 0 /Q), that is, only special periods appear eg eγ eγ 61

65 If X 0 is not affine, one might try to break up γ into subchains γ j with Q-semi-algebraic representatives Γ j that are already supported on Uj an for open affine subsets i : U j X 0 Finding a divisor D j on U j such that Γ j Dj an would give us a singular chain [i 1 Γ j ] H sing (Uj an, D an j ; Q) and so would allow us to reduce the general case to the affine one γ j, ω 0 = [i 1 Γ j ], i ω 0 If γ and ω 0 are top-dimensional of real dimension 2d, the procedure described above will definitely not work: A boundary Γ j has real codimension one, but D an has real codimension two for D 0 a divisor on X 0 On the other hand, the top-dimensional case is simpler, since the relative algebraic derham cohomology group involved is in fact an absolute one H 2d dr (X 0, D 0 /Q) = H 2d dr (X 0/Q), because of the long exact sequence in algebraic derham cohomology and the vanishing of H p dr (D 0/Q) for p dim R (D an ) = 2d 2 Now, generalizing a result of Huisman [Hui, Thm 51, p 7] to varieties over Q(i), we find that every algebraic derham cohomology class in H dr( X0 Q Q(i)/Q(i) ) can be realized by a rational differential form on π Q(i)/Q X 0 with coefficients in Q, where π Q(i)/Q X 0 is the restriction of scalars according to Weil with respect to the field extension Q(i)/Q of X 0 The set P := P a = P p = P n is not only a Q-algebra, but, at least conjecturally, comes with a triple coproduct imposing on Spec P[ 1 2πi ] the structure of a torsor, this notion being defined as follows 72 Torsors Heuristically, a torsor is a group that has forgotten its identity, more formally: Definition 721 (Torsor, [K, p 61]) A torsor is a non-empty set X together with a map (,, ) : X X X X satisfying (i) (a, a, c) = c (ii) (a, b, b) = a (iii) ((a, b, c), d, e) = (a, (b, c, d), e) = (a, b, (c, d, e)) Example 722 A group G becomes a torsor by setting (,, ) : G G G G (a, b, c) a b 1 c 62

66 Conversely, a torsor X becomes a group by choosing a distinguished element e X ( the identity ) and setting a b := (a, e, b) a 1 := (e, a, e) Example 723 The fibre X of a principal G-bundle is a torsor: For any a, b, c X we can work out the difference of a and b, which is an element g G, and let g act on c in order to define (a, b, c) Example 724 Let C be a category and E, F two isomorphic objects of C The set of isomorphisms Iso(E, F ) comes naturally with a right action of the automorphism group Aut(E) and a left action of Aut(F ); both actions being simply transitive This can be encoded in a single map, thus turning Iso(E, F ) into a torsor: Iso(E, F ) Iso(E, F ) Iso(E, F ) Iso(E, F ) (a, b, c) a b 1 c These examples are generic in the sense that we could have defined a torsor alternatively as a principal homogenous space over a group (Example 723), or as a category containing just two isomorphic objects (Example 724) 73 Fourth Definition of a Period: Effective Periods We pick up our discussion of periods and show how periods could give us a torsor The usual tools for proving identities between integrals are the change of variables and the Stoke s formula These rules are formalized by the following definition Definition 731 (Effective Periods, [K, def 20, p 62]) The Q-vector space P + of effective periods has as generators the quadruples (X 0, D 0, ω 0, γ), as considered in the definition of an abstract period (cf Definition 521), modulo the following relations (linearity in both ω 0 and γ) (X 0, D 0, q 1 ω 0,1 + q 2 ω 0,2, γ) q 1 (X 0, D 0, ω 0,1, γ) + q 2 (X 0, D 0, ω 0,2, γ) for q 1, q 2 Q, (X 0, D 0, ω 0, q 1 γ 1 + q 2 γ 2 ) q 1 (X 0, D 0, ω 0, γ 1 ) + q 2 (X 0, D 0, ω 0, γ 2 ) for q 1, q 2 Q (change of variables) If f : (X 1, D 1 ) (X 2, D 2 ) is a morphism of pairs defined over Q, γ 1 (X1 an, Dan 1 ; Q) and ω 2 Ω d X 2 then H sing d (Stoke s formula) (X 1, D 1, f ω 2, γ 1 ) (X 2, D 2, ω 2, f γ 1 ) Write D 0 = n i=1 D i as union of its prime divisors, then n (X 0, D 0, dω 0, γ) (D 0,, ω 0, γ) = i, i=1(d D j D i, ω 0, γ Di an ), j i where : H sing d (X an, D an ; Q) H sing d 1 (Dan ; Q) is the boundary operator 63 (38)

67 Here the term γ Di an in the Stoke s formula is defined in an ad-hoc fashion: We consider D an as a Q-semi-algebraic set (by Lemma 614) D an R N and pick a representative Γ γ which is a rational combination Γ = k a kγ k of Q-semi-algebraic simplices Γ k (by Proposition 622) Then the images of the Γ k are Q-semi-algebraic sets G k := im Γ k R N by Fact 613 Applying Proposition 623 to the system {G k, Di an } yields triangulations of the G k s, ie we can write G k = l k,l where the k,l are (homeomorphic images of) oriented closed simplices of dimension d 1 whose interiors are disjoint They enjoy the additional property that each k,l lies entirely in one of the Di an (because the same k,l triangulate also the Di an s) The induced triangulation on G i k := G k Di an gives us a singular chain Γ i k on Dan i and hence a singular homology class γ i k Hsing d 1 (Dan i, i j D an i D an j ; Q) which we call γ D an i We will denote the equivalence class represented by the quadruple (X 0, D 0, ω 0, γ) by [X 0, D 0, ω 0, γ] Proposition 732 We have an evaluation morphism ev : P + [X 0, D 0, ω 0, γ] Proof On page 44, we have already seen that this map is well-defined on the level of representatives ẽv : (X 0, D 0, ω 0, γ) ω, so it remains to show that the map ev factors through the relations imposed in Definition 731 (linearity) ẽv(x 0, D 0, q 1 ω 0,1 + q 2 ω 0,2, γ) = q 1 ω 1 + q 2 ω 2 = γ = q 1 ω 1 + q 2 ω 2 = q 1 ẽv(x 0, D 0, ω 0,1, γ) + q 2 ẽv(x 0, D 0, ω 0,2, γ), γ γ ẽv(x 0, D 0, ω 0, q 1 γ 1 + q 2 γ 2 ) = ω = q 1 γ 1 +q 2 γ 2 = q 1 ω + q 2 ω = q 1 ẽv(x 0, D 0, ω 0, γ 1 ) + q 2 ẽv(x 0, D 0, ω 0, γ 2 ), γ 1 γ 2 γ C ω γ 64

68 (change of variables) ẽv(x 1, D 1, f ω 2, γ 1 ) = f ω 2 = γ 1 ω 2 = ẽv(x 2, D 2, ω 2, f γ 1 ), f γ 1 (Stoke s formula) ẽv(x 0, D 0, dω 0, γ) = = i γ dω = γ D an i γ ω = i ω = ẽv(d i, i j D i D j, ω 0, γ D i ) We state some easy properties of P + Proposition 733 (Basic properties of P +, [K, p 63]) 1 The set P + is a Q-algebra 2 Q im(ev) Proof (cf [K, p 63]) First fact: Let [X i, D i, ω i, γ i ] P +, i = 1, 2 and define their product to be [X 1 Q X 2, D 1 Q X 2 X 1 Q D 2, p 1ω 1 p 2ω 2, γ 1 γ 2 ] P +, where p i : X 1 Q X 2 X i, i = 1, 2, are the natural projections formula ( ) ( ) p 1ω 1 p 2ω 2 = ω 1 ω 2, γ 1 γ 2 γ 1 γ 2 By the Fubini ie the map ev becomes a Q-algebra-morphism Second fact: Let α Q, p Q[t] its minimal polynomial and X 0 := Spec Q[t]/(p) = Spec Q(α) We may regard α as a closed point of X via Spec C X C C[t]/(p) α t Any closed point x of X gives an element of zeroth homology {x} H sing 0 (X an ; Q) Now t Ω 0 X 0 and we get ev[x,, t, {α}] = α Note that the Q-algebra morphism ev : P + P is surjective as a consequence of the definition of abstract periods (cf Definition 521) Conjecture 734 ([K, p 62]) It is conjectured that the map ev : P + P is injective, ie that all identities between integrals can be proved using standard rules only 65

69 This would give us a Q-algebra-isomorphism P + = P For the remainder of this section, we assume Conjecture 734 We adjoin formally (2πi) 1 to P + and define a coproduct on P P := P + [ 1 (2πi) ]? = P[ 1 (2πi) ] Definition 735 (The triple coproduct on P, [K, p 63]) Let α := [X 0, D 0, ω 0, γ] be an effective period represented by an quadruple (X 0, D 0, ω 0, γ) We write P for the period matrix of (X 0, D 0 ) as defined in Definition 511 and assume ev(α) = P i,j, the entry in P at position (i, j) We define (P i,j ) := k,l P i,k (P 1 ) k,l P l,j P Q P Q P Here we used Remark 512 about the determinant of P : det P being a period in P times a power of (2πi) Thus P 1 has coefficients in P = P + [ 1 2πi ] =? P[ 1 2πi ] We adopt the convention ( ) Pi,j (2πi) n := P i,k (2πi) n (2πi)n (P 1 ) k,l P l,j (2πi) n for n N 0 k,l Extending the map in a Q-linear way to all of P yields the triple coproduct : P P Q P Q P Remark 736 It is far from being obvious that is well-defined In his paper [K], Kontsevich states that the proof of the correctness of Definition 735 relies on an unpublished result by M Nori [K, Thm 6, p 63] From Nori s theorem, also the following proposition would follow Proposition 737 Modulo Conjecture 734 and Nori s theorem, the triple coproduct on P induces the structure of a torsor on Spec P We calculate a triple coproduct in Example 81 66

70 8 Examples 81 First Example: A 1 C \ {0} First part (cf [K, p 63]): Let X 0 := A 1 Q \ {0} = Spec Q[t, t 1 ] be the affine line with the point 0 deleted and D 0 := {1, α} with α 0, 1 a divisor on X 0 Figure 4: The algebraic pair (X 0, D 0 ) Then the singular homology of the pair (X an, D an ) = (C, {1, α}) (cf Definition 481) is generated by a small loop σ turning counter-clockwise around 0 once and the interval [1, α] In order to compute the algebraic derham cohomology of (X 0, D 0 ) (cf Definition 326), we first note that H dr (X 0, D 0 /Q) = h Γ(X 0 ; Ω X 0,D 0 /Q ), since X 0 is affine and the sheaves Ω p X 0,D 0 /Q are quasi-coherent, hence acyclic for the global section functor by [Ha, Thm III35, p 215] We spell out the complex Γ(X 0 ; Ω X 0,D 0 /Q) in detail (cf page 15) 0 Γ(X 0 ; Ω 1 X 0,D 0 /Q ) = Γ( X 0 ; Ω 1 X 0 /Q j d Γ(X 0 ; O X0 ) = Q[t, t 1 ] ) i O Dj = Q[t, t 1 ]dt Q Q 1 α and observe that the map evaluation-at-a-point is surjective with kernel Q[t, t 1 ] Q Q 1 α f(t) ( f(1), f(α) ) (t 1)(t α)q[t, t 1 ] = span Q {t n+2 (α + 1)t n+1 + αt n n Z} Differentiation maps this kernel to span Q {(n + 2)t n+1 (n + 1)(α + 1)t n nαt n 1 n Z}dt 67

71 Therefore we get H 1 dr (X 0, D 0 ; Q) = Γ(X 0 ; Ω X0,D 0 /Q) / Γ(X 0 ; O X0 ) = Q[t, t 1 ]dt Q Q / d(q[t, t 1 ]) 1 α = Q[t, t 1 ]dt/ span Q {(n + 2)t n+1 (n + 1)(α + 1)t n nαt n 1 }dt By the last line, we see that the class of t n dt in H 1 dr (X 0, D 0 ; Q) for n 1 is linearly dependent of t n 1 dt and t n 2 dt, and t n+1 dt and t n+2 dt, hence linearly dependent of dt t and dt by an induction argument Hence H1 dr (X 0, D 0 ; Q) is spanned by dt t and 1 α 1 dt We obtain the following period matrix P for (X 0, D 0 ) (cf Definition 511) 1 α 1 dt dt t [1, α] 1 ln α σ 0 2πi Let us the compute the triple coproduct of ln α (cf Definition 735) from this example We have ( ) 1 P 1 ln α 2πi = 1 0 2πi and thus get for the triple coproduct (cf [K, p 63]) (39) (ln α) = ln α 1 ln α 2πi 2πi 1 2πi 2πi ln α (40) Second part: Now consider the degenerate configuration α = 1, ie D = 2 {1} with D 1 = D 2 = {1} Although D 0 is not a divisor with normal crossings anymore, the machinery developed in Section 3 still works Figure 5: The algebraic pair (X 0, D 0 ) Repeating the calculations done for the general case gives H 1 dr (X 0, D 0 /Q) = Q dt t Q (1 D D2 ), where c Di is the constant function equal to c on the irreducible component D i of D, i = 1, 2 Computing the smooth singular cohomology group H 1 (X an, D an ; Q) (cf Definition 4113) in a similar fashion gives H 1 (X an, D an ; Q) = Q σ Q (D 1 D 2 ), and we find that the period matrix of (X 0, D 0 ) is precisely the limit of (39) for α 1 68

72 Third part: Finally, we let 1 D1 + 0 D2 dt t D 1 D σ 0 2πi D 0 := {1, α, β} with α 0, 1 and β 0, 1, α, but keep X 0 := A 1 Q \ {0} = Spec Q[t, t 1 ] Figure 6: The algebraic pair (X 0, D 0 ) Then H sing 1 (X an, D an ; Q) is generated by the loop σ from the first part and the intervals [1, α] and [α, β] Hence the differential forms dt t, dt and 2t dt give a basis of H 1 dr (X 0, D 0 ; Q): If they were linearly dependent, the period matrix P would not be of full rank dt t dt 2t dt σ 2πi 0 0 [1, α] ln α α 1 α 2 1 ( ) [α, β] ln β α β α β 2 α 2 Observe that det P = 2πi(α 1)(β α)(β 1) 0 We can compute the triple coproduct of ln α again We have 1 2πi 0 0 P 1 = ln β(α 2 1) ln α(β 2 1) α+β 2πi(β α)(α 1)(β 1) (α 1)(β 1) ln β(α 1)+ln α(β 1) 2πi(β α)(α 1)(β 1) and therefore get for the triple coproduct (ln α) = ln α 1 2πi 2πi 1 (α 1)(β 1) α+1 (α β)(β 1) 1 (α β)(β 1), + (α 1) ln β(α2 1) + ln α(β 2 1) 2πi 2πi(β α)(α 1)(β 1) α + β + (α 1) (α 1)(β 1) ln α ( ) α + 1 β + (α 1) (α β)(β 1) ln α + (α 2 ln β(α 1) ln α(β 1) 1) 2πi(β α)(α 1)(β 1) 2πi + (α 2 1) + (α 2 1) 1 (α 1)(β 1) ln α 1 (α β)(β 1) ln = ln α 1 ln α 2πi 1 2πi ln α, 2πi 2πi ie we obtain the same result as in (40) 69 ( ) β α

73 82 Second Example: Quadratic Forms Let Q(x) : Q 3 Q x = (x 0, x 1, x 2 ) x A x T be a quadratic form with A Q 3 3 being a regular, symmetric matrix The zero-locus of Q(x) X 0 := {x QP 2 Q(x) = 0} is a quadric or non-degenerate conic We are interested in its affine piece X 0 := X 0 {x 0 0} Q 2 QP 2 We show that we can assume Q(x) to be of a particular nice form A vector v Q 3 is called Q-anisotropic, if Q(v) 0 Since char Q 2, there exist such vectors; just suppose the contrary: and A would be degenerate In particular Q(1, 0, 0) = 0 gives A 11 = 0, Q(0, 1, 0) = 0 gives A 22 = 0, Q(1, 1, 0) = 0 gives 2 A 12 = 0 Q(1, λ, 0) = Q(1, 0, 0) + 2λQ(1, 1, 0) + λ 2 Q(0, 1, 0) will be different form zero for almost all λ Q Hence, we can assume that (1, 0, 0) is anisotropic after applying a coordinate transformation of the form x 0 := x 0, x 1 := λx 0 + x 1, x 2 := x 2 After another affine change of coordinates, we can also assume that A is a diagonal matrix by [Se, Ch IV, 1, 2, Thm 1, p 30] An inspection of the proof of this result reveals that we can choose this coordinate transformation such that the x 0 -coordinate is left unaltered (Just take for e 1 the anisotropic vector (1, 0, 0) in the proof) Such a transformation does not change the isomorphy-type of X 0, and we can take X 0 to be cut out by an equation of the form ax 2 + by 2 = 1 for a, b Q with affine coordinates x := x 1 x 0 and y := x 2 x 0 Since X 0 is affine, and hence the sheaves Ω p X 0 /Q are Γ(X 0;?)-acyclic by [Ha, Thm III35, p 215], we can compute its algebraic derham cohomology (cf Definition 323) by H dr (X 0/Q) = h Γ(X 0 ; Ω X 0 /Q ); so we write down the complex Γ(X 0 ; Ω X 0 /Q) in detail 0 Γ(X 0 ; Ω 1 X 0 /Q ) = Q[x, y]/(ax2 + by 2 1){dx, dy} / (axdx + bydy) d Γ(X 0 ; O X0 ) = Q[x, y]/(ax 2 + by 2 1) Obviously, H 1 dr (X 0/Q) can be presented with generators x n y m dx and x n y m dy for m, n N 0 modulo numerous relations We easily get 70

74 y m dy = d ym+1 m+1 0 x n dx = d xn+1 n+1 0 n 1 x n y m dy = n m+1 xn 1 y m+1 dx + d xn y m+1 m+1 n m+1 xn 1 y m+1 dx for n 1 n 2 x n y 2m dx = x n( ) 1 ax 2 m b dx x n y 2m+1 dx = x n( ) 1 ax 2 my b dx xy dx = x2 2 dy + d x2 y 2 by2 1 2a dy = b 2a y2 dy 1 2a dy 0 x n y dx = b a xn 1 y 2 dy + x n y dx + b a xn 1 y 2 dy x n y dx = b a xn 1 y 2 dy + xn 1 y 2a d(ax 2 + by 2 1) = b a xn 1 y 2 dy + d ( (x n 1 y)(ax 2 +by 2 1) 2a b a xn 1 y 2 dy = ( x n+1 ) xn 1 a dy = ( (n + 1)x n y + n 1 a xn 2 y ) dx + d ( x n+1 y xn 1 a y) n 1 (n+2)a xn 2 y dx for n 2 Thus we see that all generators are linearly dependent of y dx ) H 1 dr (X 0/Q) = h 1 Γ(X 0 ; Ω X 0 /Q ) = Q y dx What about X, the base change to C of X 0? We use the symbol branch of the square root Over C, the change of coordinates for the principal u := ax i by, v := ax + i by gives X = Spec C[x, y]/(ax 2 + by 2 1) = Spec C[u, v]/(uv 1) = Spec C[u, u 1 ] = A 1 C \ {0} Hence the first singular homology group H sing (X an ; Q) of X an is generated by σ : [0, 1] X an, s u = e 2πis, ie a circle with radius 1 turning counter-clockwise around u = 0 once 71

75 The period matrix (cf Definition 511) consists of a single entry v u y dx = σ σ 2i b d u + v 2 a v du u dv Stokes = σ = 1 2i ab = π ab = 4i ab du u σ π discriminant The denominator squared is nothing but the discriminant of the quadratic form Q (cf [Se, Ch IV, 1, 1, p 27]) disc Q := det A Q /Q 2 This is an important invariant, that distinguishes some, but not all isomorphy classes of quadratic forms Since disc Q is well-defined modulo (Q ) 2, it makes sense to write H 1 dr (X 0/Q) = Q π disc Q H 1 sing(x an ; Q) Q C 83 Third Example: Elliptic Curves In this example, we give a brief summary of Chapter VI in Silverman s classical textbook [Sil] on elliptic curves His Chapter VI deals with elliptic curves over C, but some results generalize easily to the case of elliptic curves over Q An elliptic curve E 0 is an one-dimensional, nonsingular, complete scheme defined over a field k, which has genus 1, where the geometric genus p g is defined as p g (E 0 ) := dim k Γ(E 0 ; Ω 1 E 0 /k ) For simplicity, we assume k = Q It can be shown, using the Riemann-Roch theorem, that such an elliptic curve E 0 can be given as the zero locus in QP 2 of a Weierstraßequation (cf [Sil, Ch III, 3, p 63]) Y 2 Z = 4X 3 60G 4 XZ 2 140G 6 Z 3 (41) with coefficients G 4, G 6 Q and projective coordinates X, Y and Z The base change E := E 0 Q C of E 0 gives us a complex torus E an (see [Sil, Ch VI, 5, Thm 51, p 161]), ie an isomorphism in the complex analytic category (cf Subsection 21), with being a lattice of full rank (cf Figure 7) E an = C/Λω1, ω 2 (42) Λ ω1, ω 2 := ω 1 Z ω 2 Z for ω 1, ω 2 C linearly independent over R, 72

76 Figure 7: The lattice Λ ω1, ω 2 C Thus all elliptic curves over C are diffeomorphic to the standard torus S 1 S 1 (cf Figure 8), but carry different complex structures as the parameter τ := ω 2 /ω 1 varies We can describe the isomorphism (42) quite explicitly using periods Let α and β be a basis of H sing 1 (E an ; Z) = H sing 1 (S 1 S 1 ; Z) = Z α Z β The Q-vector space Γ(E 0 ; Ω 1 E 0 /Q) is spanned by the so-called invariant differential [Sil, p 48] ω = d(x/z) Y/Z One can show (cf [Sil, Ch VI, 5, Prop 52, p 161]), that the (abstract) periods (cf Definition 521) ω 1 := ω and ω 2 := ω are R-linearly independent and that the map α E an C/Λ ω1, ω 2 P P O β ω modulo Λ ω1, ω 2 (43) is an isomorphism (here O = [0 : 1 : 0] denotes a base-point on E) Note that under the natural projection π : C C/Γ ω1, ω 2 any meromorphic function f on the torus C/Γ ω1, ω 2 lifts to a doubly-periodic function π f on the complex plane C with periods ω 1 and ω 2 f(x + nω 1 + mω 2 ) = f(x) for all n, m Z and x C This example is possibly the origin of the term period for the elements of the sets P p, P a, and P n The inverse map C/Λ ω1, ω 2 E an for the isomorphism (43) can be described in terms of the Weierstraß- -function of the lattice Λ := Λ ω1, ω 2 (z) = (z, Λ) := 1 z (z ω) 2 1 ω 2 ω Λ ω 0 Figure 8: The standard torus S 1 S 1 73

77 and takes the form (cf [Sil, Prop 36, p 158]) C/Λ ω1, ω 2 E an CP 2 an z [ (z) : (z) : 1] The defining coefficients G 4, G 6 of E 0 can be recovered from Λ ω1, ω 2 Eisenstein series (cf [Sil, Ch VI, 3, Thm 35, p 157]) by computing G 2k := ω Λ ω 0 ω 2k for k = 2, 3 Thus the periods ω 1 and ω 2 determine the elliptic curve E 0 uniquely However, they are not invariants of E 0, since they depend on the chosen Weierstraß-equation (41) of E 0 It is proved in [Sil, p 50], that a change of coordinates which preserves the shape of (41), must be of the form X = u 2 X, Y = u 3 Y, Z = Z for u Q In the new parameterization X, Y, Z, we have G 4 = u 4 G 4, G 6 = u 6 G 6, ω = u 1 ω ω 1 = u 1 ω 1 and ω 2 = u 1 ω 2 Hence τ = ω 2 /ω 1 is a true invariant of E 0, which allows us to distinguish nonisomorphic elliptic curves E over C If we consider both τ and the image of ω 1 in C /Q, we can even distinguish elliptic curves over Q We may ask whether the isomorphy-type of E 0 is already encoded in its cohomology The answer is yes if we consider also the pure Hodge structure living on cohomology By the Hodge theorem [GH, p 116], the (hypercohomology) spectral sequence H p (E an ; Ω q E an ) H n dr (Ean ; C) splits, yielding a decomposition of the first classical derham cohomology group (cf Subsection 31) H 1 dr (Ean ; C) = Γ(E an ; Ω 1 E an) H1 (E an ; O E an) This Hodge decomposition already lives on the algebraic derham cohomology group H 1 dr (E 0/Q) (cf Definition 323) H 1 dr (E 0/Q) = Γ(E 0 ; Ω 1 E 0 /Q ) H1 (E 0 ; O E0 ) (44) In order to see this, observe that the morphism of spectral sequences H p (E 0 ; Ω q E 0 /Q ) Q C H n dr (E 0; Q) Q C H p (E an ; Ω q E an ) H n dr (Ean ; C) is indeed an isomorphism by the Base Change Proposition 351, the GAGA-theorem 223 and the Comparison Theorem 4101 Hence the spectral sequence H p (E 0 ; Ω q E 0 /Q ) Hn dr (E 0; Q) 74

78 splits as well and we obtain the decomposition (44) Pairing H sing 1 (E an ; Z) = Z 2 with some ω Γ(E an ; Ω 1 E an) = Q ω gives us Λ u ω 1,u ω 2 C, where we write ω = u ω with u Q That is, we can recover Λ ω1, ω 2 up to a rational multiple from the integral singular homology H sing 1 (E an ; Z) of E an plus the algebraic derham cohomology of E 0 equipped with its Hodge decomposition (44), ie we can recover the isomorphy-type of E 0 from this data 84 Fourth Example: A ζ-value Earlier we raised the question of how to write ζ(2) as a period and found the identity (cf Proposition 535) dx dy ζ(2) = (1 x) y 0 x y 1 The problem was that this identity did not give us a valid representation of ζ(2) as a naïve period (cf Definition 532), since the pole locus of the integrand and the domain of integration are not disjoint We show how to circumvent this difficulty First we define thus getting Y 0 := A 2 Q with coordinates x and y, Z 0 := {x = 1} {y = 0}, X 0 := Y 0 \ Z 0, D 0 := ({x = 0} {y = 1} {x = y}) \ Z 0, := {(x, y) Y an x, y R, 0 x y 1} a triangle in Y an, and dx dy ω 0 := (1 x) y, ζ(2) = with ω 0 Γ(X 0, Ω 2 X 0 /Q ) and Dan {(0, 0), (1, 1)}, see Figure 9 ω, Figure 9: The configuration Z 0, D 0, Now we blow up Y 0 in the points (0, 0) and (1, 1) obtaining π 0 : Ỹ0 Y 0 We denote the strict transform of Z 0 by Z 0, π0 ω 0 by ω 0 and Ỹ0 \ Z 0 by X 0 The strict transform πan 1 ( \ {(0, 0), (1, 1)}) will be called and (being Q-semi-algebraic hence triangulable cf Theorem 623) gives rise to a singular chain γ H sing 2 ( X an, D an ; Q) 75

79 Since π is an isomorphism away from the exceptional locus, this exhibits ζ(2) as an (abstract) period (cf Definition 521) ζ(2) = ω = ω P a = P, see Figure 10 eγ Figure 10: The configuration Z 0, D 0, We will conclude this example by writing out ω 0 and more explicitely Note that Ỹ 0 can be described as the subvariety of cut out by A 2 Q QP 1 QP 1 with coordinates ( x, ỹ; λ 0 : λ 1 ; µ 0 : µ 1 ) xλ 0 = ỹλ 1 and ( x 1)µ 0 = (ỹ 1)µ 1 With this choice of coordinates π 0 takes the form π 0 : Ỹ 0 Y 0 ( x, ỹ; λ 0 : λ 1 ; µ 0 : µ 1 ) ( x, ỹ) and we have X 0 := Ỹ0 \ ({λ 0 = 0} {µ 1 = 0}) We can embed X 0 into affine space X 0 A 4 Q ( x, ỹ; λ 0 : λ 1 ; µ 0 : µ 1 ) ( x, ỹ, λ 1 λ 0, µ 0 µ 1 ) and so have affine coordinates x, ỹ, λ := λ 1 λ 0 and µ := µ 0 µ 1 on X 0 Now, on X 0 \ π 1 0 (1, 1), the form ω 0 is given by ω 0 = d x dỹ (1 x) ỹ = d(λỹ) dỹ (1 x) ỹ = dλ dỹ 1 x, while on X 0 \ π0 1 (0, 0) we have ω 0 = d x dỹ (1 x) ỹ = d x d(ỹ 1) (1 x) ỹ = d x d(µ( x 1)) (1 x) ỹ = d x dµ ỹ The region is given by = {( x, ỹ, λ, µ) X an x, ỹ, λ, µ R, 0 x ỹ 1, 0 λ 1, 0 µ 1} 76

80 85 Fifth Example: The Double Logarithm Variation of Mixed Hodge Structures The previous example fits into a more general framework 851 Multiple Polylogarithms Define the hyperlogarithm as the iterated integral dt 1 I n (a 1,, a n ) := dt n t 1 a 1 t n a n 0 t 1 t n 1 with a 1,, a n C (cf [Z1, p 168]) These integrals specialize to the multiple polylogarithm (cf [loc cit] 1 ) ( a2 a n Li m1,,m n,,, 1 ) := ( 1) n I P m a 1 a n 1 a n (a 1, 0,, 0,, a n, 0,, 0), n }{{} m 1 1 }{{} m n 1 which is convergent if 1 < a 1 < < a n (cf [G3, 23, p 9]) Alternatively, we can describe the multiple polylogarithm as a power series (cf [G3, Thm 22, p 9]) Li m1,,m n (x 1,, x n ) = x k m 1 0<k 1 < <k n k 1 1 xkn n 1 kn mn for x i < 1 (45) Of special interest to us will be the dilogarithm Li 2 (x) = k>0 xk x k y l kl k 2 and the double logarithm Li 1,1 (x, y) = 0<k<l At first, the functions Li m1,,m n (x 1,, x n ) only make sense for x i < 1, but they can be analytically continued to multivalued meromorphic functions on C n (cf [Z1, p 2]), for example Li 1 (x) = ln(1 x) General Remarks Polylogarithms have applications in Zagier s conjecture (cf [Z1, p 2]) and are used in the study of certain motivic fundamental groups (cf [HW, p 9] and [G2]) Besides, equation (45) would suggest a connection between values of multiple polylogarithms and multiple ζ-values (cf [G1, p 617]) Indeed, we have formally ζ(2) = π2 6 1 ζ(m 1,, m n ) := k m 1 0<k 1 <<k m 1 kn mn = Li 2 (1) = lim lim I 2 (a, b) = lim b 0 a 1 lim Li 1,1 b 0 a 1 ( b a, 1 b However, such formal computations are problematic, since a limit may diverge or at least depend on the direction in which the limit is taken, because multiple polylogarithms are multivalued To deal with this problem, Zagier, Goncharov and others proposed regularization procedures, for which we have to refer to [G3, , p 18 24] 1 Note that the factor ( 1) n is missing in equation (12) of [Z1, p 168] ) 77

81 After these general remarks, we want to study a variation of mixed Q-Hodge structures: the double logarithm variation, for which multiple polylogarithms appear as coefficients For that we will combine results in [G1, p 620f], [Kj, 4, p 32f], [Z1], [Z2a] and [Z2b] 2 By computing limit mixed Q-Hodge structures, we will obtain ζ(2) as a period of a mixed Q-Hodge structure 852 The Configuration Let us consider the configuration Y := A 2 C with coordinates x and y, Z := {x = a} {y = b} with a 0, 1 and b 0, 1 X := Y \ Z D := ({x = 0} {y = 1} {x = y}) \ Z, see Figure 11 We denote the irreducible components of the divisor D as follows: D 1 := {x = 0} \ {(0, b)}, D 2 := {y = 1} \ {(a, 1)}, and D 3 := {x = y} \ {(a, a), (b, b)} By projecting from Y an onto the y- or x-axis, we get isomorphisms for the associated complex analytic spaces (cf Subsection 21) D an 1 = C \ {b}, D an 2 = C \ {a}, and D an 3 = C \ {a, b} Figure 11: The algebraic pair (X, D) 853 Singular Homology We can easily give generators for the second singular homology of the pair (X an, D an ), see Figure 12 Let α : [0, 1] C be a smooth path, which does not meet a or b We define a triangle := { ( α(s), α(t) ) 0 s t 1} 2 Note that [Z2a] plus [Z2b] is just a detailed version of [Z1] 78

82 Consider the closed curve in C { } a C b := s [0, 1], b + εe2πis which divides C into two regions: an inner one containing a b and an outer one We can choose ε > 0 small enough such that C b separates a b from 0 to 1, ie such that 0 and 1 are contained in the outer region This allows us to find a smooth path β : [0, 1] C from 0 to 1 not meeting C b We define a slanted tube S b := {( β(t) (b + εe 2πis ), b + εe 2πis) s, t [0, 1] } which winds around {y = b} and whose boundary components are supported on D1 an (corresponding to t = 0) and Dan 3 (corresponding to t = 1) The special choice of β guarantees S b Z an = Similarly, we choose ε > 0 such that the closed curve { } b 1 C a := s [0, 1] a 1 εe2πis separates b 1 a 1 form 0 and 1 Let γ : [0, 1] C be a smooth path from 0 to 1 which does not meet C a We have a slanted tube S a := {( a + εe 2πis, 1 + γ(t) (a + εe 2πis 1) ) s, t [0, 1] } winding around {x = a} with boundary supported on D an 2 and D an 3 Finally, we have a torus T := {(a + εe 2πis, b + εe 2πit ) s, t [0, 1]} The 2-form ds dt defines an orientation on the unit square [0, 1] 2 = {(s, t) s, t [0, 1]} Hence the manifolds with boundary, S b, S a, T inherit an orientation; and since they can be triangulated, they give rise to smooth singular chains By abuse of notation we will also write, S b, S a, T for these smooth singular chains The homology classes of, S b, S a and T will be denoted by γ 0, γ 1, γ 2 and γ 3, respectively Figure 12: Generators of H sing 2 (X an, D an ; Q) 79