Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi-Yau Varieties

Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi-Yau Varieties James D. Lewis Abstract We review the transcendental aspects of algebraic cycles, and explain how this relates to Calabi-Yau varieties. More precisely, after presenting a general overview, we begin with some rudimentary aspects of Hodge theory and algebraic cycles. We then introduce Deligne cohomology, as well as the generalized higher cycles due to Bloch that are connected to higher K-theory, and associated regulators. Finally, we specialize to the Calabi-Yau situation, and explain some recent developments in the field. Key words: Calabi-Yau variety, algebraic cycle, Abel-Jacobi map, regulator, Deligne cohomology, Chow group James D. Lewis University of Alberta, e-mail: lewisjd@ualberta.ca Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada 3

4 James D. Lewis CONTENTS Introduction Notation Some Hodge theory - Formalism of mixed Hodge structures Algebraic cycles - Generalized cycles A short detour via Milnor K-theory - The Gersten-Milnor complex Hypercohomology Deligne cohomology - Alternate take on Deligne cohomology - Deligne-Beilinson cohomology Examples of HZar r m (,Kr, M ) and corresponding regulators - Case m = 0 and CY threefolds - Deligne cohomology and normal functions - Case m = 1 and K3 surfaces - Torsion indecomposables - Case m = 2 and elliptic curves - Constructing K 2 () classes on elliptic curves References 1 Introduction These notes concern that part of Calabi-Yau geometry that involves algebraic cycles - typically built up from special subvarieties, such as rational points and rational curves. From these algebraic cycles, one forms various doubly indexed groups, called higher Chow groups, that mimic simplicial homology theory in algebraic topology. These Chow groups come equipped with various maps whose target space is a certain transcendental cohomology theory called Deligne cohomology. More precisely these maps are called regulators, from the higher cycle groups of S. Bloch, denoted by CH k (,m), of a projective algebraic manifold, to Deligne cohomology, viz.: cl r,m : CH r (,m) HD 2r m ( ),A(r), (1) where A R is a subring, A(r) := A(2πi) r is called the Tate twist, and as we will indicate below, some striking evidence that these regulator maps become highly

Algebraic Cycles and Transcendental Algebraic Geometry 5 interesting in the case where is Calabi-Yau. As originally discussed in [43], we are interested in the following case scenarios, with the intention of also providing an update on new developments. For the moment we will consider A = Z; however we will also consider A = Q, R later on. When m = 0, the objects of interest are the null homologous codimension 2 (= dimension 1) cycles CH 2 hom () = CH 1,hom() on a projective threefold, and where in this case, (1) becomes the Abel-Jacobi map: Φ 2 : CH 2 hom () H 3 (,C) J2 () = F 2 H 3 (,C) + H 3 (,Z(2)) {H3,0 () H 2,1 ()}, H 3 (,Z(1)) (2) defined by a process of integration, J 2 () being the Griffiths jacobian of. One of the reasons for introducing the Abel-Jacobi map is to study the Griffiths group Griff 2 () Q. If we put CH r alg () to be codimension r cycles algebraically equivalent to zero, then the Griffiths group is given by Griff r () := CH r hom ()/CHr alg (). When m = 1, the object of interest is the group { ( ) f j,z j f j C(Z j ) } CH 2 (,1) = j,cd Z j =1 j div( f j ) = 0 Image ( Tame symbol ), on a projective algebraic surface. If we mod out by the subgroup of CH 2 (,1) where the f j s C, then we arrive at the quotient group of indecomposables CH 2 ind (,1) which plays an analogous role to the Griffiths group above. Moreover if we assume that the torsion part of H 3 (,Z) is zero, then in this case (1) becomes a map: [ H cl 2,1 : CH 2 2,0 ind (,1) () Htr 1,1 () ], (3) H 2 (,Z) where Htr 1,1 () is the transcendental part of H 1,1 (), being the orthogonal complement to the subgroup of algebraic cocycles. In the case m = 2, the objects of interest are the group of symbols: { CH 2 (,2) = ξ := { f j,g j } j f j,g j C() j,p ( ( 1) ν p( f j )ν p (g j ) ( νp(g j ) f j g νp( f j ) j } ) ), (p), p = 0 (ν p = order of vanishing at p), on a smooth projective curve. In this case (1) becomes the regulator: cl 2,2 : CH 2 (,2) H 1 (,C/Z(2)). (4) As first pointed out in [43], if is a smooth projective variety of dimension d, where 1 d 3, then the maps and objects

6 James D. Lewis cl 2,2 in (4) and CH 2 (,2) Q for d = 1, cl 2,1 in (3) and CH 2 ind (,1) Q for d = 2, Φ 2 in (2) and Griff 2 () Q for d = 3, become especially interesting and generally nontrivial in the case where is a Calabi-Yau variety; moreover, in a sense that will be specified later, these maps are essentially trivial when restricted to indecomposables, for either of lower or higher order to its Calabi-Yau counterpart. Cycle constructions involving nodal rational curves and torsion points, play a prominent role here. Several recent developments in the context of algebraic cycles are included in these notes since the appearance of [43], which should be of interest to specialists. Having said this, these notes are prepared with the expressed interest in enticing a wider group of researchers into the subject. We have benefited from conversations with Matt Kerr, Bruno Kahn and i Chen. We are also grateful to Bruno for sharing with us his preprint [33]. We owe the referee a debt of gratitude for doing a splendid job in recommending improvements and catching errors in an earlier version of this paper. We are also pleased that the referee made us aware of the interesting work of Friedman-Laza ([24]), and for raising the very interesting question of how to construct normal functions over the Calabi-Yau variations of Hodge structure that they construct. 2 Notation Throughout these notes, and unless otherwise specified, = /C is a projective algebraic manifold, of dimension d. A projective algebraic manifold is the same thing as a smooth complex projective variety. If V is an irreducible subvariety of, then C(V ) is the rational function field of V, with multiplicative group C(V ). Depending on the context (which will be made abundantly clear in the text), O will either be the sheaf of germs of holomorphic functions on in the analytic topology, or the sheaf of germs of regular functions in the Zariski topology. 3 Some Hodge theory Some useful reference material for this section is [27] and [41]. Let E k = C-valued C k-forms on. (One could also use the common notation of A k () for C forms, but let s not.) We have the decomposition: E k = p+q=k E p,q, E p,q = Eq,p,

Algebraic Cycles and Transcendental Algebraic Geometry 7 where E p,q are the C (p,q)-forms which in local holomorphic coordinates z = (z 1,...,z n ), are of the form: f IJ dz I dz J, I =p, J =q f IJ are C valued C f unctions, I = 1 i 1 < < i p d, J = 1 j 1 < < j q d, dz I = dz i1 dz ip, dz J = dz j1 dz jq. One has the differential d : E k Ek+1, and we define HDR k (,C) = kerd : Ek Ek+1 de k 1. The operator d decomposes into d = +, where : E p,q E p+1,q. Further d 2 = 0 2 = 2 = 0 = +, by (p,q) type. E p,q+1 The above decomposition descends to the cohomological level, viz., Theorem 3.1 (Hodge decomposition) H k sing (,Z) Z C H k DR (,C) = p+q=k H p,q (), where H p,q () = d-closed (p,q)-forms (modulo coboundaries), and Furthermore: H p,q () = H q,p (). H p,q () E p,q,d closed E p 1,q 1 Some more terminology: Hodge filtration. Put F k H i (,C) = p k H p,i p ().. and : E p,q Now recall dim = d. Theorem 3.2 (Poincaré and Serre duality) The following pairings induced by (w 1,w 2 ) w 1 w 2, are non-degenerate: H k DR (,C) H2d k(,c) C, DR H p,q () H d p,d q () C. Therefore H k () H 2d k (), H p,q () H d p,d q ()

8 James D. Lewis Corollary 3.3 H i (,C) F r H i (,C) Fd r+1 H 2d i (,C). 3.4 Formalism of mixed Hodge structures Definition 3.5 Let A R be a subring. An A-Hodge structure (HS) of weight N Z is given by the following datum: A finitely generated A-module V, and either of the two equivalent statements below: 1 A decomposition V C = p+q=n V p,q, V p,q = V q,p, where is complex conjugation induced from conjugation on the second factor C of V C := V C. 2 A finite descending filtration V C F r F r 1 {0}, satisfying V C = F r F N r+1, r Z. Remark 3.6 The equivalence of 1 and 2 can be seen as follows. Given the decomposition in 1, put F r V C = V p,q. p+q=n,p r Conversely, given {F r } in 2, put V p,q = F p F q. Example 3.7 /C smooth projective. Then H i (,Z) is a Z-Hodge structure of weight i. Example 3.8 A(k) := (2πi) k A is an A-Hodge structure of weight 2k and of pure Hodge type ( k, k), called the Tate twist. Example 3.9 /C smooth projective. Then H i (,Q(k)) := H i (,Q) Q(k) is a Q-Hodge structure of weight i 2k. To extend these ideas to singular varieties, one requires the following terminology. Definition 3.10 An A-mixed Hodge structure (A-MHS) is given by the following datum:

Algebraic Cycles and Transcendental Algebraic Geometry 9 A finitely generated A-module V A, A finite descending Hodge filtration on V C := V A C, V C F r F r 1 {0}, An increasing weight filtration on V A Q := V A Z Q, {0} W l 1 W l V A Q, such that {F r } induces a (pure) HS of weight l on Gr W l := W l /W l 1. Theorem 3.11 (Deligne [20]) Let Y be a complex variety. Then H i (Y,Z) has a canonical and functorial Z-MHS. Remark 3.12 (i) A morphism h : V 1,A V 2,A of A-MHS is an A-linear map satisfying: h ( W l V 1,A Q ) Wl V 2,A Q, l, h ( F r V 1,C ) F r V 2,C, r. Deligne ([20] (Theorem 2.3.5)) shows that the category of A-MHS is abelian; in particular if h : V 1,A V 2,A is a morphism of A-MHS, then ker(h), coker(h) are endowed with the induced filtrations. Let us further assume that A Q is a field. Then Deligne (op. cit.) shows that h is strictly compatible 1 with the filtrations W and F, and that the functors V Gr W l V, V Grr FV are exact. (ii) Roughly speaking, the functoriality of the MHS in Deligne s theorem translates to the following yoga: the standard exact sequences in singular (co)homology, together with push-forwards and pullbacks by morphisms (wherever permissible) respect MHS. In particular for a subvariety Y, the localization cohomology sequence associated to the pair (,Y ) is a long exact sequence of MHS. Here is where the Tate twist comes into play: Suppose that Y is an inclusion of projective algebraic manifolds with codim Y = r 1. One has a Gysin map H i 2r (Y,Q) H i (,Q) which involves Hodge structures of different weights. To remedy this, one considers the induced map H i 2r (Y,Q( r)) H i (,Q(0)) = H i (,Q) via (twisted) Poincaré duality (see (5) below), which is a morphism of pure Hodge structures (hence of MHS). A simple proof of this fact can be found in 7 of [41]. Note that the morphism H i Y (,Q) Hi (,Q) is a morphism of MHS, and that accordingly H i Y (,Q) Hi 2r (Y,Q( r)) is an isomorphism of MHS (with Y still smooth). 1 Strict compatibility means that h(f r V 1,C ) = h(v 1,C ) F r V 2,C and h(w l V 1,A Q ) = h(v 1,A Q ) W l V 2,A Q for all r and l. A nice explanation of Deligne s proof of this fact can be found in [49], where a quick summary goes as follows: For any A-MHS V, V C has a C-splitting into a bigraded direct sum of complex vector spaces I p,q := F p W p+q [ F q W p+q + i 2 F q i+1 W p+q i ], where one shows that F r V C = p r q I p,q and W l V C = p+q l I p,q. Then by construction of I p,q, one has h(i p,q (V 1,C ) I p,q (V 2,C ). Hence h preserves both the Hodge and complexified weight filtrations. Now use the fact that A Q is a field to deduce that h preserves the weight filtration over A Q.

10 James D. Lewis Example 3.13 Let U be a compact Riemann surface, Σ U a finite set of points, and put U := U\Σ. According to Deligne, H 1 (U,Z(1)) carries a Z-MHS. The Hodge filtration on H 1 (U,C) is defined in terms of a filtered complex of holomorphic differentials on U with logarithmic poles along Σ ([20], but also see (9) below). One can observe the MHS as follows. Poincaré duality gives us H 1 Σ (U,Z) H 1(Σ,Z) = 0, and the localization sequence in cohomology below is a sequence of MHS: where 0 H 1 (U,Z(1)) H 1 (U,Z(1)) H 0 (Σ,Z(0)) 0, H 0 (Σ,Z(0)) := ker ( H 2 Σ (U,Z(1)) H 2 (U,Z(1)) ) Z(0) Σ 1. Put W 0 = H 1 (U,Z(1)), W 1 = Im ( H 1 (U,Z(1)) H 1 (U,Z(1)) ), W 2 = 0. Then Gr W 1 H1 (U,Z(1)) H 1 (U,Z(1)) has pure weight 1 and Gr W 0 H1 (U,Z(1)) Z(0) Σ 1 has pure weight 0. The following notation will be introduced: Definition 3.14 Let V be an A-MHS. We put and Γ A V := hom A MHS (A(0),V ), J A (V ) = Ext 1 A MHS (A(0),V ). In the case where A = Z or A = Q, we simply put Γ = Γ A and J = J A. Example 3.15 Suppose that V = V Z is a Z (pure) HS of weight 2r. Then V (r) := V Z(r) is of weight 0, and (up to the twist) one can identify ΓV with V Z F r V C = V Z V r,r := ε 1( V r,r), where ε : V V C. Example 3.16 Let V be a Z-MHS. There is the identification due to J. Carlson (see [8], [32]), W 0 V C J(V ) F 0 W 0 V C +W 0 V, where in the denominator term, V := V Z is identified with its image V Z V C (viz., quotienting out torsion). For example, if {E} Ext 1 MHS (Z(0),V ) corresponds to the short exact sequence of MHS: 0 V E α Z(0) 0, then one can find x W 0 E and y F 0 W 0 E C such that α(x) = α(y) = 1. Then x y V C descends to a class in W 0 V C / {F 0 W 0 V C +W 0 V }, which defines the map from Ext 1 MHS (Z(0),V ) to W 0V C / {F 0 W 0 V C +W 0 V }.

Algebraic Cycles and Transcendental Algebraic Geometry 11 4 Algebraic cycles For the next two sections, the reader may find it helpful to consult [44]. Recall /C smooth projective, dim = d. For 0 r d, put z r () (= z d r ()) = free abelian group generated by subvarieties of codim r (= dim d r) in. Example 4.1 (i) z d () = z 0 () = { M j=1 n j p j n j Z, p j }. (ii) z 0 () = z d () = Z{} Z. (iii) Let 1 := V (z 2 2 z 0 z 3 1 z 0z 2 1 ) P2, and 2 := V (z 2 2 z 0 z 3 1 z 1z 2 0 ) P2. Then 3 1 5 2 z 1 (P 2 ) = z 1 (P 2 ). (iv) codim V = r 1, f C(V ). div( f ) := ( f ) := ( f ) 0 ( f ) z r () (principal divisor). (Note: div( f ) is easy to define, by first passing to a normalization Ṽ of V, then using the fact that the local ring OṼ, of regular functions at is a discrete valuation ring for a codimension one point on Ṽ, together with the proper pushforward associated to Ṽ V.) Divisors in (iv) generate a subgroup, z r rat() z r (), which defines the rational equivalence relation on z r (). Definition 4.2 is called the r-th Chow group of. CH r () := z r ()/z r rat(), Remark 4.3 On can show that ξ z r rat() w z r (P 1 ), each component of the support w flat over P 1 (, such that ξ = w[0] w[ ]. (Here w[t] := pr 2, pr 1 (t) w P ) 1.) If one replaces P 1 by any choice of smooth connected curve Γ (not fixed!) and 0, by any 2 points P, Q Γ, then one obtains the subgroup z r alg () zr () of cycles that are algebraically equivalent to zero 2. There is the fundamental class map (described later) z r () H 2r (,Z) whose kernel is denoted by z r hom (). More precisely, the target space and map requires some twisting, viz., z r () H 2r (,Z(r)). To explain the role of twisting here, we illustrate this with three case scenarios. 2 The fact that a smooth connected Γ will suffice (as opposed to a [connected] chain of curves) in the definition of algebraic equivalence follows from the transitive property of algebraic equivalence (see [41] (p. 180)).

12 James D. Lewis Let f : Y be a morphism of smooth projective varieties, where dimy = dim 1. One has a commutative diagram of cycle class maps: z r 1 (Y ) H 2(r 1) (Y,Z(r 1)) f f z r () H 2r (,Z(r)) Thus from the perspective of (mixed) Hodge theory, this diagram is natural, as the right hand vertical arrow is a morphism of (M)HS. Let U/C be a smooth quasi-projective variety of dimension d, and Y U a closed algebraic subset. Using the twisted Poincaré duality theory formalism in this situation (see [32] (p. 82, p. 92)), Poincaré duality gives us an isomorphism of MHS: H i Y (U,Z( j)) H 2d i (Y,Z(d j)) := H 2d i (Y,Z)( j d), where H i (Y,Z) := H BM i (Y,Z) is Borel-Moore homology 3. For example if U = Y = is smooth projective, then H i (,Z( j)) is a pure HS of weight i 2 j, and H a (,Z(b)) := H a (,Z)( b) is known to be a pure HS of weight 2b a, hence H 2d i (Y,Z(d j)) has weight 2(d j) (2d i) = i 2 j. Thus is an isomorphism of HS. H i (,Z( j)) H 2d i (,Z(d j)), (5) Remark 4.4 Although tempting, from a purist point of view, it would be a mistake to interpret H a (,Z(b)) = H a (,Z)(b). This would imply that the Poincaré duality isomorphism in (5) would not preserve weights, and hence not an isomorphism of (M)HS in the sense given in Remark 3.12. Let O be the sheaf of analytic functions on. Recall the exponential short exact sequence of sheaves 0 Z O exp(2πi ( )) O 0, where O O is the sheaf of units. It is well-known that H 1 (,O ) CH1 (), and hence there is an induced Chern class map CH 1 () H 2 (,Z). But this is not so natural as there is no canonical choice of i. Instead, one considers 0 Z(1) O exp O 0, 3 We remind the reader that for singular homology H sing (U, Z) and ignoring twists, Poincaré duality gives the isomorphism Hc(U,Z) i H sing 2d i (U,Z), where Hi c(u,z) is cohomology with compact support; whereas H i (U,Z) H2d i BM (U,Z).

Algebraic Cycles and Transcendental Algebraic Geometry 13 and accordingly the induced cycle class map CH 1 () H 2 (,Z(1)). One has inclusions: Definition 4.5 Put (i) CH r alg () := zr alg ()/zr rat(), (ii) CH r hom () := zr hom ()/zr rat(), z r rat() z r alg () zr hom () zr (). (iii) Griff r () := z r hom () /z r alg () = CHr hom ()/CHr alg (), called the Griffiths group. The Griffiths group is known to be trivial in the cases r = 0, 1, d. 4.6 Generalized cycles The basic idea is this: ( CH r () = Coker cd V =r 1 In the context of Milnor K-theory, this is just ( ) K2 M Tame (C(V )) cd V =r 2 } {{ } building a complex on the left cd V =r 1 ) C(V ) div z r (). K M 1 (C(V )) div cd V =r K M 0 (C(V )). For a field F, one has the Milnor K-groups K M (F), where K0 M(F) = Z, KM 1 (F) = F and { }/ Steinberg relations K2 M (F) = Symbols {a,b} a,b {a 1 a 2,b} = {a 1,b}{a 2,b} F {a,b} = {b,a} 1. {a,1 a} = 1, a 1 One has a Gersten-Milnor resolution of a sheaf of Milnor K-groups on, which leads to a complex whose last three terms and corresponding homologies (indicated at ) for 0 m 2 are: cd Z=r 2 K M 2 (C(Z)) T cd Z=r 1 C(Z) div cd Z=r Z (6) CH r (,2) CH r (,1) CH r (,0)

14 James D. Lewis where div is the divisor map of zeros minus poles of a rational function, and T is the Tame symbol map. The Tame symbol map T : K2 M (C(Z)) K1 M (C(D)), cd Z=r 2 cd D=r 1 is defined as follows. First K M 2 (C(Z)) is generated by symbols { f,g}, f, g C(Z). For f, g C(Z), T ( { f,g} ) ( = ( 1) ν D( f )ν D (g) f ν D (g) D g ν D( f ) where ( )D means restriction to the generic point of D, and ν D represents order of a zero or pole along an irreducible divisor D Z. Example 4.7 Taking cohomology of the complex in (6), we have: (i) CH r (,0) = z r ()/z r rat() =: CH r (). (ii) CH r (,1) is represented by classes of the form ξ = j ( f j,d j ), where codim D j = r 1, f j C(D j ), and div( f j ) = 0; modulo the image of the Tame symbol. (iii) CH r (,2) is represented by classes in the kernel of the Tame symbol; modulo the image of a higher Tame symbol. Example 4.8 (i) = P 2, with homogeneous coordinates [z 0,z 1,z 2 ]. P 1 = l j := V (z j ), j = 0,1,2. Let P = [0,0,1] = l 0 l 1, Q = [1,0,0] = l 1 l 2, R = [0,1,0] = l 0 l 2. Introduce f j C(l j ), where ( f 0 ) = P R, ( f 1 ) = Q P, ( f 2 ) = R Q. Explicitly, f 0 = z 1 /z 2, f 1 = z 2 /z 0 and f 2 = z 0 /z 1. Then ξ := 2 j=0 ( f j,l j ) CH 2 (P 2,1) represents a higher Chow cycle. / P l 1 / l 0 Q / l 2 R This cycle turns out to be nonzero. 4 Consider the line P 1 0 := V (z 0 +z 1 +z 2 ) P 2, and set q j = P 1 0 l j, j = 0,1,2. Then q 0 = [0,1, 1], q 1 = [1,0, 1], q 2 = [1, 1,0], and accordingly f j (q j ) = 1. These Chow groups are known to satisfy a projective bundle formula (see [6], p. 269) which implies that CH 2 (P 2,1) {P 1 } CH 1 (Spec(C),1), ), D 4 A special thanks to Rob de Jeu for supplying us this idea.

Algebraic Cycles and Transcendental Algebraic Geometry 15 CH 2 (P 1 0,1) {P 1 P 1 0} P 2 CH 1 (Spec(C),1), where P 2 Spec(C), and P 1 0 Spec(C) are the structure maps, and P1 P 2 is a choice of line. It is well-known that CH 1 (Spec(C),1) = C ([6], see Example 5.3 below), and thus via restriction we have the isomorphisms: moreover under this isomorphism, CH 2 (P 2,1) CH 2 (P 1 0,1) C ; ξ 2 f j (q j ) = 1 C. j=0 Hence ξ CH 2 (P 2,1) is a nonzero 2-torsion class. 5 (ii) Again = P 2. Let C be the nodal rational curve given by z 2 2 z 0 = z 3 1 +z 0z 2 1 (in affine coordinates (x,y) = (z 1 /z 0,z 2 /z 0 ) C 2, C is given by y 2 = x 3 + x 2 ). Let C P 1 be the normalization of C, with morphism π : C C. Put P = (0,0) C (node) and let {R,Q} = π 1 (P). Choose f C( C) = C(C), such that ( f ) C = R Q. Then ( f ) C = 0 and hence ( f,c) CH 2 (P 2,1) defines a higher Chow cycle. R Q / / / P / Exercise 4.9 Show that ( f,c) = 0 CH 2 (P 2,1). 5 Matt Kerr informed us of an alternate and slick approach to this example via the definition given in Example 4.7(ii). Namely one need only add Tame{z 1 /z 0,z 2 /z 0 } = ( f0 1,l 0 ) + ( f1 1,l 1 ) + ( f2 1,l 2 ) to ξ to get the 2-torsion class ( 1,l 0 ), which is the same as ξ in CH 2 (P 2,1).

16 James D. Lewis 5 A short detour via Milnor K-theory This section provides some of the foundations for the previous section. In the first part of this section, we follow closely the treatment of Milnor K-theory provided in [5], which allows us to provide an abridged definition of the higher Chow groups CH r (,m), for 0 m 2. The reader with pressing obligations who prefers to work with concrete examples may skip this section, without losing sight of the main ideas presented in this paper. Let F be a field, with multiplicative group F, and put T (F ) = n 0 T n (F ), the tensor product of the Z-module F. Here T 0 (F ) := Z, F = T 1 (F ), a [a]. If a 0,1, set r a = [a] [1 a] T 2 (F ). The two-sided ideal R generated by the {r a } s is graded, and we put: K M F = T (F ) R = n 0 K M n F, (Milnor K theory). For example, K 0 (F) = Z, K 1 (F) = F, and K2 M (F) is the abelian group generated by symbols {a,b}, subject to the Steinberg relations: Furthermore, one can also show that: {a 1 a 2,b} = {a 1,b}{a 2,b} {a,1 a} = 1, for a 0,1 {a,b} = {b,a} 1 (2.1) {a,a} = { 1,a} = {a,a 1 } = {a 1,a},and {a, a} = 1. Quite generally, one can argue that K M n (F) is generated {a 1,...,a n }, a 1,...,a n F, subject to: (i) (a 1,...,a n ) {a 1,...,a n }, is a multilinear function from F F K M n (F), if a i + a i+1 = 1 for some i < n. (ii) {a 1,...,a n } = 0, Next, let us assume given a field F with discrete valuation ν : F Z, with corresponding discrete valuation ring O F := {a F ν(a) 0}, and residue field k(ν). One has maps T : K M (F) K 1 M (k(ν)). Choose π F such that ν(π) = 1, and note that F = O F πz. For example, if we write a = a 0 π i, b = b 0 π j K1 M(F), then T (a) = i Z = K0 M (k(ν)) and T {a,b} = ( 1) i j a j b i k(ν) = K M 1 (k(ν)) (Tame symbol).

Algebraic Cycles and Transcendental Algebraic Geometry 17 5.1 The Gersten-Milnor complex The reader may find [46] particularly useful regarding the discussion in this subsection. Let O be the sheaf of regular functions on, with sheaf of units O. As in [34], we put Kr, M := ( O )/ O J, (Milnor sheaf), }{{} r times where J is the subsheaf of the tensor product generated by sections of the form: { τ1 τ r τi + τ j = 1, for some i and j, i j }. For example, K1, M = O. Introduce the Gersten-Milnor complex (a flasque resolution of Kr, M, see [23], [38]): We have cd Z=r 2 K M r, K M r (C()) K M 2 (C(Z)) cd Z=r 1 cd Z=1 Kk 1 M (C(Z)) K M 1 (C(Z)) cd Z=r K M 0 (C(Z)) = Z, K M 1 (C(Z)) = C(Z), K M 2 (C(Z)) = { symbols { f,g} / Steinberg relations }. The last three terms of this complex then are: cd Z=r 2 K M 2 (C(Z)) T cd Z=r 1 C(Z) div cd Z=r K M 0 (C(Z)) 0. Z 0 where div is the divisor map of zeros minus poles of a rational function, and T is the Tame symbol map T : K2 M (C(Z)) K1 M (C(D)), defined earlier. codim Z=r 2 Definition 5.2 For 0 m 2, Example 5.3 codim D=r 1 CH r (,m) = H r m Zar (,K M r, ). CH 1 (,1) H 0 Zar (,K M 1,) H 0 Zar (,O ) C.

18 James D. Lewis Remark 5.4 The higher Chow groups CH r (W,m) were introduced in [6], and are defined for any non-negative integers r and m, and quasi-projective variety W over a field k. The formula in Definition 5.2 is only for smooth varieties. 6 Hypercohomology An excellent reference for this is the chapter on spectral sequences in [27]. The reader familiar with hypercohomology can obviously skip this section. Let (S 0,d) be a (bounded) complex of sheaves on. One has a Cech double complex ( C (U,S ), d, δ ), where U is an open cover of. The k-th hypercohomology is given by the k-th total cohomology of the associated single complex ( M := i+ j= C i (U,S j ), D = d ± δ ), viz., H k (S ) := lim U H k (M ). Associated to the double complex are two filtered subcomplexes of the associated single complex, with two associated Grothendieck spectral sequences abutting to H k (S ) (where p + q = k): E p,q 2 := H p δ (,H q d (S )) E p,q 2 := H p d (Hq δ (,S )) The first spectral sequence shows that quasi-isomorphic complexes yield the same hypercohomology: Alternate take. Two complexes of sheaves K 1, K 2 are said to be quasi-isomorphic if there is a morphism h : K 1 K 2 inducing an isomorphism on cohomology h : H (K 1 ) H (K 2 ). Take a complex of acyclic sheaves (K,d) (viz., H i>0 (,K j ) = 0 for all j) quasi-isomorphic to S. Then by the second spectral sequence: H k (S ) := H i( H 0 (,K ) ). For example if L, is an acyclic resolution of S, then the associated single complex K = i+ j= L i, j is acyclic and quasi-isomorphic to S. Example 6.1 Let (Ω,d), (E,d) be the complexes of sheaves of holomorphic and C-valued C forms respectively. By the holomorphic and C Poincaré lemmas, one has quasi-isomorphisms: (C 0 ) (Ω,d) (E,d),

Algebraic Cycles and Transcendental Algebraic Geometry 19 where the latter two are Hodge filtered, using an argument similar to that in (11) below. The first spectral sequence of hypercohomology shows that H k (,C) H k (C 0 ) H k ((F p )Ω ) H k ((F p )E ). The second spectral sequence of hypercohomology applied to the latter term, using the known acyclicity of E, yields H k (F p E ) kerd : F p E k F p E k df p E k 1 F p H k DR(), where the latter isomorphism is due to the Hodge to de Rham spectral sequence. 7 Deligne cohomology A standard reference for this section is [22] (also see [31]). For a subring A R, we introduce the Deligne complex A D (r) : A(r) O Ω 1 Ω r 1. }{{} call this Ω <r Definition 7.1 Deligne cohomology is given by the hypercohomology: HD i (,A(r)) = Hi (A D (r)). Example 7.2 When A = Z, we have a quasi-isomorphism Z D (1) O [ 1], hence H 2 D (,Z(1)) H1 (,O ) =: Pic() CH1 (). H 1 D (,Z(1)) H0 (,O ) C CH 1 (,1). Example 7.3 Alternate take on HD 1 (,Z(1)). Look at the Cech double complex: C 0 (U,Z(1)) C 0 (U,O ) δ δ C 1 (U,Z(1)) C 1 (U,O ) So a class in H 1 D (,Z(1)) is represented (after a suitable refinement) by (λ := {λ αβ }, f := { f γ }) ( Γ (U α U β,z(1)),γ (U γ,o ) ), with f β f α =: δ( f ) αβ = λ αβ. Note that exp( f ) H 0 (,O ) determines the isomorphism H1 D (,Z(1)) H 0 (,O ) C.

20 James D. Lewis Definition 7.4 The product structure on Deligne cohomology H k D (,Z(i)) Hl D (,Z( j)) Hk+l D (,Z(i + j)), is induced by the multiplication of complexes µ : Z D (i) Z D ( j) Z D (i + j) defined by x y, if degx = 0, µ(x,y) := x dy, if degx > 0 and degy = j > 0, 0, otherwise. Example 7.5 For example, this product structure implies that H 1 D (,Z(1)) H1 D (,Z(1)) = {0} H2 D (,Z(2)). Recall from Hodge theory, one has the isomorphism: H i (Ω r ) F r H i (,C). This together with the short exact sequence of complexes: implies that 0 Ω r Ω Ω <r 0, H i (Ω <r ) Hi (,C) F r H i (,C). Thus applying H ( ) to the short exact sequence: yields the short exact sequence: 0 Ω <r [ 1] A D (r) A(r) 0, H i 1 (,C) 0 H i 1 (,A(r)) + F r H i 1 HD i (,C) (,A(r)) Γ A( H i (,A(r)) ) 0. }{{} ) =J A (H i 1 (,A(r)) If we consider A = Z, and i = 2r, then (7) becomes: 0 J r () H 2r D (,Z(r)) Hgr () 0, where J r () = J ( H 2r 1 (,Z(r)) ) is the Griffiths jacobian, and where the Hodge group Hg r () in untwisted form can be identified with: {w H 2r (,Z) w H r,r (,C)}, via ker ( H 2r (,Z) H 2r (Ω <r ) ) ). In particular, Hg r () includes the torsion classes in H 2r (,Z(r)). (7)

Algebraic Cycles and Transcendental Algebraic Geometry 21 Next, if A = Z and i 2r 1, then from Hodge theory, H i (,Z(r)) F r H i (,C) is torsion. In particular, there is a short exact sequence: 0 H i 1 (,C) F r H i 1 (,C) + H i 1 (,Z(r)) Hi D (,Z(r)) Hi tor(,z(r)) 0, where H i tor(,z(r)) is the torsion subgroup of H i (,Z(r)). The compatibility of Poincaré and Serre duality yields the isomorphism: H i 1 (,C) F r H i 1 (,C) + H i 1 (,Z(r)) Fd r+1 H 2d i+1 (,C) H 2d i+1 (,Z(d r)). Next, if A = R and i = 2r 1, then H i tor(,r(r)) = 0; moreover if we set π r 1 : C = R(r) R(r 1) R(r 1) to be the projection, then we have the isomorphisms: HD 2r 1 H 2r 2 (,C) (,R(r)) F r H 2r 2 (,C) + H 2r 2 (,R(r)) π r 1 H 2r 2 (,R(r 1)) ( π r 1 F r H 2r 2 (,C) ) =: H r 1,r 1 (,R(r 1)) { H d r+1,d r+1 (,R(d r + 1)) }. 7.6 Alternate take on Deligne cohomology Let h : (A,d) (B,d) be a morphism of complexes. We define Cone ( A h B ) by the formula [ ( Cone A h B )] q := A q+1 B q, δ(a,b) = ( da,h(a) + db). Example 7.7 Cone ( A(r) F r Ω ε l Ω ) [ 1] is given by: A(r) O d Ω d d Ω r 2 (0,d) ( Ω r Ω r 1 ) δ (Ω r+1 Ω r ) δ δ (Ω d Ω d 1 ) Ω d Using the holomorphic Poincaré lemma, one can show that the natural map

22 James D. Lewis is a quasi-isomorphism 6. Thus A D (r) Cone ( A(r) F r Ω H k D (,A(r)) Hr( Cone ( A(r) F r Ω ε l Ω ) [ 1], ε l Ω ) [ 1] ). Let D be the sheaf of currents acting on C compactly supported (2d )- forms. Further, let D p,q be the sheaf of currents acting on C compactly supported(d p,d q)- forms. One has a decomposition D = p+q= D p,q, with a morphism of complexes E D ( induced by ω (2πi) d ω ( )), and with E p,q D p,q, compatible with both and. Likewise, let C = C 2d,(A(r)) be the sheaf of (Borel-Moore) chains of real codimension. Identifying the constant sheaf A(r) with the complex A(r) 0 0, we have quasi-isomorphisms A(r) C (A(r)), E D where the latter is (Hodge) filtered. Observe that D ()) [ 1] is a subcomplex of Cone ( C (,A(r)) Fr D ε l () D ()) [ 1]. Hence the cone complex description of: H i D (,A(r)) Hi( Cone ( C (,A(r)) F r D () ε l D () ) [ 1] ), yields the exact sequence 7 : H i 1 (,A(r)) F r H i 1 (,C) H i 1 (,C) (8) H i D (,A(r)) Hi (,A(r)) F r H i (,C) 7.8 Deligne-Beilinson cohomology The formulation of Deligne cohomology in Definition 7.1 above, which incidentally can be defined in the same way for any complex manifold (and is also called analytic Deligne cohomology), works well for projective algebraic manifolds, but not so Ω r 1 δ 6 Indeed first consider (a,b) Ω r ( da,db a) Ω r+1 Ω r. Then δ(a,b) = (0,0) da = 0 & a = db a = db. Therefore kerδ/im(0,d) Ω r 1 /dω r 2 = H r 1 (A D (r)). Next, for j 1, (a,b) Ω r+ j r+ j 1 Ω, δ(a,b) = 0 (a,b) = δ( b,0). 7 The reader familiar with Deligne homology will see this definition as the same thing up to twist. Indeed this definition already incorporates Poincaré duality.

Algebraic Cycles and Transcendental Algebraic Geometry 23 well for smooth open U. First of all, the naive Hodge filtration on U, viz., Ω r U is the wrong choice. For example, if W is a Stein manifold, then H q (W,Ω i W ) = 0 for all i and where q 1. This tells us, via the Grothendieck spectral sequences associated to hypercohomology, that H j (W,C) H0 (W,Ω j W ) d closed dh 0 (W,Ω j 1 W ). (Note: If W is a smooth affine variety, then by Grothendieck, one can use algebraic differential forms.) We hardly expect H j (W,C) = F j H j (W,C) to be the case in general. Secondly, analytic Deligne cohomology fails to take into consideration the underlying algebraic structure of U. For instance HD 1 (U,Z(1)) = H0 (U,O U,an ), i.e. the non-zero analytic functions on U. It would be preferable to recover the non-zero algebraic functions on U instead. Beilinson s remedy is to incorporate Deligne s logarithmic complex into the picture. By a standard reduction, we may assume that j : U = \Y, where Y is a normal crossing divisor 8 with smooth components. We define Ω Y to be the de Rham complex of meromorphic forms on, holomorphic on U, with at most logarithmic poles along Y. One has a filtered complex F r Ω Y = Ω r Y, with Hodge to de Rham spectral sequence degenerating at E 1. This gives F r H i (U,C) = H i (F r Ω Y ) H i (Ω Y ) = H i (U,C), (9) and defines the correct Hodge filtration. The weight filtration is characterized in terms of differentials with residues along Y [ ], where Y [ ] is the simplicial complex made up of the intersections of the irreducible components of Y. Definition 7.9 Deligne-Beilinson cohomology is given by H i D (U,A(r)) := Hi (A D (r)), where A D (r) := Cone ( R j A(r) F r Ω Y ε l R j Ω U) [ 1]. Here ε and l are the natural maps obtained after a choice of (the direct image of) injective resolutions of A(r) and Ω U. One shows that this is independent of the good compactifications of U. One has a short exact sequence: 0 where H i 1 (U,C) F r H i 1 (U,C) + H i 1 (U,A(r)) Hi D (U,A(r)) Fr H i (U,A(r)) 0, (10) 8 Y is a normal crossing divisor, which in local analytic coordinates (z 1,...,z d ) on, Y is given by z 1 z l = 0, and so Ω 1 Y has local frame { dz 1 /z 1,...,dz l /z l,dz l+1,...,dz d }.

24 James D. Lewis F r H i (U,A(r)) := ker ( F r H i (U,C) H i (U,A(r)) ε l H i (U,C) ). We would like a more earthly description of HD i (U,A(r)). First observe that there are filtered quasi-isomorphisms where (F r,ω Y ) (F r,e Y ) (F r,d Y ), (11) F r D Y = {F r Ω Y } Ω D. To see this, one uses the argument in [31]. By a spectral sequence argument, it is enough to show that the associated graded pieces in (11) are quasi-isomorphic, viz., Ω r Y Ω r Y O E 0, Ω r Y O D 0,, where the differential is now 1. One now applies the lemma together with the flatness of Ω r Y over O, and using O as -linear. According to [39], D Y admits the interpretation of the space of currents acting on those (compactly supported) forms on which vanish holomorphically on Y. Let C i (,A(r)) be the chains of real codimension i in, and CY i (,A(r)) the subspace of chains supported on Y. Put C i (,Y,A(r)) := C i (,A(r)) C i Y (,A(r)). One has a map of complexes: ( C (,Y,A(r)),d ) ( D Y (),d ), which induces a quasi-isomorphism C (,Y,A(r)) C D Y (). Definition 7.10 Deligne-Beilinson cohomology is given by H i D (U,A(r)) := Hi( Cone ( C (,Y,A(r)) F r D Y () ε l (D Y () ) [ 1] ). Example 7.11 Let us compute HD 1 (U,Z(1)). First of all {ξ } H1 D (U,Z(1)) is represented by a D-closed triple: ξ = (a,b,c) ( C 1 (,Y,Z(1)) F 1 D 1 Y () D 0 Y () ), where da = 0, db = 0 and a b = dc. Note that -regularity implies that b Ω 1 Y () d closed. Let ˆΩ U 1 be the sheaf of d-closed holomorphic 1-forms on U, and let s make the identification C = C/Z(1). From the short exact sequence: 0 C O U d log ˆΩ 1 U 0,

Algebraic Cycles and Transcendental Algebraic Geometry 25 and the relation a b = dc, it follows that b ker ( H 0 (U, ˆΩ 1 U) H 1 (U,C ) ), and hence b = d log f for some f O U (U). Since b Ω 1 Y (), it follows that f O U,alg (U). Thus in Deligne cohomology9 {ξ } = ( 2πiT ξ,ω ξ,r ξ ), where T ξ := δ f 1 (R ) is given by integration along f 1 [,0], and Ω ξ = [d log f ], R ξ = [log f ] are the obvious defined currents. Corollary 7.12 is an isomorphism. cl 1,1 : CH 1 (U,1) := O U,alg (U) H 1 D (U,Z(1)), Remark 7.13 We observe in passing the following. We deduce from (10) the short exact sequence: 0 H 0 (U,C) F 1 H 0 (U,C) + H 0 (U,Z(1)) H1 D (U,Z(1)) Γ ( H 1 (U,Z(1)) ) 0, which in turn from Corollary 7.12 yields the short exact sequence: 0 C CH 1 (U,1) d log Γ ( H 1 (U,Z(1)) ) 0. Remark 7.14 Let U/C be a smooth quasi-projective variety. If H D,an (U,Z( )) denotes that analytic Deligne cohomology, then we know that HD,an 2 (U,Z(1)) H 1 (U,O U,an ), the holomorphic isomorphism classes of holomorphic line bundles over U. For Deligne-Beilinson cohomology, and using the fact that H 1 (U,Z(1)) = W 0 H 1 (U,Z(1)), it follows that there is a short exact sequence.: but in general 0 J ( H 1 (U,Z(1)) ) H 2 D (U,Z(1)) α F 1 H 2 (U,Z(1)) 0, Γ ( H 2 (U,Z(1)) ) = F 0 W 0 H 2 (U,Z(1)) F 1 H 2 (U,Z(1)), 9 For compactly supported ω EU,c 2d 1, and f O U (U), U d f f ω = d ( log f ω ) log f dω = 2πi ω + d[log f ](ω), U U\ f 1 [,0] f 1 [,0] where we use the principal branch of log.

26 James D. Lewis where the shift F 0 F 1 is really the same filtration, but the latter is in untwisted terminology. To remedy this, let us put H 2 H (U,Z(1)) = α 1( Γ ( H 2 (U,Z(1)) )). This turns out to be the same thing as the image H 2 D (U,Z(1)) H2 D (U,Z(1)), where U is any smooth projective compactification of U. Then H 2 H (U,Z(1)) amounts to a special instance of Beilinson s absolute Hodge cohomology (see [2]). We then have the following: Proposition 7.15 Let U/C be a smooth quasi-projective variety. Then: Proof. First recall that There is a commutative diagram: H 2 H (U,Z(1)) H1 Zar(U,O U,alg ) H 1 Zar(U,O U,alg ) = H1 Zar(U,K M 1,U) = CH 1 (U). 0 CH 1 hom (U) CH1 (U) CH 1 (U) CH 1 hom (U) 0 Φ 1 cl 1 0 J ( H 1 (U,Z(1)) ) H 2 H (U,Z(1)) Γ ( H 2 (U,Z(1)) ) 0 It suffices to show that Φ 1 is an isomorphism. Let U be a smooth projective compactification of U. We may assume that Y := U\U is a divisor. With regard to the short exact sequence: 0 H 1 (U,Z(1)) H 1 (U,Z(1)) H 2 Y (U,Z(1)) 0, it is clear that J ( H 2 Y (U,Z(1)) ) = 0, and hence the following diagram finishes the proof: CH 1 Y (U) CH 1 hom (U) CH1 hom (U) 0 Φ 1 Γ H 2 Y (U,Z(1)) J ( H 1 (U,Z(1)) ) J ( H 1 (U,Z(1)) ) 0

Algebraic Cycles and Transcendental Algebraic Geometry 27 8 Examples of H r m Zar (,Kr, M ) and corresponding regulators The reader is encouraged to consult for example [40] and [48] (as well as works due to Bloch, Beilinson, Esnault and Goncharov), for various earlier incarnations of regulator type currents for higher Chow cycles. A complete description of the Beilinson/Bloch regulator in terms of polylogarithmic type currents for complex varieties can be found in [36] and [37]. 8.1 Case m = 0 and CY threefolds In this case we recall that The fundamental class map: H r Zar(,K M r, ) = CH r (). cl r : CH r () H 2r DR(,C) H 2d 2r DR (,C), can be defined in a number of equivalent ways: (i) (See [21].) The d log map Kr, M Ω r, { f 1,..., f r } j d log f j, induces a morphism of complexes in the Zariski topology {Kr, M r 0} Ω [r], and thus using GAGA, CH r () = HZar(,K r r, M ) = H r( {Kr, M 0} ) H r( Ω r [r] ) = H 2r (Ω r ) = F r HDR(,C). 2r (ii) Let V be a subvariety of codimension r in, and {w} HDR 2d 2r (,C), (de Rham cohomology). Define cl r (V )(w) = 1 (2πi) d r δ 1 V := (2πi) d r w, V and extend to CH r () by linearity, where V = V \V sing. Note that dim R V = 2d 2r. The easiest way to show that cl r is well-defined (finite volume, closed current) is to first pass to a desingularization of V above, and apply a Stokes theorem argument. The proof of a more direct approach can be found, for example, in [27]. (One way to connected (i) and (ii) is as follows. If we write Γ for H 0 (, ), then there is a diagram that commutes up to sign:

28 James D. Lewis Γ K M r (C()) Γ cd Y =1 K M r 1 (C(Y )) Γ cd V =r K M 0 (C()) d log r (2πi) d Y d log r 1 (2πi) d 1 V d log 0 (2πi) d r where Γ F r D r d Γ F r D r+1 ( d log r { f1,..., f r } ) r = d log f j, j=1 d d V Γ F r D 2r d log 0 (2πi) d r = 1 (2πi) d r δ V. From the aforementioned filtered quasi-isomorphism Ω D, the prescriptions in (i) and (ii) can be seen as almost tautologies.) (iii) Thirdly one has a fundamental class generator {V } H 2d 2r (V,Z(d r)) H 2r V (,Z(r)) H 2d 2r(,Z((d r)) H 2r (,Z(r)). In summary we have cl r : CH r () Hg r (). This map fails to be surjective in general for r > 1 (see [41]). Conjecture 8.2 (Hodge Q ) Next, the Abel-Jacobi map: is defined as follows. Recall that J r () = cl r : CH r () Q Hg r () Q, is sur jective. Φ r : CH r hom () Jr (), H 2r 1 (,C) F r H 2r 1 (,C) + H 2r 1 (,Z(r)) Fd r+1 H 2d 2r+1 (,C) H 2d 2r+1 (,Z(d r)), is a compact complex torus, called the Griffiths jacobian. Prescription for Φ r : Let ξ CH r hom (). Then ξ = ζ bounds a 2d 2r + 1 real dimensional chain ζ in. Let {w} F d r+1 H 2d 2r+1 (,C). Define: Φ r (ξ )({w}) = 1 (2πi) d r w ζ (modulo periods). That Φ r is well-defined follows from the fact that F l H i (,C) depends only on the complex structure of, namely F l H i (,C) Fl E,d closed i d ( F l E i 1 ), where we recall that E i are the C complex-valued i-forms on.

Algebraic Cycles and Transcendental Algebraic Geometry 29 Alternate take for Φ r : Let ξ CH r hom (). First observe that H 2r 1 ξ (,Z) H 2d 2r+1 ( ξ,z) = 0, as dim R ξ = 2d 2r. Secondly there is a fundamental class map ξ {ξ } H 2d 2r ( ξ,z(d r)) H ξ 2r (,Z(r)) (Poincaré duality). Further, since ξ is nulhomologous, we have by duality [ξ ] H 2r ξ (,Z(r)) := ker ( H 2r ξ (,Z(r)) H2r (,Z(r)) ). Hence ξ determines a morphism of MHS, Z(0) H 2r ξ (,Z(r)). From the short exact sequence of MHS, 0 H 2r 1 (,Z(r)) H 2r 1 (\ ξ,z(r)) H 2r ξ (,Z(r)) 0, we can pullback via this morphism to obtain another short exact sequence of MHS, 0 H 2r 1 (,Z(r)) E Z(0) 0. Then Φ r (ξ ) := {E} Ext 1 MHS( Z(0),H 2r 1 (,Z(r)) ). This class {E} is easy to calculate in J r (), in terms of a membrane integral. Note that via duality, E H 2r 1 (\ ξ,z(r)) H 2d 2r+1 (, ξ,z(d r)), and that if ζ is a real 2d 2r + 1 chain such that ζ = ξ on, then {ζ } H 2d 2r+1 (, ξ,z). One can show that the class x W 0 E corresponding to the current 1 (2πi) d r, ζ maps to 1 Z(0). Now choose y F 0 W 0 E C also mapping to 1 Z(0). By Hodge type alone, the current corresponding to x y in the Poincaré dual description of J r () is the same as for x = 1 (2πi) d r ζ, which is precisely the classical description of the Griffiths Abel-Jacobi map. This next result is a consequence of the work of Griffiths (see [29], as well as 14 of [41]). Theorem 8.3 If F r 1 H 2r 1 (,C) H 2r 1 (,Q(r)) = 0, then there is an induced map Φ r : Griff r () J r (). In particular Φ r ( CH r alg () ) = 0 J r (). This is the case for a general CY threefold with r = 2. Example 8.4 We define the cycle class map cl r : CH r () HD 2r (,Z(r)). Recall the short exact sequence: 0 J r () H 2r D (,Z(r)) Hgr () 0. Let ξ CH r () with support ξ. One has a similar LES as in (8):

30 James D. Lewis H 2r 1 ξ (,Z(r)) F r H 2r 1 ξ (,C) H 2r 1 ξ (, C) HD, ξ 2r (,Z(r)) H2r ξ (,Z(r)) Fr H ξ 2r x y (,C) H ξ 2r (,C) Via Poincaré duality, one has cycle class maps ξ [ (2πi) r d( )] ( {ξ },δ ξ ker H 2r ξ (,Z(r)) Fr H ξ 2r (,C) H2r ξ (,C)) ; moreover recall that H 2r 1 (,C) = 0 (weak purity). Thus we have a class [ξ ] ξ (,Z(r)). Now use the forgetful map H 2r D, ξ HD, ξ 2r (,Z(r)) H2r D (,Z(r)), to define cl r (ξ ) HD 2r (,Z(r)). From the injection HD, ξ 2r (,Z(r)) H2r ξ (,Z(r)) Fr H ξ 2r (,C), and the aforementioned forgetful map, in terms of the cone complex, cl r (ξ ) is represented by ( (2πi) r d {ξ },(2πi) r d δ ξ,0 ). If ξ hom 0, then ξ = ζ, (2πi) r d δ ξ = ds for some S F r D 2r 1 (). So D((2πi) r d ζ,s,0) + ( (2πi) r d {ξ },(2πi) r d δ ξ,0 ) ) = ((0,0,(2πi) r d S. For ω F d r+1 H 2d 2r+1 (,C), (2πi) r d ζ ω S(ω) = by Hodge type. This is the Griffiths Abel-Jacobi map. Both maps (cl r, Φ r ) can be combined to give with commutative diagram: 1 (2πi) d r ω, ζ cl r,0 : CH r () = CH r (,0) H 2r D (,Z(r)), 0 CH r hom () CHr () CHr () CH r hom () 0 Φ r cl r,0 cl r 0 J r () H 2r D (,Z(r)) Hgr () 0. ζ

Algebraic Cycles and Transcendental Algebraic Geometry 31 8.5 Deligne cohomology and normal functions Suppose that ξ CH r () is given and that Y is a smooth hypersurface. Then there is a commutative diagram CH r () CH r (Y ) HD 2r (,Z(r)) H2r D (Y,Z(r)); Further, if we assume that the restriction ξ Y CH hom (Y ) is null-homologous, then cl r,0 (ξ ) HD 2r(,Z(r)) Jr (Y ) HD 2r(Y,Z(r)). Next, if Y = 0 { t } t S is a family of smooth hypersurfaces of, then such a ξ determines a holomorphically varying map ν ξ (t) J r ( t ), called a normal function. The class cl r (ξ ) = δ(ν ξ ) Hg r () is called the topological invariant of ν ξ, i.e. ν ξ determines cl r (ξ ). In [37], these ideas are extended in complete generality to the situation of the higher Chow groups, where the notion of arithmetic normal functions are introduced. Example 8.6 (Griffiths famous example ([29])) Let: = V (z 5 0 + z5 1 + z5 2 + z5 3 + z5 4 + z5 5 ) P5 be the Fermat quintic fourfold. Consider these 3 copies of P 2 : L 1 := V (z 0 + z 1,z 2 + z 3,z 4 + z 5 ), L 2 := V (z 0 + ξ z 2,z 2 + ξ z 3,z 4 + z 5 ), L 3 := V (z 0 + ξ z 1,z 2 + ξ z 3,z 4 + ξ z 5 ). where ξ is a primitive 5-th root of unity. Then L 1 (L 2 L 3 ) = 1 0, hence ξ := [L 2 L 3 ] is a non-zero class in H 2,2 (,Z(2)). Further, if { t } t U P 1 is a general pencil of smooth hyperplane sections of, and if t U, then it is well known that ξ t CH 2 hom ( t) by a theorem of Lefschetz. Since δ(ν ξ ) = [L 2 L 3 ] 0, it follows that ν ξ (t) is non-zero for most t U. Therefore for general t U, Griff 2 ( t ) contains an infinite cyclic group by Theorem 8.3. The upshot is that if: ( Y = V z 5 0 + z5 1 + z5 2 + z5 3 + z5 4 + ( 4 ) ) 5 a j z j P 4, j=0 for general a 0,...,a 4 C, then Griff 2 (Y ) 0 contains an infinite cyclic subgroup. H. Clemens was the first to show that the Griffiths group of a general quintic threefold in P 4 is (countably) infinite dimensional, when tensored over Q. Later it was shown by C. Voisin that the same holds for general CY threefolds. The idea is to make use of the rational curves on such threefolds.

32 James D. Lewis Theorem 8.7 (See [4], [28], [30], [29], [14], [50]) Let P 4 be a (smooth) threefold of degree d. If d 4, then Φ 2 : CH 2 hom () J 2 () is an isomorphism. Now assume that is general. If d 6 then Im ( Φ 2 ) is torsion. If d = 5, then Im ( Φ2 ) Q is countably infinite dimensional. Theorem 8.8 ([50]) If is a general Calabi-Yau threefold, then Im ( Φ 2 ) is countably infinite dimensional, when tensored over Q. In particular, since Φ 2 (CH 2 alg ()) = 0, it follows that Griff 2 (;Q) is (countably) infinite dimensional over Q. 8.9 Case m = 1 and K3 surfaces Recall the Tame symbol map T : K2 M (C(Z)) Then: codim Z=r 2 codim D=r 1 K M 1 (C(D)). { CH r (,1) = HZar r 1 (,K r, M j ( f j,d j ) } j div( f j ) = 0 ) T ( Γ ( codim Z=r 2 K2 M(C(Z)))). We recall: Definition 8.10 The subgroup of CH r (,1) represented by C CH r 1 () is called the subgroup of decomposables CH r dec (,1) CHr (,1). The space of indecomposables is given by CH r ind (,1) := CHr (,1) CH r dec (,1). The map is given by a map cl r,1 : CH r hom (,1) H2r 1(,Z(r)), D cl r,1 : CH r hom (,1) Fd r+1 H 2d 2r+2 (,C) H 2d 2r+2 (,Z(d r)), defined as follows. Assume given a higher Chow cycle ξ = N i=1 ( f i,z i ) representing a class in CH r hom (,1). Then via a proper modification, we can view f i : Z i P 1 as a morphism, and consider the 2d 2r + 1-chain γ i = fi 1 ([,0]). Then N i=1 div( f i) = 0 implies that γ := N i=1 γ i defines a 2d 2r + 1-cycle. Since ξ is null-homologous, it is easy to show that γ bounds some real dimensional 2d 2r + 2-chain ζ in, viz., ζ = γ. For ω F d r+1 H 2d 2r+2 (,C), the current defining cl r,1 (ξ ) is given by:

Algebraic Cycles and Transcendental Algebraic Geometry 33 cl r,1 (ξ )(ω) = [ 1 N (2πi) i=1 d r+1 ω log f i Z i \γ i ] 2πi ω, ζ where we choose the principal branch of the log function. (This is different branch from the one chosen in [40], for this regulator.) One can easily check that the current defined above is d-closed. Namely, if we write ω = dη for some η F d r+1 E 2d 2r, then by a Stokes theorem argument, both integrals above contribute to periods which cancel. The details of this argument can be found in [25], but quite generally can be found in [36]. Using the description of real Deligne cohomology given above, and the regulator formula, we arrive at the formula for the real regulator r r,1 : CH r (,1) (,R(r)) = H r 1,r 1 (,R((r 1)) H d r+1,d r+1 (,R(d r+1)). Namely: H 2r 1 D 1 r r,1 (ξ )(ω) = (2πi) d r+1 ω log f j. j Z j Example 8.11 Suppose that is a surface. Then we have cl 2,1 : CH 2 hom (,1) {H2,0 () H 1,1 ()}. H 2 (,Z) The corresponding transcendental regulator is defined to be and real regulator Φ 2,1 : CH 2 hom (,1) H2,0 () H 2 (,Z), Φ 2,1 (ξ )(ω) = ω. ζ There is an induced map r 2,1 : CH 2 (,1) H 1,1 (,R(1)) H 1,1 (,R(1)), r 2,1 (ξ )(ω) = 1 2πi j Z j log f j ω. r 2,1 : CH 2 ind (,1) H1,1 tr (,R(1)). If is a K3 surface, then CH 2 hom (,1) = CH2 (,1), hence there is an induced map Φ 2,1 : CH 2 ind (,1) H2,0 () H 2 (,Z). Theorem 8.12 (i) ([45]) Let P 3 be a smooth surface of degree d. If d 3, then r 2,1 : CH 2 (,1) H 1,1 (,R(1)) is surjective; moreover CH 2 ind (,1;Q) = 0. Now

34 James D. Lewis assume that is general. If d 5, then Im(r 2,1 ) is trivial, i.e. its image in the transcendental part of H 1,1 (,R(1)) is zero. (ii) [Hodge-D-conjecture for K3 surfaces ([10])] Let be a general member of a universal family of projective K3 surfaces, in the sense of the real analytic topology. Then r 2,1 : CH 2 (,1) R H 1,1 (,R(1)), is surjective. (iii) ([12]) Let /C be a general algebraic K3 surface. Then the transcendental regulator Φ 2,1 is non-trivial. Quite generally, if is a general member of a general subvariety of dimension 20 l, describing a family of K3 surfaces with general member of Picard rank l, with l < 20, then Φ 2,1 is non-trivial. Remark 8.13 (i) Regarding part (iii) of Theorem 8.12, one can ask whether Φ 2,1 can be non-trivial for those K3 surfaces with Picard rank 20, (which are rigid and therefore defined over Q)? In [12], some evidence is provided in support of this. (ii) One of the key ingredients in the proof of the above theorem is the existence of plenty of nodal rational curves on a general K3 surface. Indeed, there is the following result: Theorem 8.14 ([9]) For a general K3 surface, the union of rational curves on is a dense subset in the analytic topology. Remark 8.15 It is well known that for an elliptic curve E defined over an algebraically closed subfield k C, the torsion subgroup E tor (C) E(k). An analogous result holds for rational curves on a K3 surface. Quite generally, the following result which may be common knowledge among experts, seems worthwhile mentioning: Proposition 8.16 Assume given /C a smooth projective surface with Pg() := dimh 2,0 () > 0. If we write /C = k k C, viz., /C obtained by base change from a smooth projective surface k defined over an algebraically closed subfield k C, and if C /C is a rational curve, then C is likewise defined over k. Proof. By a standard spread argument, there is a smooth projective variety S/k of dimension 0, and a k-family C S of rational curves containing C as a general member, with embedding h: C Pr S h S k Pr S Since Pg() > 0, there are only at most a countable number of rational curves on /C, and hence Pr (h(c )) = Pr (h(prs 1 (t))) for any t S(C). Now use the fact that S(k) /0.