Higher Abel-Jacobi Maps

Mark L. Green*

1991 Mathematics Subject Classification: 14C25, 14C30, 14D07, 32G20, 32J25 Keywords and Phrases: Chow group, algebraic cycle, Abel-Jacobi map, Gauss-Manin connection, Hodge structure

For a smooth projective variety X, the structure of the Chow group CH^{p}(X) representing codimension p algebraic cycles modulo rational equivalence, is still basically a mystery when p > 1, even for 0-cycles on a surface. For any p, one has the (rational) cycle class map

ψ_{0}: CH^{p}(X) ⊗ Q → Hdg^{p}(X) ⊗ Q ⊆ H^{2p}(X, Q),

conjecturally surjective by the Hodge conjecture. By the work of Griffiths, we have the (rational) Abel-Jacobi map

ψ_{1} = AJ_{X}^{p}: ker(ψ_{0}) → J^{p}(X) ⊗ Q.

A number of beautiful results have been proved using this invariant (e.g. [Gri]), but through the work of Mumford-Roitman ([Mu],[Ro]) it was realized that the kernel of ψ_{1} can be infinite-dimensional (for 0-cycles on a surface with H^{2,0}(X) ≠ 0), while through the work of Griffiths and Clemens the image of ψ_{1} may fail to be surjective [Gri] or even finitely generated [Cl] (for 1-cycles on a general quintic 3-fold) or yet for not dissimilar geometric situations, by work of Voisin [Vo1] and myself [Gre1], the image of ψ_{1} may be 0 (for 1-cycles on a general 3-fold of degree ≥ 6). At present, there is no explicit description, even conjecturally, for what ker(ψ_{1}) and im(ψ_{1}) look like. Eventually it came to be understood through the work of Beilinson, Bloch, Deligne, and Murre, among others (see [Ja] for a discussion) that there ought to be a filtration

CH^{p}(X)⊗Q = F^{0}CH^{p}(X)⊗Q ⊇ F^{1}CH^{p}(X)⊗Q ⊇ ⋯ ⊇ F^{p+1}CH^{p}(X)⊗Q = 0

with

F^{1}CH^{p}(X) ⊗ Q = ker(ψ_{0})

and

F^{2}CH^{p}(X) ⊗ Q = ker(ψ_{1}).

* Research partially supported by the National Science Foundation.

Documenta Mathematica · Extra Volume ICM 1998 · II · 267–276

This filtration has been constructed in some cases and various geometric candidates for it have been put forward, for example, by S. Saito [Sa] and Jannsen. One suggestion as to what the graded pieces of this filtration should look like is given by Beilinson’s conjectural formula (see [Ja])

Gr^{m}CH^{p}(X) ⊗ Q ≅ Ext_{MM}^{m}(1, h^{2p−m}(X)(p)),

where MM is the conjectural category of mixed motives.

One case that stands out as being well-understood is the case of the relative Chow group CH^{2}(P^{2}, T), where T ⊂ P^{2} is the triangle z_{0}z_{1}z_{2} = 0, roughly described as 0-cycles on P^{2} − T = C^{∗} × C^{∗}, modulo divisors of meromorphic functions f on curves C ⊂ P^{2} such that f = 1 on C ∩ T. There is a series of maps

ψ_{0}: CH^{2}(P^{2}, T) → Z;
ψ_{1}: ker(ψ_{0}) → C^{∗} ⊕ C^{∗};
ψ_{2}: ker(ψ_{1}) → K_{2}(C).

Recall

K_{2}(C) = \frac{C^{∗} ⊗_{Z} C^{∗}}{\{Steinberg relations\}},

where the Steinberg relations are generated by {a ⊗ (1 − a) | a ∈ C − {0, 1}}. It is known (Bloch [Bl], Suslin [Su]) that these are all surjective and ψ_{2} is an isomorphism. These all have simple algebraic descriptions—ψ_{0} is degree; ψ_{1}(a, b) = a⊕b; ψ_{2}(a, b) = {a, b}. The essential tool in proving this is the Suslin reciprocity theorem ([Su], see also [To]).

Another illustrative example (see [Gre2]) is CH^{2}(P^{2}, E), where E is a smooth plane cubic. Here we have a series of maps

ψ_{0}: CH^{2}(P^{2}, E) → Z; ψ_{1}: ker(ψ_{0}) → 0;

ψ_{2}: ker(ψ_{1}) → \frac{C^{∗} ⊗_{Z} E}{θ̃(J^{4})} .

If a, b ∈ P^{2} − E, and L is the line through a and b, which meets E in {p_{1}, p_{2}, p_{3}}, then

ψ_{2}((a) − (b)) = ∑_{i=1}^{3}(\frac{a − p_{i}}{b − p_{i}} ⊗ p_{i}) ∈ C^{∗} ⊗_{Z} E.

Using p_{1} + p_{2} + p_{3} = 0 on E (having taken 0 to be an inflection point), this has an alternative expression

ψ_{2}((a) − (b)) = \frac{(a − p_{2})(b − p_{1})}{(a − p_{1})(b − p_{2})} ⊗ p_{2} + \frac{(a − p_{3})(b − p_{1})}{(a − p_{1})(b − p_{3})} ⊗ p_{3},

which involves cross-ratios on L and is more clearly coordinate-independent. Once again, all three maps are surjective, and ψ_{2} is an isomorphism. ψ_{0} = deg, ψ_{1} = 0, inserted to preserve the pattern. To explain the notation in ψ_{2}, for a ∈ E −div(θ), let

θ̃(a) = θ(a) ⊗ a ∈ C^{∗} ⊗_{Z} E.

Extend the definition of θ̃ to Z_{E}, the group ring of E, by linearity. In Z_{E}, let J be the augmentation ideal {∑_{i} n_{i}(a_{i}) | n_{i} ∈ Z, a_{i} ∈ E, ∑_{i} n_{i} = 0}. Although θ̃(a) depends on the lifting of a to C, on J^{4} it does not depend on the choice of lifting of the elements. Thus θ̃(J^{4}) is well-defined and constitutes a generalization of the Steinberg relations; this group has been given a motivic intepretation by Goncharov and Levin [GL].

These examples provide a model for the general case—one should think of (P^{2}, T) and (P^{2}, E) as analogous to a complete surface with h^{2,0} = 1. For a general X, one has

\begin{matrix}ψ_{0}: CH^{2}(X) → Hdg^{2}(X); \\ ψ_{1}: ker(ψ_{1}) → J^{2}(X); \end{matrix}

where ψ_{0} is the cycle class map to the Hodge classes on X, and ψ_{1} is the Abel-Jacobi map. We have constructed part of the missing map ψ_{2} in the case of 0-cycles on a surface, using a construction that has the potential to work more generally.

The regulator map for a curve

X − D⟶^{(f,g)}C^{∗} × C^{∗}

is a homomorphism r: π_{1}(X − D) → C/(2πi)^{2}Z = C/Z(2) given by

r(γ) = ∫_{γ} log(f)\frac{dg}{g} − log(g(p)) ∫_{γ} \frac{df}{f} ,

where p is a base-point on γ; the answer does not depend on p. If γ = ∂U for U a disc in C^{∗} × C^{∗}, then

r(γ) = ∫_{U} \frac{df}{f} ∧ \frac{dg}{g} .

This formula generalizes to a definition in the more general situation of a nonsingular curve C and a map f: C → X to a smooth projective surface X. If

μ ∈ ker(H^{2}(X, Z)⟶^{f^{∗}}H^{2}(C, Z)),

then f^{∗}μ = dd^{c}g for g ∈ A^{0}(C), unique up to adding a constant. If γ ∈ ker(H_{1}(C, Z) → H_{1}(X, Z)), so that γ = ∂Γ in X, then we define

e_{X,C}(μ, γ) = ∫_{Γ} μ − ∫_{γ} f^{∗}(d^{c}g) ∈ C/Z,

which does not depend on any of the choices. These quantities are known as membrane integrals. More intrinsically, e_{X,C} is the extension class of the extension of mixed Hodge structures (see [Ca])

0 → coker(H^{1}(X) → H^{1}(C)) → H^{2}(X, C) → ker(H^{2}(X) → H^{2}(C)) → 0.

Denote the term on the left H^{1}(C)_{new} and the term on the right H^{2}(X)_{C}; now

e_{X,C} ∈ \frac{Hom_{C}(H^{2}(X)_{C}, H^{1}(C)_{new})}{Hom_{Z}(H^{2}(X)_{C}, H^{1}(C)_{new}) + F^{0}Hom_{C}(H^{2}(X)_{C}, H^{1}(C)_{new})}.

The class e_{X,C} may also be obtained from the image under AJ_{X×C} of the graph of f minus some terms to make it homologous to 0 on X × C.

We may write

H^{2}(X) = ker(NS(X) → H^{2}(C)) ⊕ H^{2}(X)_{tr},

which decomposes

e_{X,C} = (e_{X,C})_{alg} ⊕ (e_{X,C})_{tr}.

The class (e_{X,C})_{alg} contains the same information as the map

ker(NS(X) → H^{2}(C)) → \frac{J^{1}(C)}{Alb(X)}

given by

L ↦ f^{∗}L.

If Z ∈ Z^{2}(X) and ψ_{0}(Z) = 0, ψ_{1}(Z) = 0, then if we lift Z to Z̃ ∈ Z^{1}(C) such that f_{∗}Z̃ = Z and deg(Z) = 0 on each component of C, then AJ_{C}(Z̃) is represented by the extension class e_{C,Z̃} of the extension of mixed Hodge structures

0 → coker(H^{0}(C) → H^{0}(|Z̃|)) → H^{1}(C, |Z̃|) → H^{1}(C)_{new} → 0;

and the divisor Z̃ gives a map coker(H^{0}(C) → H^{0}(|Z̃|)) → 1 and then

e_{C,Z̃} ∈ \frac{Hom_{C}(H^{1}(C)_{new}, 1)}{Hom_{Z}(H^{1}(C)_{new}, 1) + F^{0}Hom_{C}(H^{1}(C)_{new}, 1)}.

The two extensions of MHS fit together to give a 2-step extension of MHS of H^{2}(X)_{tr} by 1, which unfortunately cannot be used directly. By standard identifications, we may think of

(e_{X,C})_{tr} ∈ (R/Z) ⊗_{Z} Hom_{Z}(H^{2}(X)_{tr}, H^{1}(C)_{new})

and

e_{C,Z̃} ∈ (R/Z) ⊗_{Z} Hom_{Z}(H^{1}(C)_{new}, 1).

The tensor product followed by contraction gives an element

e_{X,C,Z̃} ∈ (R/Z) ⊗_{Z} (R/Z) ⊗_{Z} Hom_{Z}(H^{2}(X)_{tr}, 1).

If we let U_{2}^{2}(X) = {e_{X,C,Z̃} | f_{∗}Z̃ = 0 as a 0−cycle on X} and

J_{2}^{2}(X) = \frac{(R/Z) ⊗_{Z} (R/Z) ⊗_{Z} Hom_{Z}(H^{2}(X)_{tr}, 1)}{U_{2}^{2}(X)},

Documenta Mathematica · Extra Volume ICM 1998 · II · 267–276

then Z ↦ [e_{X,C,Z̃}] gives a well-defined invariant

ψ_{2}^{2}: ker(ψ_{1}) → J_{2}^{2}(X)

that is independent of the choices of C and Z̃, and which depends only on the rational equivalence class of Z on X. It is necessary to allow reducible curves C. Claire Voisin [Vo2] has shown that for surfaces with h^{2,0} ≠ 0, the map ψ_{2} has infinite-dimensional image, and also that it need not be injective, so that our ψ_{2}^{2} is only part of the story.

An explanation of the role played by the extension class (e_{X,C})_{alg} comes from Beilinson’s conjectural formula:

Gr^{m}CH^{p}(X) ⊗ Q ≅ Ext_{MM}^{m}(1, h^{2p−m}(X)(p)),

where MM is the conjectural category of mixed motives. The map

f_{∗}: Gr^{1}CH^{1}(C) → Gr^{1}CH^{1}(X)

is followed by a map

f_{∗}^{+1}: ker(f_{∗}) → Gr^{2}CH^{2}(X).

In terms of Beilinson’s formula, this is a map

Ext_{MM}^{1}(1, ker(H^{1}(C) → H^{3}(X))) → Ext_{MM}^{2}(1, coker(H^{0}(C) → H^{2}(X)))

which (see [Ja]) factors through a map

f_{∗}^{+1}: Ext_{MM}^{1}(1, ker(H^{1}(C) → H^{3}(X))) → Ext_{MM}^{2}(1, H^{2}(X)_{tr})).

It is reasonable to expect that it is given by Yoneda product with an element

e ∈ Ext_{MM}^{1}(ker(H^{1}(C) → H^{3}(X)), H^{2}(X)_{tr}).

The philosophical point here is that e should come from

(e_{X,C})_{tr} ∈ Ext_{MHS}^{1}(ker(H^{1}(C) → H^{3}(X)), H^{2}(X)_{tr}).

In fact, one would conjecture that the map

ker(J^{1}(C) → J^{2}(X)) → CH^{2}(X)

is zero if and only if (e_{X,C})_{tr} is torsion—the only if direction has been shown [Gre2].

The question then becomes how to use (e_{X,C})_{tr}. One answer is given by ψ_{2}^{2} above. Another piece of the puzzle is to apply the arithmetic Gauss-Manin connection ∇ to (e_{X,C})_{tr}.

An invariant which complements the one above was obtained in joint work with Phillip Griffiths [GG]. By work of Katz [Ka2] and Grothendieck [Gro], there is for any smooth projective variety X defined over C the arithmetic Gauss-Manin connection

∇_{X/Q}: H^{k}(X, C) → Ω_{C/Q}^{1} ⊗_{C} H^{k}(X, C).

To capture this abstractly, we define an arithmetic Hodge structure (AHS) to be a complex vector space V with a finite descending filtration F^{•}V and a Qlinear connection ∇: V → Ω_{C/Q}^{1} ⊗_{C} V satisfying ∇^{2} = 0 (flatness) and ∇F^{p}V ⊆ Ω_{C/Q}^{1} ⊗_{C} F^{p−1}V (Griffiths transversality) for all p.

A short exact sequence 0 → A⟶^{f}B⟶^{g}C → 0 of AHS (exact on each F^{p}) has extension class

e ∈ Ext_{AHS}^{1}(C, A) = H^{1}(Ω_{C/Q}^{•} ⊗_{C} F^{−•}Hom_{C}(C, A), ∇_{Hom_{C}(C,A)}).

To obtain this, let ϕ ∈ F^{0}Hom_{C}(C, B) be a lifting of g. Now

g ∘ ∇_{Hom_{C}(C,B)}ϕ = 0,

so

∇_{Hom_{C}(C,B)}ϕ = f ∘ e

for a unique e ∈ F^{−1}Hom_{C}(C, A). The class of e in

H^{1}(Ω_{C/Q}^{•} ⊗_{C} F^{−•}Hom_{C}(C, A), ∇_{Hom_{C}(C,A)})

is independent of the choice of ϕ.

A 2-step exact sequence 0 → A → B → C → D → 0 of AHS has a well-defined injective map from the Yoneda Ext

Ext_{AHS}^{2}(D, A) → H^{2}(Ω_{C/Q}^{•} ⊗_{C} F^{−•}Hom_{C}(D, A), ∇_{Hom_{C}(D,A)}).

This is obtained by composing the extension class of the two 1-step extensions 0 → A → B → E → 0, 0 → E → C → D → 0 it breaks into, and then using the natural map

H^{1}(Ω_{C/Q}^{•} ⊗_{C} F^{−•}Hom_{C}(E, A), ∇_{Hom_{C}(E,A)})⊗_{C}
H^{1}(Ω_{C/Q}^{•} ⊗_{C} F^{−•}Hom_{C}(D, E), ∇_{Hom_{C}(D,E)}) →
H^{2}(Ω_{C/Q}^{•} ⊗_{C} F^{−•}Hom_{C}(D, A), ∇_{Hom_{C}(D,A)}).

This is exactly the obstruction to finding an AHS V with an additional increasing filtration W_{•}V by sub-AHS with W_{0} = 0, W_{1} ≅ A, W_{2} ≅ B, W_{3} = V, V/W_{1} ≅ C, and V/W_{2} ≅ D and realizing the given 1-step extensions. The extension class is injective on Ext_{AHS}^{2}(D, A), but it is not clear that all extension classes can occur.

This approach has much in common with the work of Carlson and Hain [CH]. This extension class theory fits in well with the pre-existing arithmetic cycle

class map (see [EP] and work of Srinivas [Sr])

η: CH^{p}(X) ⊗_{Z} Q → H^{2p}(Ω_{X/Q}^{≥p})

whose graded pieces are

η_{m}: ker(η_{m−1}) → H^{m}(Ω_{C/Q}^{•} ⊗_{C} F^{p−•}H^{2p−m}(X, C), ∇_{X/Q});

one expects these to be consistent with the conjectural Bloch-Beilinson-Deligne-Murre filtration on CH^{p}(X) ⊗_{Z} Q. For 0-cycles on a surface, we are able to show that η_{2} is the element of Ext_{AHS}^{2}(H^{2}(X), 1) coming from the 2-step extension of AHS using Z ⊂ C ⊂ X analogous to the construction above in the mixed Hodge structure case. A parallel construction was found independently by Asakura and Saito [AS], who have used it for some interesting geometric applications.

I would like to close by listing a few open problems that particularly appeal to me and which seem especially relevant to the next phase of the study of algebraic cycles.

(i) Hodge-theoretic formula for ∇_{X/Q} In cases of smooth projective varieties over C where Torelli’s theorem holds (i.e. X is determined by Hodge-theoretic data on H^{∗}(X), at least theoretically ∇_{X/Q} is determined on H^{i}(X) by the Hodge structure of H^{i}(X). It would be helpful to have a formula for this. Such a formula, involving Eisenstein series, was found by Katz [Ka1] for elliptic curves. For abelian varieties and K-3 surfaces, it would be very revealing to have a formula for ∇_{X/Q}. It would also be interesting to have an example where Torelli’s theorem fails and ∇_{X/Q} is different for two X’s with the same Hodge structure; the alternative to this is the very attractive prospect that there is a general Hodge-theoretic formula for ∇_{X/Q}.

One facet of this question is the conjecture of Deligne (see [DMOS]), subordinate to the Hodge conjecture, that for X defined over C, a Hodge class ξ necessarily satisfies

∇_{X/Q}ξ = 0.

One possible “explanation” why this might be true is that a formula as alluded to above exists—such a formula would be expected to have the property that if H^{i}(X) = H_{1} ⊕ H_{2} as Hodge structures, then H_{1}, H_{2} would be ∇_{X/Q}-stable.

A related question is to ask whether Gr^{m}CH^{p}(X) ⊗ Q is determined by the Hodge structure of H^{2p−m}(X), or whether one definitely needs further information contained in the motive h^{2p−m}(X), e.g ∇_{X/Q}.

(ii) Gr^{2}CH^{2}(A) for an abelian surface A If Z_{A} is the group ring of A with augmentation ideal J, then

S_{Z}^{2}A ≅ \frac{J^{2}}{J^{3}}

maps surjectively to Gr^{2}CH^{2}(A) by

a ⊗ b ↦ ((a) − (0)) ∗ ((b) − (0)).

Thus

Gr^{2}CH^{2}(A) = \frac{S_{Z}^{2}A}{U}

for some subgroup U ⊂ S_{Z}^{2}A. Describe U in terms of the Hodge structure of A. For A = E_{1} × E_{2} a product of elliptic curves,

Gr^{2}CH^{2}(E_{1} × E_{2}) = \frac{E_{1} ⊗_{Z} E_{2}}{U′}

for some subgroup U′ ⊆ E_{1} ⊗ E_{2}. Somekawa has given a description of U′, but not in explicit Hodge-theoretic terms. The subgroups U, U′ may be thought of as generalized Steinberg relations, as in the example given earlier of Gr^{2}CH^{2}(P^{2}, E). This is an excellent test case.

(iii) Higher regulators for K-groups The Borel regulator map

r: K_{3}(C)^{ind} = Gr_{2}K_{3}(C) → C^{∗}

is, by the work of Goncharov [Go] the Abel-Jacobi map

CH^{2}(P^{3}, T_{2})_{hom} → J^{2}(P^{3}, T_{2}),

where T_{2} is the tetrahedron {z_{0}z_{1}z_{2}z_{3} = 0}. One should think of (P^{3}, T_{2}) as the analogue of a 3-fold with trivial canonical bundle. Conjecturally, r is injective when tensored with Q. Its image has the same qualitative properties that Clemens showed the image of AJ_{X}^{2} possesses for the general quintic 3-fold—zero-dimensional, but not finitely generated even over Q. One should think of r as a “toy model” model for the Abel-Jacobi map for codimension 2 cycles, in much the same was as K_{2}(C) is the toy model for Gr^{2}CH^{2}(X) for a surface with H^{2,0} ≠ 0. The toy model for the higher Abel-Jacobi maps

Gr^{m}CH^{p}(X) ⊗ Q → J_{m}^{p}(X)

should be maps, injective when tensored with Q,

r_{m}^{p}: Gr_{p}K_{2p−m}(C) → \frac{⊗_{Z}^{m}C^{∗}}{U_{m}^{p}}

for some subgroup U_{m}^{p} ⊂ ⊗_{Z}^{m}C^{∗}. For m = 1 these are the Borel regulators, while for m = p they are the isomorphisms to Milnor K-theory. Can these maps, or something like them, be constructed?

(iv) Explicit Suslin Reciprocity Theorem The Suslin Reciprocity Theorem [Su], used for example to compute CH^{2}(P^{2}, T) above, gives the vanishing of certain elements of ∧_{Z}^{m}C^{∗} in K_{m}(C), but does not explicitly produce the elements of the Steinberg ideal that makes them vanish. On some level, these are produced by the proof, but the transfer map or norm map N is not geometrically explicit. For example, given a rational curve Y ⊂ P^{2} and f ∈ C(Y) such that f|_{Y ∩T} = 1, we know not only that div(f) ∈ Z_{0}(C^{∗} × C^{∗}) maps to 1 in K_{2}(C), but in fact if we map it to ∧_{Z}^{2}C^{∗}, if div(f) = ∑_{i} n_{i}p_{i}, it maps to the product of the Steinberg symbols (CR(p_{i}) ∧ (1 − CR(p_{i})))^{n_{i}}, where CR(p_{i}) is the cross-ratio of p_{i} and one point each from the intersections of Y with each of the lines in T, with the product taken over all choices and all i (Goncharov,[Gre2]). For Y of higher genus, there is no comparably satisfying formula. In general, it would be nice to have as simple a version of N as possible for this type of geometric situation.

(v) Definition of F^{2}CH^{p}(X) ⊗ Q

Nori showed [No] that, for p ≥ 3, Z ≡_{AJ} 0 does not imply NZ ≡_{alg} 0 for some N > 0. However, one might hope that for any p, Z ≡_{AJ} 0 implies that there exists a codimension 1 subvariety Y ⊂ X such that Z ⊂ Y and NZ ≡_{hom} 0 on Y for some N > 0. In many ways, a natural definition for F^{2}CH^{p}(X) ⊗ Q is those Z such that both NZ ≡_{AJ} 0 and there exists a codimension 1 subvariety Y ⊂ X such that Z ⊂ Y and NZ ≡_{hom} 0 on Y for some N > 0. This conjecture would make the two plausible definitions of F^{2} the same. It also fits in with what is needed to make the construction of ψ_{2}^{2} go through for codimension 2 cycles in general. In particular, it would imply that for p = 2, Z ≡_{AJ} 0 implies NZ ≡_{alg} 0 for some N > 0.

The general conjecture would be that if Z ∈ F^{m}CH^{p}(X)⊗Q, then there exists a codimension 1 subvariety Y ⊂ X such that Z ⊂ Y and Z ∈ F^{m−1}CH^{p−1}(Y)⊗Q. This is what is needed to carry out the construction of higher Abel-Jacobi maps for arbitrary codimension.

References

[AS] Asakura, M. and Saito, S., “Filtration on Chow groups and higher Abel-Jacobi maps,” preprint.

[Bl] Bloch, Spencer, “An elementary presentation for K-groups and motivic cohomology,” Motives (Seattle, WA, 1991), 239–244, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.

[Ca] Carlson, James A., “Extensions of mixed Hodge structures,” Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 107–127, Sijthoff & Noordhoff, Alphen aan den Rijn— Germantown, Md., 1980.

[CH] Carlson, James A.; Hain, Richard M., “Extensions of variations of mixed Hodge structure,” Actes du Colloque de Théorie de Hodge (Luminy 1987), Astérisque (1989), no. 179-180, 9, 39-65.

[Cl] Clemens, H., “Homological equivalence, modulo algebraic equivalence, is not finitely generated,” Inst. Hautes Études Sci. Publ. Math. No. 58 (1983), 19–38 (1984).

[DMOS] Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Philosophical Studies Series in Philosophy, 20. Springer-Verlag, Berlin-New York, 1982.

[EP] Esnault, Hélène; Paranjape, Kapil H., “Remarks on absolute de Rham and absolute Hodge cycles,” C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 1, 67–72.

[Go] Goncharov, A. B., “Chow polylogarithms and regulators,” Math. Res. Lett. 2 (1995), no. 1, 95–112.

[GL] Goncharov, A. and Levin, A., “Zagier’s conjecture on L(E, 2),” preprint.

[Gre1] Green, Mark L., “Griffiths’ infinitesimal invariant and the Abel-Jacobi map,” J. Differential Geom. 29 (1989), no. 3, 545–555.

[Gre2] Green, Mark L., “‘What comes after the Abel-Jacobi map?,” preprint.

[GG] Green, Mark and Griffiths, Phillip, in preparation.

[Gri] Griffiths, Phillip, “On the periods of certain rational integrals I, II,” Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496–541.

[Gro] Grothendieck, A., “On the de Rham cohomology of algebraic varieties,” Inst. Hautes Études Sci. Publ. Math. No. 29 (1966), 95–103.

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[Ka1] Katz, Nicholas M., “p-adic interpolation of real analytic Eisenstein series,” Ann. of Math. (2) 104 (1976), no. 3, 459–571.

[Ka2] Katz, Nicholas M., “Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin,” Inst. Hautes Études Sci. Publ. Math. No. 39 (1970), 175–232.

[Mu] Mumford, D., “Rational equivalence of 0-cycles on surfaces,” J. Math. Kyoto Univ. 9 (1968), 195–204.

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[Sa] Saito, Shuji, “Motives and filtrations on Chow groups,” Invent. Math. 125 (1996), no. 1, 149–196.

[Sr] Srinivas, V., “Gysin maps and cycle classes for Hodge cohomology,” Proc. Indian Acad. Sci. Math. Sci. 103 (1993), no. 3, 209-247.

[Su] Suslin, A. A., “Reciprocity laws and the stable rank of rings of polynomials,” (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1394–1429.

[To] Totaro, Burt, “Milnor K-theory is the simplest part of algebraic Ktheory,” K-Theory 6 (1992), no. 2, 177–189.

[Vo1] Voisin, Claire, “Une remarque sur l’invariant infinitésimal des fonctions normales,” C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 4, 157– 160.

[Vo2] Voisin, Claire, “Some results on Green’s higher Abel-Jacobi map,” preprint.

Mark L. Green

Department of Mathematics U.C.L.A. Los Angeles, CA. 90095 USA mlg@math.ucla.edu