An Abel-Jacobi Invariant for Cobordant Cycles

Gereon Quick^{1}

Received: April 14, 2015 Revised: January 2, 2016

Communicated by Max Karoubi

Abstract. We discuss an Abel-Jacobi invariant for algebraic cobordism cycles whose image in topological cobordism vanishes. The existence of this invariant follows by abstract arguments from the construction of Hodge filtered cohomology theories in joint work of Michael J. Hopkins and the author. In this paper, we give a concrete description of the Abel-Jacobi map and Hodge filtered cohomology groups for projective smooth complex varieties.

2010 Mathematics Subject Classification: 14F42, 19E15, 55N22; 14F43, 14F35, 14C25, 32Q15. Keywords and Phrases: Abel-Jacobi map, algebraic cycles, cobordism, Hodge filtered spaces, Deligne cohomology.

1. Introduction

The Abel-Jacobi map Φ is a fundamental invariant which appears in various forms in algebraic geometry. Let us mention three important examples. For an elliptic curve E over the complex numbers, Φ is the map that identifies the group of complex valued points E(ℂ) with a complex torus ℂ/Λ, where Λ is a lattice defined by the periods of E. For general smooth projective complex varieties, Φ is a fundamental tool in Lefschetz’ proof of the Hodge conjecture for (1, 1)-classes. In his seminal work [9], Griffiths showed that the Abel-Jacobi map can be used to detect cycles which may have codimension bigger than one and are homologous to zero. According to Deligne, one way to define the Abel-Jacobi map is the following. Let H_{D}^{2p}(X; ℤ(p)) denote the 2pth Deligne cohomology of X with coefficients in ℤ(p), let Hdg^{2p}(X) be the group of integral Hodge classes in H^{2p}(X; ℤ), and let CH^{p}(X) be the pth Chow group of cycles of codimension p modulo rational equivalence. Let CH_{hom}^{p}(X) be the subgroup of CH^{p}(X) of cycles which are homologous to zero. Then there is a commutative diagram

1The author was supported in part by the German Research Foundation under QU 317/1-2 and RO 4754/1-1.

(1) \xymatrix{ & CH_{hom}^{p}(X) \ar[d]_{Φ} \ar[r] & CH^{p}(X) \ar[d]_{_{cl_{D}}} \ar[dr]^{cl} \\ 0 \ar[r] & J^{2p−1}(X) \ar[r] & H_{D}^{2p}(X; ℤ(p)) \ar[r] & Hdg^{2p}(X) \ar[r] & 0.}

The bottom row of this diagram is an exact sequence, and the homomorphism Φ is induced by the Deligne-cycle map cl_{D} and the fact that CH_{hom}^{p}(X) is the kernel of the cycle map cl. The group J^{2p−1}(X) is the pth intermediate Jacobian of Griffiths defined by

J^{2p−1}(X) = H^{2p−1}(X; ℂ)/ (F^{p}H^{2p−1}(X; ℂ) + H^{2p−1}(X; ℤ)) .

The purpose of this paper is to study an analog of the Abel-Jacobi map when we replace the role of Chow groups and cohomology with algebraic and complex cobordism, respectively. In [19], Levine and Morel constructed algebraic cobordism as the universal oriented cohomology theory on the category Sm_{k} of smooth quasi-projective schemes over a field k of characteristic zero. For X ∈ Sm_{k}, the algebraic cobordism ring of X is denoted by Ω^{∗}(X). For a given p ≥ 0, Ω^{p}(X) is generated by prime cycles of the form f: Y → X where Y is a smooth scheme over k and f is a projective k-morphism. In the case k = ℂ, taking complex points induces a natural homomorphism of rings

φ_{MU}: Ω^{∗}(X) → MU^{2∗}(X) := MU^{2∗}(X(ℂ))

to the complex cobordism, represented by the Thom spectrum MU, of the space of complex points X(ℂ). More recently, Michael J. Hopkins and the author [12] constructed natural generalizations of Deligne-Beilinson cohomology on Sm_{ℂ} for any topological spectrum E. We remark that a version of Hodge filtered complex K-theory had already been defined and studied by Karoubi in [16] and [17]. For E = MU, we obtain logarithmic Hodge filtered complex bordism groups. For n, p ∈ ℤ and X ∈ Sm_{ℂ}, they are denoted by MU_{log}^{n}(p)(X). Taking the sum over all n and p, MU_{log} is equipped with a ring structure. Moreover, it was shown in [12, §7.2] that MU_{log}^{2∗}(∗)(−) defines an oriented cohomology theory on Sm_{ℂ}. The universality of Ω^{∗}(−) (proven in [19]) then implies that, for every X ∈ Sm_{ℂ}, there is a natural ring homomorphism

φ_{MU_{log}} : Ω^{∗}(X) → MU_{log}^{2∗}(∗)(X).

Furthermore, we can generalize diagram (1) in the following way. For a given p, let Ω_{top}^{p}(X) be the kernel of the map φ_{MU}. Then for every smooth projective complex variety X, there is a natural commutative diagram

(2) \xymatrix{ & Ω_{top}^{p}(X) \ar[d]_{Φ_{MU}} \ar[r] & Ω^{p}(X) \ar[d]_{φ_{MU_{log}}} \ar[dr]^{φ_{MU}} \\ 0 \ar[r] & J_{MU}^{2p−1}(X) \ar[r] & MU_{log}^{2p}(p)(X) \ar[r] & Hdg_{MU}^{2p}(X) \ar[r] & 0.}

The bottom row of this diagram is again exact. The group Hdg_{MU}^{2p}(X) is the subgroup of elements in MU^{2p}(X) which are mapped to Hodge classes in cohomology. The group J_{MU}^{2p−1}(X) is a complex torus which is determined by the Hodge structure of the cohomology and the complex cobordism of X. We consider J_{MU}^{2p−1}(X) as a natural generalization of Griffiths’ intermediate Jacobian. The map Φ_{MU} is induced by φ_{MU_{log}} and can be considered as an analog of the Abel-Jacobi map. By its definition via diagram (2), Φ_{MU} can be used to detect algebraic cobordism cycles which are topologically cobordant. The main goal of this paper is to describe how one can associate to an element in Ω_{top}^{p}(X) an element in J_{MU}^{2p−1}(X). We will achieve this goal by providing a concrete description of the elements in Hodge filtered cohomology groups for projective smooth complex varieties. We would like to add a few more words on the relationship between diagrams (1) and (2). By [19] and [12], there is a natural commutative diagram

(3)  \xymatrix{Ω^{p}(X) \ar[d]_{θ} \ar[r] & MU_{log}^{2p}(p)(X) \ar[d]_{ϑ_{log}} \ar[r] & MU^{p}(X) \ar[d]_{ϑ} \\ CH^{p}(X) \ar[r] & H_{D}^{2p}(X; ℤ(p)) \ar[r] & H^{2p}(X; ℤ)}

The composite Ω^{p}(X) → H^{2p}(X; ℤ) is the canonical homomorphism induced by the transformation of oriented cohomology theories. Let Ω_{hom}^{p}(X) be the subgroup of elements in Ω^{p}(X) which are mapped to zero under the map Ω^{p}(X) → H^{2p}(X; ℤ). It is clear from diagram (3) that we have a natural inclusion

Ω_{top}^{p}(X) ⊂ Ω_{hom}^{p}(X).

We also have an induced map θ_{hom} : Ω_{hom}^{p}(X) → CH_{hom}^{p}(X). An Abel-Jacobi map for Ω^{p}(X) which would correspond to Ω^{p}(X) → H^{2p}(X; ℤ) would factor through θ_{hom}. But θ_{hom} has a huge kernel to which such a map would be insensitive. The map Φ_{MU} however is a much finer invariant for algebraic cobordism. In fact, Φ_{MU} is able to detect at least some elements in the kernel of θ. In [12, §7.3], we considered simple examples of algebraic cobordism classes in Ω_{top}^{∗}(X) which lie in the kernel of θ. These classes can be obtained from the cycles which Griffiths constructed in [9] to show that the Griffiths group can be infinite. We hope that this paper will help finding new types of examples.

Now let E be another complex oriented cohomology theory and let E_{log}^{∗}(∗) be the associated logarithmic Hodge filtered cohomology theory. Let us assume that E is equipped with maps MU → E → Hℤ. This data induces an intermediate row in diagram (3). The natural map Ω^{p}(X) → E_{log}^{2p}(p)(X) factors through a quotient of Ω^{p}(X) which is determined by the formal group law of E. This yields the subgroup Ω_{E,top}^{p}(X) of those elements in Ω^{p}(X) which vanish under the natural map

Ω^{p}(X) → E^{2p}(X) = E^{2p}(X(ℂ)).

This subgroup satisfies

Ω_{top}^{p}(X) ⊆ Ω_{E,top}^{p}(X) ⊆ Ω_{hom}^{p}(X).

We do not discuss the difference of these groups in the present paper. However, we would like to remark that there is another interesting subgroup of Ω^{p}(X) which is given by algebraic cobordism cycles which are algebraically equivalent to zero in the sense of the work of Krishna and Park in [18]. This group sits between Ω_{top}^{p}(X) and Ω_{hom}^{p}(X). Moreover, we expect the topological triviality relation we consider in this paper to be strictly coarser than the one in [18]. It would be very interesting to understand the difference between these relations in more detail.

We will now give a brief overview of the organization of the paper. In order to understand the map Φ_{MU} we provide a concrete representation of elements in logarithmic Hodge filtered cohomology theories which may be of independent interest. An element in MU_{log}^{n}(p)(X) can be represented by pairs of elements consisting of holomorphic forms and cobordism elements which are connected by a homotopy. This resembles the way one can view elements in Deligne cohomology for complex manifolds (see e.g. [8] or [24]) and elements in differential cohomology theories for smooth manifolds as in [13]. In order to obtain this representation we first discuss some facts about homotopy pullbacks for simplicial presheaves. Then we define logarithmic Hodge filtered spaces and study their global sections for smooth projective varieties. The above mentioned representation is then an immediate consequence of the construction. In the fourth section we use this representation to describe the Abel-Jacobi invariant for topologically trivial algebraic cobordism cycles.

Acknowledgements. We are very grateful to Aravind Asok, Dustin Clausen, Mike Hopkins, Marc Levine and Kirsten Wickelgren for many helpful discussions. We would also like to thank the anonymous referee for very helpful suggestions and comments.

2. Homotopy pullbacks of simplicial presheaves

We briefly recall some basic facts about simplicial presheaves. Then we will discuss the construction of homotopy pullbacks of simplicial presheaves which will be needed in the next section.

2.1. Simplicial presheaves. Let T be an essentially small site with enough points. The following two examples of such sites will occur in this paper:

• The category Man_{ℂ} of complex manifolds and holomorphic maps which we consider as a site with the Grothendieck topology defined by open coverings.

• The category Sm_{ℂ,Nis} = Sm of smooth complex algebraic varieties (separated schemes of finite type over ℂ) with the Nisnevich topology. We recall that a distinguished square in Sm_{ℂ,Nis} is a cartesian square of the form

(4)  \xymatrix{U ×_{X} V \ar[r] \ar[d] & V \ar[d]^{p} \\ U \ar[r]^{j} & X}

such that p is an étale morphism, j is an open embedding and the induced morphism p^{−1}(X − U) → X − U is an isomorphism, where the closed subsets are equipped with the reduced induced structure. The Nisnevich topology is the Grothendieck topology generated by coverings of the form (4) (see [21, §3.1]).

We denote by sPre = sPre(T) the category of simplicial presheaves on T, i.e., contravariant functors from T to the category sS of simplicial sets. Objects in sPre will also be called spaces. There are several important model structures on the category sPre (see [14], [2], [5]). We start with the projective model structure on sPre. A map F → G in sPre is called

• an objectwise weak equivalence if F(X) → G(X) is a weak equivalence in sS (equipped with the standard model structure) for every X ∈ T;

• a projective fibration if F(X) → G(X) is a Kan fibration in sS for every X ∈ T;

• a projective cofibration if it has the left lifting property with respect to all acyclic fibrations.

In order to obtain a local model structure, i.e., one which respects the topology on the site T, we can localize the projective model structure at the hypercovers in T (see [14], [2], [5]). We briefly recall the most important notions. A map f : F → G of presheaves on T is called a generalized cover if for any map X → F from a representable presheaf X to F there is a covering sieve R ↪ X such that for every element U → X in R the composite U → X → F lifts through f. Dugger and Isaksen [7, §7] give the following characterization of local acyclic fibrations and hypercovers. A map f : F → G of simplicial presheaves on T is a local acyclic fibration if for every X ∈ T and every commutative diagram

\xymatrix{∂Δ^{n} ⊗ X \ar[r] \ar[d] & F \ar[d] \\ Δ^{n} ⊗ X \ar[r] & G}

there exists a covering sieve R ↪ X such that for every U → X in R, the diagram one obtains from restricting from X to U

\xymatrix{∂Δ^{n} ⊗ U \ar[r] \ar[d] & F \ar[d] \\ Δ^{n} ⊗ U \ar@{.>}[ur] \ar[r] & G}

has a lifting Δ^{n} ⊗ U → F. Note that this implies in particular that the map F_{0} → G_{0} of presheaves is a generalized cover.

Definition 2.1. Let X be an object of T and let U be a simplicial presheaf on T with an augmentation map U → X in sPre. This map is called a hypercover of X if it is a local acyclic fibration and each U_{n} is a coproduct of representables.

If U → X is a hypercover, then the map U_{0} → X is a cover in the topology on T. Moreover, the map U_{1} → U_{0} ×_{X} U_{0} is a generalized cover. In general, for each n, the face maps combine such that U_{n} is a generalized cover of a finite fiber product of different U_{k} with k < n. Since the projective model structure on sPre is cellular, proper and simplicial, it admits a left Bousfield localization with respect to all maps

{hocolim U_{∗} → X}

where X runs through all objects in T and U runs through the hypercovers of X. The resulting model structure is the local projective model structure on sPre (see [2] and [5]). The weak equivalences, fibrations and cofibrations in the local projective model structure are called local weak equivalences, local projective fibrations and local projective cofibrations, respectively. We denote the corresponding homotopy category by hosPre. Note that the local weak equivalences are precisely those maps F → G in sPre such that the induced map of stalks F_{x} → G_{x} is a weak equivalence in sS for every point x in T. Dugger, Hollander and Isaksen showed that the fibrations in the local projective model structure on sPre have a nice characterization (see [6, §§3+7]). Let U → X be a hypercover in sPre and let F be a projective fibrant simplicial presheaf. Since each U_{n} is a coproduct of representables, we can form a product of simplicial sets ∏_{a} F((U_{n})^{a}) where a ranges over the representable summands of U_{n}. The simplicial structure of U defines a cosimplicial diagram in sS

∏_{a} F(U_{0}^{a}) ⇉ ∏_{a} F(U_{1}^{a}) ⟶_{⟶}^{⟶} ⋯

The homotopy limit of this diagram is denoted by holim_{Δ} F(U).

Following [6, Definition 4.3] we say that a simplicial presheaf F satisfies descent for a hypercover U → X if there is a projective fibrant replacement F → F′ such that the natural map

(5)  F′(X) → holim_{Δ} F′(U)

is a weak equivalence. It is easy to see that if F satisfies descent for a hypercover U → X, then the map (5) is a weak equivalence for every projective fibrant replacement F → F′. By [6, Corollary 7.1], the local projective fibrant objects in sPre are exactly those simplicial presheaves which are projective fibrant and satisfy descent with respect to all hypercovers U → X. For our final applications we will need the following facts whose proofs can be found in [23].

Lemma 2.2. Let F be a simplicial presheaf that satisfies descent with respect to all hypercovers. Then every fibrant replacement F → F_{f} in the local projective model structure is an objectwise weak equivalence, i.e., for every object X ∈ T the map

F(X) → F_{f}(X)

is a weak equivalence of simplicial sets.

Proposition 2.3. Let F be a simplicial presheaf that satisfies descent with respect to all hypercovers and let X be an object of T. Then, for every projective fibrant replacement g : F → F′, the natural map

Hom_{hosPre}(X, F) → π_{0}(F′(X))

is a bijection.

2.2. Homotopy pullbacks of simplicial presheaves. We briefly recall the construction of homotopy pullbacks in sPre (see [11, §13.3] for more details) and will then show that its local and global versions are homotopy equivalent. Let sPre be equipped with any of the above model structures. We fix a functorial factorization E of every map f: X → Y into

X ⟶^{i_{f}} E(f) ⟶^{p_{f}} Y

where i_{f} is an acyclic cofibration and p_{f} is a fibration. The homotopy pullback

of the diagram X ⟶^{f} Z ⟵^{g} Y is the pullback of E(f) ⟶^{p_{f}} Z ⟵^{p_{g}} E(g). The homotopy pullback satisfies the following invariance. If we have a diagram

\xymatrix{X \ar[d] \ar[r]^{f} & Z \ar[d] & Y \ar[l]_{g} \ar[d] \\ X′ \ar[r]^{f′} & Z′ & Y′ \ar[l]_{g′}}

in which the three vertical maps are weak equivalences, then the induced map of homotopy pullbacks

E(f) ×_{Z} E(g) → E(f′) ×_{Z′} E′(g′)

is a weak equivalence as well.

We will need the following fact about pulling back local weak equivalences along maps which are merely projective fibrations.

Lemma 2.4. Let f: X → Z be a projective fibration and g: Y → Z be a local weak equivalence in sPre. Then the induced map f′: X ×_{Z} Y → X is a local weak equivalence as well.

Proof. For a point x of sPre and a map h: V → W of simplicial presheaves, let h_{x}: V_{x} → W_{x} denote the induced map of stalks at x. With the notation of the lemma, we need to show that f′_{x} is a weak equivalence in sS for every point x in T. Since x preserves finite limits, we have (X ×_{Z} Y)_{x} = X_{x} ×_{Z_{x}} Y_{x} and f′_{x} equals the induced map in the corresponding pullback diagram in sS. Since the standard model structure on sS is right proper, it thus suffices to show that g_{x} is a Kan fibration. But every map in sPre which is an objectwise Kan fibration is also a stalkwise Kan fibration. We will provide a proof of this fact for completeness. Given a point x of T, we need to check that the map of sets induced by g

(6) Hom_{sS}(Δ[n], Y_{x}) → Hom_{sS}(Λ_{k}[n], Y_{x}) ×_{Hom_{sS}(Λ_{k}[n],Z_{x})} Hom_{sS}(Δ[n], Z_{x})

is surjective for all n ≥ 1 and 0 ≤ k ≤ n. Now for a simplicial presheaf W and a finite simplicial set K, we can consider the functor X ↦ Hom_{sS}(K, W(X)) as a presheaf of sets on T. We denote this presheaf by Hom(K, W). The stalk of this presheaf at x is exactly the set Hom_{sS}(K, W_{x}). Hence the map (6) is surjective if and only if the map of presheaves of sets

(7)  Hom(Δ[n], Y) → Hom(Λ_{k}[n], Y) ×_{Hom(Λ_{k}[n],Z)} Hom(Δ[n], Z)

induces a surjective map of stalks at x. But, since g is a projective fibration, g(X) is a Kan fibration for every object X ∈ T. Hence, by the definition of Hom(−, −), the induced map (8) Hom(Δ[n], Y)(X) → Hom(Λ_{k}[n], Y)(X) ×_{Hom(Λ_{k}[n],Z)(X)} Hom(Δ[n], Z)(X)

is surjective. Since forming stalks preserves objectwise epimorphisms, this implies that g_{x} is a Kan fibration. □

The following result is probably a well-known fact. We include its proof for completeness and lack of a reference.

Lemma 2.5. The homotopy pullback of a diagram in sPre in the projective model structure is stalkwise weakly equivalent to the homotopy pullback of the diagram in the local projective model structure.

Proof. Let X ⟶^{f} Z ⟵^{g} Y be a diagram in sPre. Consider the diagram

(9)  [figure: X; i_{f}^{proj}; f; Z; Y; g; i_{g}^{proj}; E^{proj}(f); p_{f}^{proj}; Z; E^{proj}(g); p_{g}^{proj}; E^{loc}(p_{f}^{proj}); Z; E^{loc}(p_{g}^{proj})]

where the superscripts proj and loc indicate whether we take functorial replacements in the projective and local projective model structure, respectively. We denote the projective homotopy pullback of the initial diagram by P^{proj} = E^{proj}(f) ×_{Z} E^{proj}(g). The vertical maps i_{f}^{proj} and i_{g}^{proj} are projective acyclic cofibrations. Since left Bousfield localization does not change cofibrations and since objectwise weak equivalences are in particular local weak equivalences, i_{f}^{proj} and i_{g}^{proj} are local acyclic cofibrations as well. Hence the composition of the vertical maps in (9) are local acyclic cofibrations. Thus, E^{loc}(p_{f}^{proj}) ×_{Z} E^{loc}(p_{g}^{proj}) computes the local homotopy pullback P^{loc}

of X ⟶^{f} Z ⟵^{g} Y. Hence we need to show that the induced map

(10)  E^{proj}(f) ×_{Z} E^{proj}(g) → E^{loc}(p_{f}^{proj}) ×_{Z} E^{loc}(p_{g}^{proj})

is a local weak equivalence. This map equals the composition (11) E^{proj}(f) ×_{Z} E^{proj}(g) → E^{loc}(p_{f}^{proj}) ×_{Z} E^{proj}(g) → E^{loc}(p_{f}^{proj}) ×_{Z} E^{loc}(p_{g}^{proj}).

Hence in order to show that (10) is a local weak equivalence, it suffices to show that the two maps in (11) are both local weak equivalences. For this, it suffices to show that the pullback of a local weak equivalence along a projective fibration is again a local weak equivalence which has been checked in Lemma 2.4. □

Remark 2.6. The result of Lemma 2.5 does not depend on the fact that we use the projective model structure. The same proof (after replacing the superscript proj with inj ) would also work if we used the injective model structure on sPre. More precisely, the homotopy pullback of a diagram in sPre in the injective model structure is stalkwise weakly equivalent to the homotopy pullback of the diagram in the local injective model structure.

The lemma shows that we can calculate the set of homotopy classes of maps into a homotopy pullback via global sections in the following way.

Proposition 2.7. Let X ⟶^{f} Z ⟵^{g} Y be a diagram in sPre, and let P denote the homotopy pullback of this diagram in the local projective model structure. We assume that all three simplicial presheaves X, Y and Z satisfy descent for all hypercovers. For an object X ∈ T, let Q(X) denote the homotopy pullback in sS of the diagram of simplicial sets X (X) ⟶^{f(X)} Z(X) ⟵^{g(X)} Y(X). Then there is a natural bijection for every X ∈ T

Hom_{hosPre}(X, P) ≅ π_{0}(Q(X)).

Proof. Let X ↦ X′ be a functorial projective fibrant replacement in sPre. The invariance property of homotopy pullbacks implies that the homotopy pullback of X ⟶^{f} Z ⟵^{g} Y is stalkwise equivalent to the homotopy pullback

of the induced diagram X′ ⟶^{f′} Z′ ⟵^{g′} Y′. It also implies that, for every X ∈ T, Q(X) is equivalent to the homotopy pullback Q′(X) of the diagram

X′(X) ⟶^{f′(X)} Z′(X) ⟵^{g′(X)} Y′(X) in sS. Hence we can assume from now on that X, Y and Z are also projective fibrant. Now consider the diagram

(12)  [figure: X; f; Z; Y; g; E^{proj}(f); p_{f}^{proj}; Z; E^{proj}(g); p_{g}^{proj}]

where E^{proj} is a functorial replacement in the projective model structure as before. Let P^{proj} = E^{proj}(f) ×_{Z} E^{proj}(g) denote the homotopy pullback of X ⟶^{f} Z ⟵^{g} Y calculated in the projective model structure. By definition of pullbacks in sPre, we have P^{proj}(X) = E^{proj}(f)(X) ×_{Z(X)} E^{proj}(g)(X) for every X ∈ T. The invariance property of homotopy pullbacks implies that P^{proj}(X) is equivalent to the homotopy pullback Q(X) of the diagram

X (X) ⟶^{f(X)} Z(X) ⟵^{g(X)} Y(X) in sS. (In fact, we could compute Q(X) as P^{proj}(X).) Moreover, since X, Y and Z satisfy descent for all hypercovers and since homotopy pullbacks commute with homotopy limits in sS, we see that P^{proj} satisfies descent for all hypercovers as well. By Lemma 2.2, this implies that P^{proj} is local projective fibrant. Finally, by Lemma 2.5, P^{proj} is equivalent to the homotopy pullback in the local projective model structure. Hence, by Proposition 2.3, for every X ∈ T, there are natural bijections

Hom_{hosPre}(X, P) ≅ Hom_{hosPre}(X, P^{proj}) ≅ π_{0}(P^{proj}(X)) ≅ π_{0}(Q(X)).

□

3. Logarithmic Hodge filtered function spaces

We construct spaces which represent logarithmic Hodge filtered cohomology groups. In particular, we will show that we can represent elements in logarithmic Hodge filtered complex bordism groups as triples consisting of a class in complex bordism, a holomorphic form with suitable coefficients and a homotopy that connects both in an appropriate sense.

3.1. Hodge filtration on forms and Eilenberg-MacLane spaces. Let C^{∗} be a cochain complex of presheaves of abelian groups on T. For any given n, we denote by C^{∗}[n] the cochain complex given in degree q by C^{q}[n] := C^{q+n}. The differential on C^{∗}[n] is the one of C^{∗} multiplied by (−1)^{n}. The hypercohomology H^{∗}(U, C^{∗}) of an object U of T with coefficients in C^{∗} is the graded group of morphisms Hom(ℤ_{U}, aC^{∗}) in the derived category of cochain complexes of sheaves on T, where aC^{∗} denotes the complex of associated sheaves of C^{∗}. We will denote by K(C^{∗}, n) the Eilenberg-MacLane simplicial presheaf corresponding to C^{∗}[−n]. The following result is a version of Verdier’s hypercovering theorem due to Ken Brown.

Proposition 3.1. ([3, Theorem 2], see also [21], [15]) Let C^{∗} be a cochain complex of presheaves of abelian groups on T. Then for any integer n and any object U of T, one has a canonical isomorphism

H^{n}(U; C^{∗}) ≅ Hom_{hosPre(T)}(U, K(C^{∗}, n)).

Now let T be the site Sm of smooth complex varieties with the Nisnevich topology. We would like to find a simplicial presheaf which represents Hodge filtered complex cohomology. To have a good filtration on holomorphic forms requires a compact variety. By the work of Hironaka, we know that every smooth complex variety does have a nice compactification. Following Deligne [4] and Beilinson [1], we will use this fact to construct simplicial presheaves on Sm whose global sections are isomorphic to the Hodge filtered cohomology groups of X. Let \overline{Sm} be the category whose objects are smooth compactifications, i.e., pairs (X, \overline{X}) = (X ⊂ \overline{X}) consisting of a smooth variety X embedded as an open subset of a projective variety \overline{X} and having the property that \overline{X} − X is a normal crossing divisor which is the union of smooth divisors. A map from (X, \overline{X}) to (Y, \overline{Y}) is a commutative diagram

\xymatrix{X \ar[r] \ar[d] & \overline{X} \ar[d] \\ Y \ar[r] & \overline{Y} .}

By Hironaka’s theorem [10], every smooth variety over ℂ admits a smooth compactification. Moreover, for a given smooth variety X, the category C(X) of all smooth compactifications of X is filtered (see [4]). The forgetful functor

u : \overline{Sm} → Sm
(X, \overline{X}) ↦ X

induces a pair of adjoint functors on the categories of simplicial presheaves

u^{∗} : sPre(\overline{Sm}) ↔ sPre(Sm) : u_{∗}.

The left adjoint u^{∗} is given by sending a simplicial presheaf F on \overline{Sm} to the simplicial presheaf

X ↦ u^{∗}F(X) = colim_{C(X)} F(\overline{X}).

For a smooth complex variety X, let Ω_{X}^{p} denote the sheaf of holomorphic pforms on X. Let \overline{X} be a smooth compactification of X and let D := \overline{X} − X denote the complement of X. Let Ω1 ; \overline{X}⟨D⟩ be the locally free sub-module of j_{∗}Ω_{X}^{1} generated by Ω_{X}^{1} and by \frac{dz_{i}}{z_{i}} where z_{i} is a local equation for an irreducible local component of D. The sheaf Ωp ; \overline{X}⟨D⟩ of meromorphic p-forms on \overline{X} with at most logarithmic poles along D is defined to be the locally free subsheaf ⋀^{p} Ω1 ; \overline{X}⟨D⟩ of j_{∗}Ωp ; \overline{X}. The Hodge filtration on the complex cohomology of X can be defined as the image

(13)  F^{p}H^{n}(X; ℂ) := Im (H^{n}(\overline{X}; Ω∗≥p ; \overline{X}⟨D⟩) → H^{n}(X; ℂ)).

This definition is independent of the compactification \overline{X} (see [4]). We denote by \overline{Ω}^{∗} the presheaf of differential graded ℂ-algebras on \overline{Sm} that sends a pair X ⊂ \overline{X} with D := \overline{X} − X to Ω∗ ; \overline{X}⟨D⟩(\overline{X}). For any given integer

p ≥ 0, we denote by \overline{Ω}^{∗≥p} the presheaf on \overline{Sm} that sends a pair X ⊂ \overline{X} to

Ω∗≥p ; \overline{X}⟨D⟩(\overline{X}).

Let

Ω∗≥p ; \overline{X}⟨D⟩ → A∗≥p ; \overline{X}⟨D⟩

be any resolution by cohomologically trivial sheaves which is functorial in the pair X ⊂ \overline{X} and which induces a commutative diagram

\xymatrix{Ω∗≥p ; \overline{X}⟨D⟩(\overline{X}) \ar[r] \ar[d] & Ω_{X}^{∗≥p}(X) \ar[d] \\ A∗≥p ; \overline{X}⟨D⟩(\overline{X}) \ar[r] & A_{X}^{∗≥p}(X)}

where A_{X}^{∗≥p} denotes a functorial resolution by cohomologically trivial sheaves of Ω_{X}^{∗≥p}. For example, A∗≥p ; \overline{X}⟨D⟩ and A_{X}^{∗≥p} could be the Godemont resolutions ([4, §3.2.3]) or the logarithmic Dolbeault resolution ([22, §8]). Even though A_{X}^{∗} and A∗≥p ; \overline{X}⟨D⟩ are double complexes, we will only consider their total complexes. We denote the presheaf of complexes on \overline{Sm} that sends a pair (X, \overline{X}) to

A∗≥p ; \overline{X}⟨D⟩(\overline{X}) by F^{p}Ā^{∗}, and let

\overline{Ω}^{∗≥p} → F^{p}Ā^{∗}

be the associated map of complexes of presheaves on \overline{Sm}. Now let V_{∗} be an evenly graded ℂ-algebra such that each V_{2j} is a finite dimensional complex vector space. We will write

F^{p}H^{n}(X; V_{∗}) := ⨁_{j} F^{p+j}H^{n+2j}(X; V_{2j})

for the graded Hodge filtered cohomology groups. The functor X ↦ F^{p}H^{n}(X; V_{∗}) is representable in hosPre(Sm) in the following way. For (X, \overline{X}) ∈ \overline{Sm} and j ∈ ℤ, let F^{p+j}Ā^{∗}(\overline{X}; V_{2j})[−2j] denote the corresponding complex with coefficients in V_{2j} shifted by degree 2j. We write

(14)  F^{p}Ā^{∗}(\overline{X}; V_{∗}) = ⨁_{j} F^{p+j}Ā^{∗}(\overline{X}; V_{2j})[−2j].

Let F^{p}Ā^{∗}(V_{∗}) denote the corresponding presheaf on \overline{Sm}. Let K(F^{p}Ā^{∗}(V_{∗}), n) be the associated Eilenberg-MacLane simplicial presheaf. Note that (14) induces an isomorphism

(15)  K(F^{p}Ā^{∗}(V_{∗}), n) ≅ ⋁_{j} K(F^{p+j}Ā^{∗}(V_{2j}), n + 2j).

For every smooth complex variety X, [23, Theorem 3.5] shows that there is a natural isomorphism

Hom_{hosPre(Sm)}(X, u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n)) ≅ F^{p}H^{n}(X; V_{∗}).

A crucial fact for the proof of [23, Theorem 3.5] is that the simplicial presheaf u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n) satisfies Nisnevich descent. This implies that any projective fibrant replacement of u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n) is already local projective fibrant. As a consequence we obtain that, for every smooth complex variety X, there is a natural isomorphism

(16)  π_{0}(u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n)(X)) ≅ F^{p}H^{n}(X; V_{∗}).

Finally, we point out that, for every n and p, the map of presheaves of complexes

F^{p}Ā^{∗}(V_{∗})[−n] → A^{∗}(V_{∗})[−n]

induces a morphism of simplicial presheaves

(17)  u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n) → K(A^{∗}(V_{∗}), n).

3.2. The singular functor for complex manifolds. As a short digression, we need to consider simplicial presheaves on complex manifolds as well. In this subsection, we let T be the site Man_{ℂ}. Let Δ^{n} be the standard topological n-simplex

Δ^{n} = {(t_{0}, …, t_{n}) ∈ ℝ^{n+1}|0 ≤ t_{j} ≤ 1, ∑ t_{j} = 1}.

For topological spaces Y and Z, the singular function complex Sing_{∗}(Z, Y) is the simplicial set whose n-simplices are continuous maps

f: Z × Δ^{n} → Y.

We denote the simplicial presheaf

M ↦ Sing_{∗}(M, Y) =: Sing_{∗}Y (M)

on Man_{ℂ} by Sing_{∗}Y. Since, for any CW -complex Y, Sing_{∗}Y satisfies descent, the criterion of [6] implies that Sing_{∗}Y is a fibrant object in the local projective model structure on sPre (see also [12, Lemma 2.3]).

Let V_{∗} be an evenly graded complex vector space, and let K(V_{∗}, n) be an associated Eilenberg-MacLane space in the category of CW -complexes. Then the simplicial presheaf Sing_{∗}K(V_{∗}, n) represents the functor of cocycles with coefficients in V_{∗}, i.e., for every M ∈ Man_{ℂ}, there is a natural isomorphism of abelian groups

Z^{n}(M; V_{∗}) ≅ Hom_{sPre}(M, Sing_{∗}K(V_{∗}, n)).

Since M is a representable presheaf, we have a natural bijection of sets

Hom_{sPre}(M, Sing_{∗}K(V_{∗}, n)) = Sing_{0}K(V_{∗}, n)(M).

Moreover, M is a cofibrant object in the local projective model structure on sPre. Hence there is a natural bijection

(18)  Hom_{hosPre}(M, Sing_{∗}K(V_{∗}, n)) = π_{0}(Sing_{∗}K(V_{∗}, n)(M)).

3.3. Logarithmic Hodge filtered function spaces. In this subsection we will work with both sites, Sm_{ℂ} and Man_{ℂ}. For X ∈ Sm_{ℂ}, we denote by X_{an} ∈ Man_{ℂ} the associated complex manifold whose underlying set is X(ℂ). This defines a functor

ρ^{−1} : Sm_{ℂ} → Man_{ℂ}, X ↦ ρ^{−1}(X) := X_{an}.

Composition with ρ^{−1} induces a functor

ρ_{∗}: sPre(Man_{ℂ}) → sPre(Sm_{ℂ}).

Note that ρ_{∗} is the right adjoint in a Quillen pair of functors between the corresponding local projective model structures. We can now construct logarithmic Hodge filtered spaces whose global sections yield generalized Hodge filtered cohomology groups. The idea to define Hodge filtered spaces is similar to the way that differential function spaces were defined for presheaves on the category of smooth manifolds in [13]. Let n, p be integers and V_{∗} an evenly-graded complex vector space. Let Y be a CW-complex and let ι ∈ Z^{n}(Y; V_{∗}) by a cocycle on Y. A cocycle corresponds to a map of CW-complexes

Y → K(V_{∗}, n)

and induces a map of simplicial presheaves on Man_{ℂ}

Sing_{∗}Y → Sing_{∗}K(V_{∗}, n).

Let | · | denote the geometric realization of simplicial sets. Using the canonical map K(V_{∗}, n) → |K(A^{∗}(V_{∗}), n)| we can form the following diagram in sPre(Sm_{ℂ})

(19)  \xymatrix{ & ρ_{∗}Sing_{∗}Y \ar[d]^{ι^{∗}} \\ u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n) \ar[r] & ρ_{∗}Sing_{∗}|K(A^{∗}(V_{∗}), n)|.}

Definition 3.2. We define the logarithmic Hodge filtered function space (Y (p), ι, n) to be the homotopy pullback of (19) in sPre(Sm_{ℂ}).

Note that (Y (p), ι, n) depends on ι only up to homotopy, i.e., if ι′ is another cocycle which represents the same cohomology class as ι then (Y (p), ι, n) and (Y (p), ι′, n) are equivalent.

Remark 3.3. Let us contemplate a little more on diagram (19). For a complex manifold M, let Z^{n}(M × Δ^{•}; V_{∗}) be the simplicial abelian group whose group of k-simplices is given by C^{∞}-n-cocycles on M × Δ^{k} with coefficients in V_{∗}. We denote the corresponding simplicial presheaf

M ↦ Z^{n}(M × Δ^{•}; V_{∗})

on Man_{ℂ} by Z^{n}(− × Δ^{•}; V_{∗}). Our chosen cocycle ι determines a map of simplicial presheaves

Sing_{∗}Y → Z^{n}(− × Δ^{•}; V_{∗}), f ↦ ι^{∗}f,

given by taking the pullback along ι. Let I denote the map given by integration of forms

I: F^{p+j}A^{n+2j}(X; V_{2j}) → C^{n+2j}(X; V_{2j}), η ↦ (σ ↦ ∫_{Δ^{n+2j}} σ^{∗}η).

We can form a diagram of simplicial presheaves

(20)  \xymatrix{ & ρ_{∗}Sing_{∗}Y \ar[d]^{ι^{∗}} \\ u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n) \ar[r]_{I} & ρ_{∗}Z^{n}(− × Δ^{•}; V_{∗}).}

The map

Sing_{∗}K(V_{∗}, n)(M) → Z^{n}(M × Δ^{•}; V_{∗})

given by pulling back a fundamental cocycle in Z^{n}(K(V_{∗}, n); V_{∗}) is a simplicial homotopy equivalence (see e.g. [13, Proposition A.12]). Hence the homotopy pullback of (20) is homotopy equivalent to the homotopy pullback of (19).

Remark 3.4. For Y = K(ℤ, n), we recover Deligne-Beilinson cohomology in the following way. Let ι: K(ℤ, n) → K(ℂ, n) be the map that is induced by the (2πi)^{p}-multiple of the inclusion ℤ ⊂ ℂ. Then the space K(ℤ, n)(p) := (K(ℤ, n)(p), ι, n) represents Deligne-Beilinson cohomology in the homotopy category of sPre(Sm_{ℂ}) in the sense that there is a natural isomorphism

H_{D}^{n}(X; ℤ(p)) ≅ Hom_{hosPre(Sm_{ℂ})}(X, K(ℤ, n)(p)).

3.4. Hodge filtered spaces and spectra. Of particular interest is the case when Y is a space in a spectrum. We refer the reader to [12] for a more detailed discussion of the maps of spectra involved. We can reinterpret the construction of [12] on the level of spaces as follows. Let E be a topological Ω-spectrum and let E_{n} be its nth space. We assume that E is rationally even, i.e., π_{∗}E ⊗ ℚ is concentrated in even degrees. Let V_{∗} be the evenly graded ℂ-vector space π_{∗}E ⊗ ℂ. Let

τ: E → E ∧ Hℂ =: E_{ℂ}

be a map of spectra which induces for every n the map

π_{2n}(E) ⟶^{(2πi)^{n}} π_{2n}(E_{ℂ})

defined by multiplication by (2πi)^{n} on homotopy groups. The choice of such a map is unique up to homotopy. For a given integer p, multiplication by (2πi)^{p} on homotopy groups determines a map

E ⟶^{(2πi)^{p}τ} E ∧ Hℂ.

Let

E ∧ Hℂ → H(π_{∗}E ⊗ ℂ)

be a map that induces the isomorphism

π_{∗}(E ∧ Hℂ) ≅ π_{∗}E ⊗ ℂ = V_{∗}.

The composite with (2πi)^{p}τ defines a map of spectra

ι: E → H(V_{∗}).

The inclusion V_{∗} ↪ A^{∗}(V_{∗}) induces a map of spectra H(V_{∗}) → H(A^{∗}(V_{∗})). Composition with E → H(V_{∗}) defines a map

(21)  ι: E → H(A^{∗}(V_{∗}))

which we also denote by ι. We call ι a p-twisted fundamental cocycle of E. This map corresponds to a family of maps of spaces which, for each n, are of the form

ι_{n}: E_{n} → K(A^{∗}(V_{∗}), n)

and which are compatible with the structure maps of the spectrum E. For given p and ι and each n, we can form the diagram in sPre(Sm_{ℂ})

(22)  \xymatrix{ & ρ_{∗}Sing_{∗}E_{n} \ar[d]^{ι_{n}^{∗}} \\ u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n) \ar[r] & ρ_{∗}Sing_{∗}|K(A^{∗}(V_{∗}), n)|.}

We will write (E_{n}(p), ι) for the homotopy pullback of (22) in sPre(Sm_{ℂ}). Note that a different choice ι′ of a p-twisted fundamental cocycle of E yields a homotopy equivalent simplicial presheaf (E_{n}(p), ι′). Therefore, we will often drop ι from the notation and write E_{n}(p) for (E_{n}(p), ι).

Definition 3.5. According to our previous terminology, we call E_{n}(p) the nth logarithmic Hodge filtered function space of E (even though it is only unique up to homotopy equivalence).

Remark 3.6. For X ∈ Sm_{ℂ}, let E_{log}^{n}(p)(X) denote the logarithmic Hodge filtered E-cohomology groups of X as defined in [12, Definition 6.4]. It follows from the definition of E_{n}(p) as a homotopy pullback of (22) that the groups Hom_{hosPre}(X, E_{n}(p)), for varying n, sit in long exact sequences analog to the one of [12, Proposition 6.5]. This shows that we have a natural isomorphism

E_{log}^{n}(p)(X) ≅ Hom_{hosPre}(X, E_{n}(p)).

Alternatively, we could have remarked that E_{n}(p) is the nth space of the fibrant spectrum E_{log}(p) of [12, §6].

3.5. The case of smooth projective varieties. If X is a projective smooth complex varieties, we obtain a more concrete description of the global sections of a Hodge filtered space. For, in this case, X is an initial object in the filtered category C(X) of all smooth compactifications of X. Hence the colimit that computes the value of u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n) at X reduces to

u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n)(X) = K(F^{p}A^{∗}(V_{∗}), n)(X).

Thus, for X projective, we have

(23)  Hom_{sPre(Sm_{ℂ})}(X, u^{∗}K(F^{p}Ā^{∗}(V_{∗}), n)) ≅ K(F^{p}A^{∗}(X; V_{∗}), n).

In terms of homotopy classes of maps, isomorphism (16) just states the fact

π_{0}K(F^{p}A^{∗}(X; V_{∗}), n) ≅ F^{p}H^{n}(X; V_{∗}).

Now let E be a rationally even topological Ω-spectrum together with the choice of a p-twisted fundamental cocycle ι. Let V_{∗} again denote π_{∗}E ⊗ ℂ. By Proposition 2.7, we can calculate the homotopy pullback of (22) objectwise. This implies that the space E_{n}(p)(X) is homotopy equivalent to the homotopy pullback of the following diagram of simplicial sets

(24)  \xymatrix{ & Sing_{∗}E_{n}(X) \ar[d]^{ι_{n}^{∗}} \\ K(F^{p}A^{∗}(X; V_{∗}), n) \ar[r] & Sing_{∗}|K(A^{∗}(V_{∗}), n)|(X).}

By Remark 3.6, this implies that the logarithmic Hodge filtered E-cohomology group E_{log}^{n}(p)(X) is isomorphic to the group of connected components of the space E_{n}(p)(X), i.e.,

E_{log}^{n}(p)(X) = π_{0}(E_{n}(p)(X)).

We can now read off from diagram (24) the following characterization of elements of E_{log}^{n}(p)(X).

Proposition 3.7. For rationally even spectrum E and a smooth projective variety, an element of E_{log}^{n}(p)(X) is given by a triple

(25)  q: X → E_{n}, ω ∈ F^{p}A^{n}(X; V_{∗})_{cl}, ξ ∈ A^{n−1}(X; V_{∗})

such that dξ = ι_{n}^{∗}q − ω, where q is a continuous map, d denotes the differential in A^{∗}(X; V_{∗}), and ω is a closed form.

Remark 3.8. In view of Remark 3.3, we can rewrite diagram (24) as follows. Let I denote again the map given by integration of forms. Following the argument in Remark 3.3, we see that E_{n}(p)(X) fits into the following homotopy cartesian square of simplicial sets

\xymatrix{E_{n}(p)(X) \ar[r] \ar[d] & Sing_{∗}E_{n}(X) \ar[d]^{ι_{n}^{∗}} \\ K(F^{p}A^{∗}(X; V_{∗}), n) \ar[r]_{I} & Z^{n}(X × Δ^{•}; V_{∗}).}

Hence we can represent an element in E_{log}^{n}(p)(X) also as a triple

q: X → E_{n}, ω ∈ F^{p}A^{n}(X; V_{∗})_{cl}, h ∈ C^{n−1}(X; V_{∗})

such that δh = ι_{n}^{∗}q − I(ω), where δ denotes the differential in C^{∗}(X; V_{∗}).

Remark 3.9. Recall from Remark 3.4 that for E = Hℤ, i.e., E_{n} = K(ℤ, n), K(ℤ, n)(p) represents Deligne-Beilinson cohomology in hosPre(Sm_{ℂ}). Then Remark 3.8 just rephrases the well-known fact (see e.g. [24, §12.3.2] or [8]) that an element in H_{D}^{n}(X; ℤ(p)) can be represented (in the notation of [24]) by a triple (a_{ℤ}^{n}, b_{F}^{n} , c_{ℂ}^{n−1}) where a_{ℤ}^{n} is an integral singular cochain of degree n, b_{F}^{n} is a form in F^{p}A^{n}(X), and c_{ℂ}^{n−1} is a complex singular cochain of degree n − 1 such that δc_{ℂ}^{n−1} = a_{ℤ}^{n} − b_{F}^{n}.

4. A generalized Abel-Jacobi invariant

In this section, we will always assume that X is a projective smooth complex variety. Let p be a fixed integer. Let MU be the Thom spectrum representing complex cobordism. Recall that the homotopy groups of MU vanish in all odd and in all negative degrees. Moreover, for j ≥ 0, π_{2j}MU is a finitely generated free abelian group. To shorten the notation we will again write

V_{∗} := π_{∗}MU ⊗_{ℤ} ℂ.

Let ι be a p-twisted fundamental cocycle of MU. The reader may find a detailed discussion of fundamental cocycles for MU in [12, §5]. Here we just recall that ι comes equipped with an isomorphism

(26)  MU^{∗}(X)_{ℂ} := MU^{∗}(X) ⊗_{ℤ} ℂ ≅ H^{∗}(X; V_{∗}) = ⨁_{j≥0} H^{∗+2j}(X; V_{2j}).

4.1. Cobordism, Jacobians, and Hodge structures. We first define the generalized Jacobian we mentioned in the introduction. By the construction of the space MU_{n}(p) as a homotopy pullback and by using isomorphism (15), we deduce that the groups MU_{log}^{n}(p)(X) sit in a long exact sequence of the form (see also [12, §4.2])

\begin{aligned}… → H^{n−1}(X; V_{∗}) & → MU_{log}^{n}(p)(X) → \\ → MU^{n}(X) ⊕ F^{p}H^{n}(X; V_{∗}) & → H^{n}(X; V_{∗}) → … \end{aligned}

For n = 2p, we can split this long exact sequence into the following short exact sequence

0 → J_{MU}^{2p−1}(X) → MU_{log}^{2p}(p)(X) → Hdg_{MU}^{2p}(X) → 0 which is the bottom row of diagram (2). The group on the left hand side is defined as

J_{MU}^{2p−1}(X) := MU^{2p−1}(X)_{ℂ}/(F^{p}H^{2p−1}(X; V_{∗}) + MU^{2p−1}(X)).

The group Hdg_{MU}^{2p}(X) is defined as the subgroup of MU^{2p}(X) that is given as the pullback

(27)  \xymatrix{Hdg_{MU}^{2p}(X) \ar[d] \ar[r] & MU^{2p}(X) \ar[d] \\ F^{p}H^{2p}(X; V_{∗}) \ar[r] & H^{2p}(X; V_{∗}).}

The space X(ℂ) has the homotopy type of a finite complex. This implies that each group MU^{k}(X) is finitely generated over ℤ. Hence we may consider MU^{k}(X) as a Hodge structure with the following filtration on MU^{k}(X)_{ℂ}. Using isomorphism (26) we set

F^{i}MU^{k}(X)_{ℂ} := ⨁_{j≥0} F^{i+j}H^{k+2j}(X; ℂ) ⊗_{ℤ} π_{2j}MU.

We can then interpret J_{MU}^{2p−1}(X) as the Jacobian associated to the Hodge structure of weight 2p − 1 on MU^{2p−1}(X):

(28) J_{MU}^{2p−1}(X) = MU^{2p−1}(X) ⊗_{ℤ} ℂ/(F^{p}MU^{2p−1}(X)_{ℂ} ⊕ MU^{2p−1}(X)).

Moreover, the canonical map MU → Hℤ induces a map of Hodge structures

(MU^{2p−1}(X), F^{∗}MU^{2p−1}(X)_{ℂ}) → (H^{2p−1}(X; ℤ), F^{∗}H^{2p−1}(X; ℂ))

which induces the natural map

J_{MU}^{2p−1}(X) → J^{2p−1}(X).

4.2. A cycle map for algebraic cobordism. Our goal in this subsection is to describe the vertical maps in diagram (2). Let Ω^{∗}(X) be the algebraic cobordism ring of X of Levine and Morel. By [19], Ω^{∗}(−) is the universal oriented cohomology theory on Sm. Moreover, in [12, Theorem 7.10], Michael J. Hopkins and the author showed that MU_{log}^{2∗}(∗)(−) is an oriented cohomology theory on Sm, and hence the universality of Ω^{∗}(−) induces a natural transformation

τ: Ω^{∗}(−) → MU_{log}^{2∗}(∗)(−).

Following [19], for given X ∈ Sm, τ is defined as follows. Recall that Ω^{p}(X) is generated by elements [f: Y → X] with f a projective morphism in Sm of relative codimension p. Let p_{Y} : Y → Spec ℂ be the structure map of Y. The class [f] is equal to f_{∗}(p_{Y}^{∗} (1_{Ω})) where f_{∗} denotes the pushforward along f, p_{Y}^{∗} is the pullback along p_{Y} in Ω^{∗}(−), and 1_{Ω} is the unit in Ω^{0}(ℂ). The image of [f: Y → X] under τ: Ω^{p}(X) → MU_{log}^{2p}(p)(X) is then defined as

τ([f: Y → X]) := f_{∗}(p_{Y}^{∗} (1_{MU_{log}}))

where now f_{∗} and p_{Y}^{∗} denote the pushforward and pullback in MU_{log}^{2∗}(∗)(−), respectively, and 1_{MU_{log}} is the unit in MU_{log}^{0}(0)(ℂ). The natural map

φ_{MU_{log}} : Ω^{∗}(X) → MU_{log}^{2∗}(∗)(X)

is defined by φ_{MU_{log}}([f]) := τ([f]). We remark that this is in fact a ring homomorphism for every X ∈ Sm (see [12, Theorem 7.10]). In order to further describe the element φ_{MU_{log}}([f]) in MU_{log}^{2p}(p)(X), we will first look at the image of [f] under the natural map

φ_{MU}: Ω^{p}(X) → MU^{2p}(X).

We know that the image of φ_{MU} lies in the subgroup Hdg_{MU}^{2p}(X). Therefore, we need a better understanding of the group Hdg_{MU}^{2p}(X). Using isomorphism (26) and the Hodge decomposition of complex cohomology, we can define subgroups of MU^{k}(X)_{ℂ}

(29)  MU^{p,q}(X)_{ℂ} := ⨁_{j≥0} H^{p+j,q+j}(X; ℂ) ⊗_{ℤ} π_{2j}MU.

Then MU^{k}(X)_{ℂ} splits into a direct sum

MU^{k}(X)_{ℂ} = ⨁_{p+q=k} MU^{p,q}(X)_{ℂ}.

Lemma 4.1. Let γ be an element of Hdg_{MU}^{2p}(X), and let c ∈ H^{2p}(X; V_{∗}) be the image of γ under isomorphism (26). Then c is given by a family of real cohomology classes (c_{j})_{j≥0} with

c_{j} ∈ H^{p+j,p+j}(X; ℂ) ⊗ π_{2j}MU.

Proof. This follows immediately from diagram (27) and the fact that the image of MU^{2p}(X) in H^{2p}(X; π_{∗}MU ⊗ ℂ) factors through H^{2p}(X; π_{∗}MU ⊗ ℝ). □

Remark 4.2. In view of our notation (29), we see that the group Hdg_{MU}^{2p}(X) can be identified with the elements in MU^{2p}(X) whose image in MU^{2p}(X)_{ℂ} lies in the subgroup MU^{p,p}(X)_{ℂ}.

Lemma 4.3. With the notation of the previous lemma, the class c can be represented by a family of closed forms ω = (ω_{j}) in F^{p}A^{2p}(X; V_{∗})_{cl} such that each ω_{j} is a real form of type (p + j, p + j).

Proof. This follows from Lemma 4.1, Hodge symmetry and the uniqueness of the Hodge decomposition of complex cohomology. □

Remark 4.4. Note that in both families (c_{j}) and (ω_{j}) there are only finitely many nonzero elements. This is due to the fact that X_{an} is a compact complex manifold.

This allows us to describe the image of φ_{MU} as follows.

Proposition 4.5. Let [f: Y → X] be a generator in Ω^{p}(X). The image of φ_{MU}(f) under isomorphism (26) is represented by a family of closed forms ω = (ω_{j}) in F^{p}A^{2p}(X; V_{∗})_{cl} such that each ω_{j} is a real form of type (p+j, p+j).

As a consequence, we can also say more about the image of

φ_{MU_{log}} : Ω^{p}(X) → MU_{log}^{2p}(p)(X).

Proposition 4.6. Let [f: Y → X] be a generator in Ω^{p}(X). The class φ_{MU_{log}}(f) can be represented by a triple

(30)  f_{an} : Y_{an} → X_{an}, ω ∈ F^{p}A^{2p}(X; V_{∗})_{cl}, ξ ∈ A^{2p−1}(X; V_{∗})

such that dξ = ι^{∗}(f_{an}) − ω and the image of f_{an} in A^{2p}(X; V_{∗}) is a family of real forms of type (p + j, p + j).

Proof. Via the Pontryagin-Thom construction, the image [f_{an}] of f in MU^{2p}(X) corresponds to a continuous map q: X → MU_{2p}. The assertion then follows from Propositions 3.7 and 4.5. □

Remark 4.7. The proposition shows to what extend the class of Y → X in MU_{log}^{2p}(p)(X) contains more information than the corresponding images in MU^{2p}(X) and F^{p}H^{2p}(X; V_{∗}): For, φ_{MU_{log}}(f) remembers the homotopy that connects the images in MU^{2p}(X)_{ℂ}. We are going to exploit this fact in the construction of the Abel-Jacobi map in the next section.

4.3. The generalized Abel-Jacobi map. Our final goal is to describe the Abel-Jacobi map

Φ_{MU}: Ω_{top}^{p}(X) → J_{MU}^{2p−1}(X)

in diagram (2). Let α = [Y → X] be a generator in Ω^{p}(X). By Remark 3.8, the image of α in MU_{log}^{2p}(p)(X) is given by a triple

(31)  q: X → MU_{2p}, ω ∈ F^{p}A^{2p}(X; V_{∗})_{cl}, c ∈ C^{2p−1}(X; V_{∗})

such that δc = I(ω) − ι_{n}^{∗}q, where δ denotes the differential in C^{∗}(X; V_{∗}), q is obtained via the Pontryagin-Thom construction and represents the class of Y in MU^{2p}(X), and ω represents the image of [Y] in H^{2p}(X; V_{∗}). Now we assume that the image of α in MU^{2p}(X) vanishes. This implies that both the cohomology class of ω and the homotopy class of q are trivial. Hence there is a form η ∈ F^{p}A^{2p−1}(X; V_{∗}) such that dη = ω, and there is a homotopy H from q to the constant map which sends all of X to the base point of MU_{2p}. We consider this homotopy as a map H: X → PMU_{2p} from X to the path space PMU_{2p} of MU_{2p}. After composition with ι: MU_{2p} → K(V_{∗}, 2p), H defines a map

ι^{∗}H: X → PK(V_{∗}, 2p).

This map, in turn, defines a cochain ι^{∗}H in C^{2p−1}(X; V_{∗}) which we, by abuse of notation, also denote by ι^{∗}H. By construction, the boundary of this cochain is δ(ι^{∗}H) = ι^{∗}q, where we also write ι^{∗}q for the cocycle corresponding to the map ι^{∗}q: X → K(V_{∗}, 2p).

Hence the triple (q, ω, c) is homotopic to the triple (0, 0, ι^{∗}H − I(η) + c), or, more precisely, they are path connected in the space MU_{2p}(p)(X) and define the same element in π_{0}(MU_{2p}(p)(X)). Thus we can assume that φ_{MU_{log}}(α) is represented by the triple (0, 0, h) with

(32)  h := ι^{∗}H − I(η) + c ∈ C^{2p−1}(X; V_{∗}) and δh = 0.

In particular, h is a cocycle in C^{2p−1}(X; V_{∗}) and represents an element [h] in H^{2p−1}(X; V_{∗}) ≅ MU^{2p−1}(X)_{ℂ}. In constructing [h], we chose a form η and a homotopy H. The class of [h] depends on these choices in the following way. First, let η′ be another form in F^{p}A^{2p−1}(X; V_{∗}) such that dη′ = ω. The difference η − η′ is then a closed form. Hence I(η − η′) is a cocycle and defines a class in F^{p}H^{2p−1}(X; V_{∗}). Second, let H′ be another homotopy from q to the constant map. Considering H and H′ as maps X → PMU_{2p}, we can compose H and H′ as paths by first walking with doubled speed along H and then with doubled speed in reversed direction of H′. Hence the two homotopies H and H′ define a map H ∗ (−H′): X → Ω(MU_{2p}) from X to the loop space of MU_{2p}. Since MU is an Ω-spectrum, Ω(MU_{2p}) is homeomorphic to MU_{2p−1}. Thus H and H′ define a map H ∗ (−H′): X → MU_{2p−1}. After taking homotopy classes, we obtain an element [H ∗ (−H′)] ∈ MU^{2p−1}(X). If we set h = ι^{∗}H −I(η)+c and h′ = ι^{∗}H′ −I(η′)+c, we see that the difference of [h] and [h′] in H^{2p−1}(X; V_{∗}) is the class of ι^{∗}[H ∗ (−H′)] − I(η − η′), where we also write ι for the map MU_{2p−1} → K(V_{∗}, 2p − 1). We thus obtain that the class [h] ∈ H^{2p−1}(X; V_{∗}) is well-defined modulo the images of the subgroups MU^{2p−1}(X) and F^{p}H^{2p−1}(X; V_{∗}). The following theorem is the summary of this discussion.

Theorem 4.8. Let α be an element in Ω^{p}(X) and h be the cocycle defined in (32). Then the image of [h] in the quotient

J_{MU}^{2p−1}(X) = MU^{2p−1}(X)_{ℂ}/(F^{p}H^{2p−1}(X; V_{∗}) + MU^{2p−1}(X))

is the image of α under Φ_{MU}.

Remark 4.9. We can interpret the class of h also in the following way. Recall that, since the underlying complex manifold of X is a Kähler manifold, the natural map H^{2p−1}(X; ℝ) → H^{2p−1}(X; ℂ)/F^{p}H^{2p−1}(X; ℂ) is an isomorphism of ℝ-vector spaces. Because of our convention on the Hodge filtration and gradings, this implies that the natural map

H^{2p−1}(X; π_{∗}MU ⊗ ℝ) → H^{2p−1}(X; π_{∗}MU ⊗ ℂ)/F^{p}H^{2p−1}(X; π_{∗}MU ⊗ ℂ)

is an isomorphism of ℝ-vector spaces, too. For h as in (32), this implies that we can assume that h defines a real class in H^{2p−1}(X; π_{∗}MU ⊗ℝ) ≅ MU^{2p−1}(X)_{ℝ} which depends only on the image of MU^{2p−1}(X). In other words, h defines a unique element in

MU^{2p−1}(X)_{ℝ}/MU^{2p−1}(X) ≅ MU^{2p−1}(X) ⊗ ℝ/ℤ which equals the image of α under the isomorphism

J_{MU}^{2p−1}(X) ≅ MU^{2p−1}(X) ⊗ ℝ/ℤ.

Remark 4.10. We see that Φ_{MU}(α) depends on how α vanishes in MU^{2p}(X) and F^{p}H^{2p}(X; V_{∗}). Hence calculating Φ_{MU}(α) is in general a difficult task. But this is not different from the classical Abel-Jacobi map. For example, the calculations for Griffiths’ famous examples are not done by just evaluating an integral of a form over some given chain, but use use more sophisticated arguments (see [9] or [25]).

Remark 4.11. Using known examples of non-trivial elements in the Griffiths group, we can produce examples of elements in Ω_{top}^{p}(X) which lie in the kernel of θ: Ω^{p}(X) → CH^{p}(X) (see [12, §7.3]). This shows that the new Abel-Jacobi invariant is able to detect certain elements in Ω_{top}^{∗}(X) which the classical invariant Φ would not see. Nevertheless, we do not yet have an example of the following type: an element in Ω_{top}^{∗}(X) which maps to the kernel of Φ, but is neither in the kernel of θ nor in the kernel of Φ_{MU}. This is a much more difficult task which requires a better understanding of the kernel of the map

MU^{2p−1}(X) ⊗ ℝ/ℤ → H^{2p−1}(X; ℝ/ℤ).

References

[1] A. Beilinson, Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036-2070.

[2] B. A. Blander, Local projective model structures on simplicial presheaves, K-Theory 24 (2001), 283-301.

[3] K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1973), 419-458.

[4] P. Deligne, Théorie de Hodge II, Pub. Math. IHES 40 (1971), 5-57.

[5] D. Dugger, Universal homotopy theories, Adv. Math. 164 (2001), 144-176.

[6] D. Dugger, S. Hollander, D. C. Isaksen, Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9-51.

[7] D. Dugger, D. C. Isaksen, Topological hypercovers and 𝔸^{1}-realizations, Math. Z. 246 (2004), 667-689.

[8] H. Esnault, E. Viehweg, Deligne-Beilinson cohomology, in: Beilinson’s conjectures on special values of L-functions, 43-91, Perspect. Math., 4, Academic Press, Boston, MA, 1988.

[9] P. A. Griffiths, On the periods of certain rational integrals, I, II. Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 1969, 496-541.

[10] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 (1964), 205-326.

[11] P. S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs, vol. 99, Amer. Math. Soc., 2003.

[12] M. J. Hopkins, G. Quick, Hodge filtered complex bordism, J. Topology 8 (2015), 147-183.

[13] M. J. Hopkins, I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geometry 70 (2005), 329-452.

[14] J. F. Jardine, Simplicial presheaves, J. Pure Appl. Algebra 47 (1987), no. 1, 35-87.

[15] J. F. Jardine, The Verdier hypercovering theorem, Canad. Math. Bull. 55 (2012), 319-328.

[16] M. Karoubi, Théorie générale des classes caractéristiques secondaires, K-Theory 4 (1990), no. 1, 55-87.

[17] M. Karoubi, Classes caractéristiques de fibrés feuilletés, holomorphes ou algébriques, in: Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part II (Antwerp, 1992), K-Theory 8 (1994), no. 2, 153-211.

[18] A. Krishna, J. Park, Algebraic cobordism theory attached to algebraic equivalence, J. K-Theory 11 (2013), no. 1, 73-112.

[19] M. Levine, F. Morel, Algebraic Cobordism, Springer Monographs in Mathematics, Springer, 2007.

[20] M. Levine, R. Pandharipande, Algebraic cobordism revisited, Invent. Math. 176 (2009), no. 1, 63-130.

[21] F. Morel, V. Voevodsky, 𝔸^{1}-homotopy theory of schemes, Publ. IHES 90 (1999), 45-143.

[22] V. Navarro Aznar, Sur la théorie de Hodge-Deligne, Invent. Math. 90 (1987), 11-76.

[23] G. Quick, Homotopy theory of smooth compactifications of algebraic varieties, New York J. Math. 19 (2013), 533-544.

[24] C. Voisin, Hodge theory and complex algebraic geometry I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002.

[25] C. Voisin, Hodge theory and complex algebraic geometry II, Cambridge Studies in Advanced Mathematics, 77, Cambridge University Press, Cambridge, 2003.

Gereon Quick Department of Mathematical Sciences NTNU NO-7491 Trondheim Norway gereon.quick@math.ntnu.no