LECTURES IN TRANSCENDENTAL ALGEBRAIC GEOMETRY: ALGEBRAIC CYCLES WITH A SPECIAL EMPHASIS ON CALABI-YAU VARIETIES

Transcription

1 LECTURES IN TRANSCENDENTAL ALGEBRAIC GEOMETRY: ALGEBRAIC CYCLES WITH A SPECIAL EMPHASIS ON CALABI-YAU VARIETIES JAMES D. LEWIS Abstract. These notes form an expanded version of some introductory lectures to be delivered at the Workshop on Arithmetic and Geometry of K3 surfaces and Calabi-Yau Threefolds, August 16-25, 2011, at the Fields Institute. 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 generalized cycles 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. Contents 0 Introduction 2 1. Notation 3 2. Some Hodge theory 3 3. Algebraic cycles (classical) 4 4. Generalized cycles 5 5. A detour via Milnor K-theory and the Gersten-Milnor complex Sheafifying everything 8 6. Hypercohomology 9 7. Deligne cohomology Examples of H r m Zar (, Kr, M ) and corresponding regulators Case m = 0 and CY threefolds Deligne cohomology and normal functions Case m = 1 and K3 surfaces Case m = 2 and elliptic curves Bloch s construction on K 2 (), for an elliptic curve 17 References Mathematics Subject Classification. 14C25, 14C30, 14C35. Key words and phrases. Calabi-Yau variety, Abel-Jacobi map, regulator, Deligne cohomology, Chow group. Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. 1

2 2 JAMES D. LEWIS 0 Introduction These lecture 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 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.: (1) cl r,m : CH r (, m) H 2r m D (, A(r) ), where A R is a subring, A(r) := A(2π 1) r is called the Tate twist, and as we will indicate below, some striking evidence that these regulator maps become highly interesting in the case where is Calabi-Yau. More specifically, we consider the following case scenarios below. 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: (2) Φ 2 : CH 2 hom() J 2 () = {H3,0 () H 2,1 ()}, H 3 (, Z) 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()/ch r alg(). When m = 1, the object of interest is the group { ( ) CH 2 j,cd Zj=1 fj, Z j f j C(Z j ) } j (, 1) = 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 (3) cl 2,1 : CH 2 2,0 () H 1,1 tr () ] ind(, 1), H 2 (, Z) where H 1,1 tr () is the transcendental part of H 1,1 (), being the orthogonal complement to the subgroup of algebraic cocyles. In the case m = 2, the objects of interest are the group of symbols: { CH 2 (, 2) = ξ := f j, g j C() } {f j, g j } ( j j,p ( 1) ν p(f j)ν p(g j) ( f νp(g j ) )(p), p ), = 0 g νp(f j )

3 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 3 [ν p = order of vanishing at p], on a smooth projective curve. If we mod out by the subgroup of symbols {f, g} where f, g C, then the group of interest is the quotient group of indecomposables CH 2 ind(, 2). In this case (1) becomes the real regulator: (4) r 2,2 : CH 2 ind(, 2) H 1 (, R). A first point we wish to make is that if is a smooth projective variety of dimension d, where 1 n 3, then the maps and objects r 2,2 in (4) and CH 2 ind(, 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. The reason for this appears to be to the abundance of special types of subvarieties on the Calabi-Yau varieties. Several recent developments in the context of algebraic cycles are included in these lecture notes. The intended target audience is advanced graduate students, post-doctoral fellows and non-specialists. 1. 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 ). 2. Some Hodge theory Some useful reference material for this section is [16] and [24]. Let E k = C-valued C k-forms on. We have the decomposition: E k = E p,q, Ep,q = Eq,p, p+q=k 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 = 1 i 1 < < i p n, I =p, J =q J = 1 j 1 < < j q n 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) = ker d : Ek Ek+1 de k 1.

4 4 JAMES D. LEWIS The operator d decomposes into d = +, where : E p,q E p,q+1. Further d 2 = 0 2 = 2 = 0 = +. Ep+1,q The above decomposition descends to the cohomological level, viz., Theorem 2.1 (Hodge decomposition). H k sing(, Q) Q 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 () Ep,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 2.2 (Poincaré and Serre Duality). The following pairings induced by (w 1, w 2 ) w 1 w 2, are non-degenerate: HDR(, k C) H 2d k DR (, C) C, H p,q () H d p,d q () C. Therefore H k () H 2d k (), H p,q () H d p,d q () Example 2.3. H i (, C) F r H i (, C) F d r+1 H 2n i (, C). For the next three sections, the reader is encouraged to consult the Lectures on Algebraic Cycles on my website: JD.html 3. Algebraic cycles (classical) 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 3.1. (i) z d () = z 0 () = { M j=1 n jp j n j Z, p j }. (ii) z 0 () = z d () = Z{} Z. (iii) Let 1 := V (z 2 2z 0 z 3 1 z 0 z 2 1) P 2, and 2 := V (z 2 2z 0 z 3 1 z 1 z 2 0) P 2. Then z 1 (P 2 ) = z 1 (P 2 ).

5 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 5 (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 of V (using the properties of DVR), together with a proper push-forward.] Divisors in (iv) generate a subgroup Definition 3.2. is called the r-th Chow group of. z r rat() z r (). (rational equivalence) CH r () := z r ()/z r rat(), Remark 3.3. On can show that ξ zrat() r 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 P1 ).] 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 zalg r () zr () of cycles that are algebraically equivalent to zero. There is a fundamental class map (described later) z r () H 2r (, Z) whose kernel is denoted by zhom r (). One has inclusions: Definition 3.4. Put (i) CH r alg() := z r alg ()/zr rat() (ii) CH r hom() := z r hom ()/zr rat() z r rat() z r alg() z r hom() z r (). (iii) Griff r () := z r hom () /z r alg () = CHr hom()/ch r alg(). (Griffiths group) The Griffiths group is known to be trivial in the cases r = 0, 1, d. 4. Generalized cycles The basic idea is this: ( CH r () = Coker cd V =r 1 In the context of Minor 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 ) K M 1 ) div z r (). (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, K1 M (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} = {a, a} = 1

6 6 JAMES D. LEWIS 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 for 0 m 2 are: cd Z=r 2 KM 2 (C(Z)) T div cd Z=r 1 C(Z) cd Z=r Z (5) CH r (, 2) CH r (, 1) CH r (, 0) 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)) (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} ) = D K M 1 ( ( 1) ν D(f)ν D (g) f ν D ) (g) 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 4.1. Taking cohomology of the complex in (5), we have: (i) CH r () := CH r (, 0) = free abelian group generated by subvarieties of codimension r in, modulo divisors of rational functions on subvarieties of codimension r 1 in. (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 (and 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 4.2. (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. 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 Exercise 4.3. Show that ξ 0. [Hint: Using a well-known fact that CH 2 (P 1, 1) C, choose a suitable line P 1 P 2 and show that ξ P 1 1 C.], D

7 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 7 (ii) Again = P 2. Let C be the nodal rational curve given by z 2 2z 0 = z z 0 z 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 / 5. A detour via Milnor K-theory and the Gersten-Milnor complex 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 [2], which provides the basic foundation for the definitions of higher Chow cycles. 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 these notes. 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 r a is graded, and we put: K M F = T (F ) R = Kn M F, n 0 (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 {a, a} = 1. (2.1) {a, a} = { 1, a} = {a, a 1 } = {a 1, a}. Quite generally, one can argue that K M n (F) is generated {a 1,..., a n },

8 8 JAMES D. LEWIS 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), (ii) {a 1,..., a n } = 0, if a i + a i+1 = 1 for some i < n. Assume given a field F with discrete valuation ν : F Z, with corresponding discrete valuation ring {a F ν(a) 0}, and residue field k(ν). One has maps T : K M (F) K M 1(k(ν)). Choose π F such that ν(π) = 1. If we write a = a 0 π i, b = b 0 π j K M 1 (F), then T (a) = i Z = K M 0 (k(ν)) and ij aj T {a, b} = ( 1) b i k(ν) = K1 M (k(ν)) (Tame symbol) Sheafifying everything. Let O be the sheaf of regular functions on, with sheaf of units O. As in [21], we put ( O Kr, M := ) O, (Milnor sheaf), J where J is the subsheaf of the tensor product generated by sections of the form: { τ1 τ k τ i + 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 [15], [22]): Kr, M Kr M (C()) Kk 1(C(Z)) M cd Z=r 2 We have K M 2 (C(Z)) cd Z=r 1 cd Z=1 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: K2 M (C(Z)) T C(Z) div cd Z=r 2 cd Z=r 1 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 : (C(Z)) (C(D)), defined earlier. codim Z=r 2 K M 2 Definition 5.2. For 0 m 2, codim D=r 1 CH r (, m) = H r m Zar (, Kr,). M Remark 5.3. The higher Chow groups CH r (W, m) are defined for any non-negative r & m. See [3], and quasi-projective variety W over a field k. K M 1

9 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 9 6. Hypercohomology An excellent reference for this is the chapter on spectral sequences in [16]. 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 (, Hq 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 H k (S ) := H i( Γ(K ) ). For example if L, is an [double complex] 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 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), where the latter two are Hodge filtered. 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 ) ker d : 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.

10 10 JAMES D. LEWIS 7. Deligne cohomology A standard reference for this section is [14]. 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: H i D(, A(r)) = H i (A D (r)). From Hodge theory, one has the isomorphisms H i (Ω r ) F r H i (, C), 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, (6) 0 H i 1 (, C) H i 1 (, A(r)) + F r H i 1 (, C) Hi D(, A(r)) H i (, A(r)) F r H i (, C) 0. If we put A = Z, and i = 2r, then (6) becomes: 0 J r () H i D(, Z(r)) Hg r () 0, where J r () is the Griffiths jacobian, and the Hodge group H 2r (, Z(r)) F r H 2r (, C) is more precisely given by Hg r () = {w H 2r (, Z(r)) w H r,r (, C), via the map H 2r (, Z(r)) H 2r (, C), induced by the inclusion Z(r) C}. Next, if A = Z and i 2r 1, then from Hodge theory, H 2r 1 (, Z(r)) F r H 2r 1 (, 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)) H i tor(, Z(r)) 0, where H i tor(, Z(r)) is the torsion subgroup of H i (, Z(k)). 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)) F d r+1 H 2d i+1 (, C) H 2d i+1 (, Z(d r)). Finally, 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)

11 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 11 to be the projection, then we have the isomorphisms: H 2r 1 H 2r 1 (, C) D (, R(r)) F r H 2r 1 (, C) + H 2r 1 (, R(r)) π r 1 H r 1,r 1 (, R) R(r 1) =: H r 1,r 1 (, R(r 1)) { H d r+1,d r+1 (, R(d r + 1))}. 8. Examples of H r m Zar (, Kr, M ) and corresponding regulators 8.1. Case m = 0 and CY threefolds. In this case, one works with the commutative diagram: codim Z=r 1 KM 1 (C(Z)) codim Z=r KM 0 (C(Z)). codim Z=r 1 C(Z) div It easily follows from this that: codim Z=r Z HZar(, r Kr,) M Γ( codim Z=r Z) = z r () div ( Γ( codim Z=r 1 C(Z) ) ) { } Free abelian group of codim r subvarieties in = =: CH r (), Divisors of rational functions on Z of codim r 1 in i.e. gives the classical description of CH r (). The fundamental class map: cl r : CH r () H 2r DR(, C) H 2d 2r DR (, C), can be defined in a number of equivalent ways: (i) The d log map K M r, Ωr, {f 1,..., f r } j d log f j, induces a morphism of complexes {K M r, 0} Ω r [r], and thus CH r () = HZar(, r Kr,) M = H r( {Kr, M 0} ) H r( Ω r [r]) = H 2r (Ω r ) = F r HDR(, 2r C). (ii) Let V be a subvariety of codimension r in, and {w} H 2d 2r DR (, C), 1 (de Rham cohomology). Define cl r (V )(w) = (2π w, and extend to 1) d r V 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 (closed current, and integrally defined) is to first pass to a desingularization of V above, and apply some standard Stokes theorem and fundamental class arguments. (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)).

12 12 JAMES D. LEWIS In summary we have cl r : CH r () Hg r () := H 2r (, Z(r)) H r,r (). This map fails to be surjective in general for r > 1 (cf. [24]). Conjecture 8.2 (Hodge). is surjective. Next, the Abel-Jacobi map: is defined as follows. Recall that J r () = cl r : CH r () Q H 2r (, Q(r)) H r,r (), Φ r : CH r hom() J r (), H 2r 1 (, C) F r H 2r 1 (, C) + H 2r 1 (, Z(r)) F d 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}) = w (modulo periods). ζ 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) F l E,d closed i d ( F l E i 1 ), where we recall that E i are the C complex-valued i-forms on. 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 very general CY threefold with r = 2. Both maps (cl r, Φ r ) can be combined to give with commutative diagram: cl r,0 : CH r () = CH r (, 0) H 2r D (, Z(r)), 0 CH r hom() CH r () Φ r cl r,0 cl r CH r () CH r hom () 0 0 J r () H 2r D (, Z(r)) Hgr () 0. The map cl r,0 can be defined using a local version of an exact sequence similar to (6), using a cone complex description of H D (, Z( )), together with a weak purity argument.

13 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 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 H 2r D CH r () CH r (Y ) (, 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)) J r (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. Roughly speaking, the class cl r (ξ) = δ(ν ξ ) H r,r (, Z(r)) is called the topological invariant of ν ξ, i.e. ν ξ determines cl r (ξ). Example 8.5 (Griffiths famous example ([18])). Let: = V (z z z z z z 5 5) P 5 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 + ξz 2, z 2 + ξz 3, z 4 + z 5 ), L 3 := V (z 0 + ξz 1, z 2 + ξz 3, z 4 + ξz 5 ). where ξ is a primitive 5-th root of unity. Then L 3 (L 1 L 2 ) = 1 0, hence ξ := [L 1 L 2 ] 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 know that ξ t CH 2 hom( t ) by a theorem of Lefschetz. Since δ(ν ξ ) = [L 1 L 2 ] 0, it follows that ν ξ (t) is non-zero for most t U. Therefore for very general t U, Griff 2 ( t ) contains an infinite cycle group by Theorem 8.3. The upshot is that if: ( Y = V z0 5 + z1 5 + z2 5 + z3 5 + z4 5 + ( 4 ) ) 5 a j z j P 4, 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. Theorem 8.6 (See [1], [17], [19], [18], [9], [30]). 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 8.7 ([30]). If is a very general Calabi-Yau threefold, then Im ( ) Φ 2 is a 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. j=0

14 14 JAMES D. LEWIS 8.8. Case m = 1 and K3 surfaces. Recall the Tame symbol map T : (C(Z)) (C(D)). Then: codim Z=r 2 K M 2 codim D=r 1 { CH r (, 1) = H r 1 Zar (, KM j r,) (f j, D j ) j div(f } j) = 0 T ( Γ( codim Z=r 2 KM 2 (C(Z)))). We recall: Definition 8.9. The subgroup of CH r (, 1) represented by C CH r 1 () is called the subgroup of decomposables CH r dec(, 1) CH r (, 1). The space of indeconposables is given by The map is given by a map CH r ind(, 1) := CHr (, 1) CH r dec(, 1). K M 1 cl r,1 : CH r hom(, 1) H 2r 1 D (, Z(r)), cl r,1 : CH r hom(, 1) F d 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 = f 1 i ([, 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: [ 1 cl r,1 (ξ)(ω) = (2π N ω log f 1) d r+1 i 2π ] 1 ω, Z i\γ i ζ i=1 where we choose the principal branch of the log function. 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. 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) H 2r 1 D (, R(r)) = H r 1,r 1 (, R((r 1)) H d r+1,d r+1 (, R(d r + 1)). Namely: 1 r r,1 (ξ)(ω) = (2π ω log f 1) d r+1 j. Z j Example Suppose that is a surface. Then we have cl 2,1 : CH 2 hom(, 1) {H2,0 () H 1,1 ()}. H 2 (, Z) j

15 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 15 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 2π 1 j Z j log f j ω. r 2,1 : CH 2 ind(, 1) H 1,1 tr (, R(1)). If is a K3 surface, then CH 2 hom(, 1) = CH 2 (, 1), hence there is an induced map Φ 2,1 : CH 2 ind(, 1) H2,0 () H 2 (, Z). Theorem (i) ([26]) 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 assume that is general. If d 5, then Im(r 2,1 ) is trivial. (ii) [Hodge-D-conjecture for K3 surfaces ([6])] Let be a general member of a universal family of projective K3 surfaces, in the sense of the real analytic topology. Then is surjective. r 2,1 : CH 2 (, 1) R H 1,1 (, R(1)), (iii) ([7]) Let /C be a very general algebraic K3 surface. Then the transcendental regulator Φ 2,1 is non-trivial. Quite generally, if is a very 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 (i) Regarding part (iii) of Theorem 8.11, one can ask whether Φ 2,1 can be non-trivial for those K3 with Picard rank 20, (which are rigid and therefore defined over Q)? In [7], 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 8.13 ([5]). For a very general K3 surface, the set of rational on is a dense subset in the strong (= analytic) topology.

16 16 JAMES D. LEWIS Case m = 2 and elliptic curves. Regulator examples on CH k (, 2) Let be a compact Riemann surface. In [25] there is constructed a real regulator (7) r : CH 2 (, 2) H 1 (, R(1)), given by [ ] ω H 1 (, R) H 1 (, R(1)) log f d log g log g d log f ω = 2 log f d log g ω, (by a Stokes theorem argument). Alternatively, up to a twist, and real isomorphism, this is the same as the regulator r 2,2 in (4). [ ( ) ( )] 1 dg df (8) 2π log f π 1 log g π 1 ω. 1 g f This latter formula has the following homological version (see [28]). Fix p, and consider any loop γ in \{ (f) (g) } based at p. Then via Poincaré duality H 1 (, R) H 1 (, R), (9) γ 1 2π Im ( γ log f dg g log g(p) γ ) df. f To show that the formulas (8) and (9) agree on CH 2 (, 2), we work out the details below. First of all, observe that: ( ) dg π 1 = π 1 (d log g) = ( ) dg 1d arg(g) ; Re = d log g. g g Next, [ Im (log f) dg ] df log g(p) γ g γ f = [ 1π 1 (log f) dg ] df log g(p) γ g γ f = log g(p) arg γ (f) + arg(f)d log g + Note that and by Stokes theorem: d ( arg(f) log g ) = log g d arg(f) + arg(f)d log g, Therefore: ( 1 2π Im log f dg log g(p) γ g γ d ( arg(f) log g ) = log g(p) arg γ (f). γ γ ) df f γ log f d arg(g). = 1 [ ] log f d arg(g) log g d arg(f) 2π γ [ ( ) ( )] 1 dg df = 2π log f π 1 log g π 1. 1 g f γ

17 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 17 Finally, formulas (8) and (9) coincide on CH 2 (, 2) by Poincaré duality, where ω := [γ], and where [γ] H 1 (, R) is the Poincaré dual of γ H 1 (, R) Bloch s construction on K 2 (), for an elliptic curve. Let be an elliptic curve and assume given f, g C() such that Σ := div(f) div(g) are points of order N in Pic(). Then T ( {f, g} N) C and 0 Pic() C. Thus there exists {h i } C() and {c i } C such that {f, g} N {h i, c i } CH 2 (, 2). 1 Note that the terms {h i, c i } do not contribute to the regulator value by the formula in (6.11) above. Clearly this construction takes advantage of the existence of a dense subset of torsion points on. Bloch ([4]) shows that the real regulator is nontrivial for general elliptic curves, and indeed A. Collino ([11]) shows that the regulator image of CH 2 (, 2) for a general elliptic curve is infinite dimensional (over Q). For curves of genus g > 1, the problem of constructing classes in CH 2 (, 2) seems to be related to the fact that under the Abel-Jacobi mapping Φ : J 1 (), p {p p 0 }, the inverse image of the torsion subgroup, Φ 1 (J 1 () tor ), is finite. This is due to Raynaud s mixed characteristic proof of the Mumford-Manin conjecture ([29]). From a different perspective, and using infinitesimal methods, A. Collino (op. cit.) proves that for a general curve of genus g > 1, the image of the regulator map CH 2 (, 2) HD 2 (, Z(2)) is torsion, hence the real regulator image is zero. On the flip side, one can arrive at an analytic proof of the following (weaker) version of the Mumford-Manin conjecture ([25]): 6.15 Let P 2 be a general curve of degree d 3, and consider the Abel- Jacobi map J 1 (). Put Γ = J 1 tor(). Then Γ is dense in d = 3. The basic idea of proof is this. Let [z 0, z 1, z 2 ] be homogeneous cordinates for P 2, and let (x, y) = (z 1 /z 0, z 2 /z 0 ) be corresponding affine cordinates. Via a degeneration argument, one can show that: [ ] log x d log y log y d log x ω 0, for a suitable choice of real 1-form ω on, and for general such P 2. It is notationally more convenient to write this as: [ ] log f d log g log g d log f ω 0, 1 This uses the Weil reciprocity theorem. Let be a compact Riemann surface, f, g C(), and for p, write T p{f, g} = ( 1) νp(g)νp(f) f ν p(g) g νp(f) «C. p Note that for p div(f) S P div(g), we have T p{f, g} = 1. Thus we can write T {f, g} = p Tp{f, g}. Weil reciprocity says that Q p Tp{f, g} = 1. Let us rewrite this as follows. If we write T {f, g} = P M j=1 (c j, p j ), where p j and c j C, then Q M j=1 c j = 1. Now fix p and let us suppose that Np j rat Np for all j. Thus there exists h j C() such that (h j ) = Np j Np. Then T {h j, c j } = (c N j, p) + (c N j, p j ). The result is that T `{f, g} N {h 1, c 1 } {h M, c M } = Q M j=1 (cn j, p) = (1, p) = 0.

18 18 JAMES D. LEWIS for some f, g C(). Now assume that Γ is dense in, and choose 0-cycles ξ f, ξ g close to div(f) and div(g) respectively, such that ξ f, ξ g are supported on Γ. Thus for some integer N > 0, we can write Nξ f = ( f), Nξ g = ( g) for some f, g C(), and for which [ ] log f d log g log g d log f ω 0, (see [25] for details). This leads us to the setting where we may assume that the divisor sets of f and g are torsion points, of order N say. Thus we can extend {f, g} to a class ξ CH 2 (, 2), for which r(ξ) 0. However, as we mentioned above, from the work of A. Collino, the real regulator image is zero for a general curve P 2 of degree 4, and more generally for a general curve of genus g > 1. Thus in our case, we have established that Γ dense in d = 3. The converse statement is obvious. References [1] Bloch, S. and Srinivas, V. Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983), [2] Bass, H. and Tate, J. The Milnor ring of a global field, in Algebraic K-theory II, Lecture Notes in Math. 342, Springer-Verlag, 1972, [3] Bloch, S. Algebraic cycles and higher K-theory, Adv. Math. 61 (1986), [4] Bloch, S. Lectures on Algebraic Cycles, Duke University Mathematics Series IV, [5] Chen, i, J. D. Lewis, Density of rational curves on K3 surfaces, submitted. [6] Chen,. and Lewis, J. D. The Hodge D conjecture for K3 and Abelian surfaces, JAG. [7] Chen,., Doran, C., Kerr, M., Lewis, J. Higher normal functions, derivatives of normal functions, and elliptic fibrations. In progress. [8] Clemens, C.H. Some results about Abel-Jacobi mappings, in Topics in Transcendental Algebraic Geometry (Princeton, 1981/82), (P. Griffiths, ed.), Ann. of Math. Stud. 106, Princeton University Press, Princeton, 1984, [9] Clemens, C.H. Homological equivalence, modulo algebraic equivalence, is not finitely generated, Publ. I.H.E.S. 58 (1983), [10] Clemens, C.H. Curves on generic hypersurfaces, Ann. Scient. Éc. Norm. Sup., 4e série, t. 19 (1989), p. 629 à 636. [11] Collino, A. Griffiths in finitesimal invariant and higher K-theory on hyperelliptic jacobians, J. Algebraic Geometry 6 (1997), [12] Collino, A. Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds, Algebraic K-theory and its applications (Trieste, 1997), World Sci. Publishing, River Edge, NJ, 1999, [13] Coombes, K. and Srinivas, V. A remark on K 1 of an algebraic surface, Math. Ann. 265 (1983), [14] Esnault, H. and Viehweg, E. Deligne-Beilinson cohomology, in Beilinson s Conjectures on Special Values of L-Functions, (Rapoport, Schappacher, Schneider, eds.), Perspect. Math. 4, Academic Press, 1988, [15] Elbaz-Vincent, P. and Müller-Stach, S. Milnor K-theory of rings, higher Chow groups and applications, Invent. Math. 148 (2002), [16] Griffiths, P. and Harris, J. Principles of Algebraic Geometry, John Wiley & Sons, New York, [17] Green, M. Griffiths infinitesimal invariant and the Abel-Jacobi map, J. Differential Geom. 29 (1989), [18] Griffiths, P.A. On the periods of certain rational integrals: I and II, Annals of Math. 90 (1969), [19] Green, M. and Müller-Stach, S. Algebraic cycles on a general complete intersection of high multi-degree of a smooth projective variety, Comp. Math. 100 (3), (1996)

19 ALGEBRAIC CYCLES AND TRANSCENDENTAL ALGEBRAIC GEOMETRY 19 [20] Jannsen, U. Deligne cohomology, Hodge-D-conjecture, and motives, in Beilinson s Conjectures on Special Values of L-Functions, (Rapoport, Schappacher, Schneider, eds.), Perspect. Math. 4, Academic Press, 1988, [21] Kato, K. Milnor K-theory and the Chow group of zero cycles, Contemporary Mathematics, Part I, , [22] Moritz Kerz, The Gersten conjecture for Milnor K-theory, Invent. math. (2008). [23] Levine, M. Localization on singular varieties, Invent. Math. 31 (1988), [24] Lewis, J.D, A Survey of the Hodge Conjecture. [25] Lewis, J.D. Real regulators on Milnor complexes, K-Theory. [26] Müller-Stach, S. Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Algebraic Geometry 6 (1997), [27] Müller-Stach, S. Algebraic cycle complexes, in Proceedings of the NATO Advanced Study Institute on the Arithmetic and Geometry of Algebraic Cycles 548, (Lewis, Yui, Gordon, Müller-Stach, S. Saito, eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, [28] Ramakrishnan, D. Regulators, algebraic cycles, and values of L functions, Contemporary Mathematics, American Mathematical Society 83 (1989), pp [29] Raynaud, M. Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), [30] Voisin, C, The Griffiths group of a general Calabi-Yau threefold is not finitely generated, Duke Math. J. 102 (1) (2000), Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 address: lewisjd@ualberta.ca