ONE-MOTIVES AND A CONJECTURE OF DELIGNE

NIRANJAN RAMACHANDRAN

Abstract. We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of these invariants presented here proves a conjecture of Deligne. Other applications include some cases of conjectures of Serre, Katz, and Jannsen on the independence of ℓ of parts of the étale cohomology of arbitrary varieties over number fields and finite fields.

Contents

1. Preliminaries 4 2. Construction of L^{n} for a simplicial pair 10 3. Hodge and de Rham realizations of L^{n} 15 4. Etale realization of L^{n} 27 5. The one-motives L^{n}(V) 31 6. Positive characteristic 34 7. Applications and related results 38 References 40

𝔚𝔢𝔫𝔫 𝔡𝔦𝔢 𝔎𝔬̈𝔫𝔦𝔤𝔢 𝔟𝔞𝔲′𝔫, 𝔥𝔞𝔟𝔢𝔫 𝔡𝔦𝔢 𝔎𝔞̈𝔯𝔯𝔫𝔢𝔯 𝔷𝔲 𝔱𝔲𝔫.

𝔉. 𝔖𝔠𝔥𝔦𝔩𝔩𝔢𝔯

Introduction. P. Deligne [14, 10.4.1] has attached one-motives to complex algebraic varieties using the theory of mixed Hodge structures. He has conjectured that these one-motives admit a purely algebraic definition. The aim of this article is to prove his conjecture (Theorem 5.7).

Recall the well known result of Riemann [12, 4.4.3], presented here in modern guise: the “Hodge realization” T_{ℤ} — this is A ↦ H_{1}(A, ℤ) — defines an equivalence from the category of complex abelian varieties to the category of torsion-free polarizable Hodge structures of type {(0, −1), (−1, 0)}. In particular, any such Hodge structure arises as the H_{1} of an essentially unique complex abelian variety.

Date: February 20, 2003.

1991 Mathematics Subject Classification. 14Fxx.

Funded in part by grants from Hewlett-Packard, Graduate Research Board (U. Maryland) and MPIM (Bonn).

Deligne [14, §10.1] has introduced the algebraic notion of a one-motive over a field k, generalizing that of an abelian variety — §1 contains the precise definitions; he has also generalized Riemann’s result by showing that the “Hodge realization” T_{ℤ} defines an equivalence from the category of one-motives over ℂ to the category of torsion-free mixed Hodge structures H of type

(∗) {(−1, −1), (−1, 0), (0, −1), (0, 0)}

with Gr_{−1}^{W}H polarizable. Thus, any such mixed Hodge structure H arises from an essentially unique one-motive I(H) over ℂ. The functor I is a quasi-inverse to T_{ℤ}.

For any complex variety V and any integer n ≥ 0, consider the largest mixed Hodge substructure t^{n}(V) of type (∗) of H^{n}(V, ℤ(1))/torsion; there exists a well-defined onemotive I^{n}(V) over ℂ whose Hodge realization is t^{n}(V); so I^{n}(V) := I(t^{n}(V)). Deligne [14, 10.4.1] has conjectured that I^{n}(V) admits a purely algebraic definition. His proof (ibid. 10.3. — Interprétation algébrique du H^{1} mixte: cas des courbes) of his conjecture for arbitrary curves suggests a precise formulation of the conjecture. Namely, we have the following (this formulation is due to the referee): Conjecture 0.1. (Deligne) For an arbitrary variety V over an arbitrary field k and integer n, define a one-motive L^{n}(V/k) and homomorphisms^{1}

T_{ℓ}(L^{n}(V/k)) → H^{n}(V × k̄, ℤ_{ℓ}(1))/torsion,
T_{DR}(L^{n}(V/k)) → H_{DR}^{n}(V/k)

from the ℓ-adic and de Rham realizations of L^{n}(V/k). The definitions of L^{n}(V/k) and the homomorphisms should be algebraic, canonical, and functorial in V and k.

Furthermore, L^{n}(V/ℂ) should be canonically isomorphic to I^{n}(V).

(Clearly, V can be replaced by a simplicial scheme.)

The prototype is A. Weil’s construction [52] of the Jacobian; his construction proves the conjecture for smooth projective curves and n = 1. The conjecture is true for smooth projective varieties (1.6): it amounts to an algebraic construction of the Picard variety and the Néron-Severi group.

The case n = 1 of (0.1) is known (up to p-isogeny in characteristic p > 0) for arbitrary varieties over perfect fields [3, 14, 43, 48]; the case n = 2 is known for complex proper surfaces [6, 7]. No general results were known for higher cohomology (i.e., for n > 2).

A natural approach to Conjecture 0.1 is to use proper hypercoverings [14, 6.2] by smooth simplicial schemes; namely, to mimic Deligne’s approach [14] to the construction of the mixed Hodge structure on H^{∗}(V, ℤ) of a complex algebraic variety V. This approach, which we follow here, gives a two-step strategy to prove (0.1): Step 1. Construct one-motives L^{n} (n ≥ 0) for smooth simplicial schemes arising from simplicial pairs (1.10) and show that they have the properties given in (0.1). Step 2. Prove cohomological descent for these one-motives; more precisely, show that the one-motives L^{n}, given by (i), of a proper hypercovering of a variety V are “independent” of the proper hypercovering; and, thus, L^{n} depend only on V.

Sections 2, 3, 4 are devoted to the first step, but only for fields of characteristic zero; the case of positive characteristic is relegated to Section 6. Our construction of the requisite one-motives L^{n}, inspired by [6], relies on the theory of the Picard scheme [5, Chapter 8]; the techniques are those of [43] but here applied to truncated simplicial schemes. The realizations of L^{n} are treated in Sections 3 (Hodge, de Rham), 4 (étale); here a crucial use is made of the validity of the Hodge conjecture for divisors (4).

1Here k̄ is an algebraic closure of k.

Section 5 is devoted to the second step. It turns out that, because an important spectral sequence [14, 8.1.19.1] degenerates only with rational coefficients, the method of proper hypercoverings only provides a theory of isogeny one-motives L^{∗}(−) ⊗ ℚ. More precisely, given two proper hypercoverings U_{•} and ′U_{•} of V, we can only show that the associated one-motives L^{n} and′L^{n} are isogenous; the isogeny one-motive L^{n}⊗ℚ depends only on V. Thus, a new ingredient is necessary to complete the second step, i.e., to endow these isogeny one-motives with integral structures. This is done, as in [40], via the integral structure on étale cohomology. Thus, we provide a complete proof (5.7) of Conjecture 0.1 for an arbitrary field of characteristic zero.

We now turn to the case of Conjecture 0.1 for a field k of characteristic p > 0; let us begin by indicating why the conjecture must be weakened slightly.

First, in [32, Appendix], A. Grothendieck notes that, for a curve C over k, the construction of Deligne [14, 10.3] provides a one-motive H_{m}^{1}(C) = L^{1}(C/k) defined over the perfection k^{perf} of k; thus, [14, 10.3] proves the case n = 1 of (0.1) only for curves over a perfect field. Second, he (loc. cit) expresses doubts about the existence of a ℤ-linear (i.e., integral) category of mixed motives over an imperfect field k; he anticipates only a ℤ[1/p]-linear category, i.e., a category of mixed motives up to pisogeny. Third, the existence of proper hypercoverings (by smooth simplicial schemes) for an arbitrary variety V over k is known only when k is perfect [9]. If one expects that the method of proper hypercoverings provides, as in characteristic zero, one-motives up to isogeny associated with V, then the methods of [40] allow a refinement to one-motives defined up to p-isogeny: controlling p-isogeny requires an integral p-adic cohomology theory for arbitrary varieties over k. These considerations^{2} lead us to a weak version of (0.1) by only requiring one-motives L^{∗}(V/k) ⊗ ℤ[1/p] (up to p-isogeny) over k^{perf}.

While the first step can be carried out in positive characteristic in the same way as in characteristic zero, the second step cannot be unless, as it seems, one assumes the Tate conjecture (5) for divisors. Note that our proof of (0.1) in characteristic zero depends on the validity of the Hodge conjecture (4) for divisors; thus, the appearance of the Tate conjecture is rather natural.

For a perfect field k, (0.1) — up to p-isogeny — holds (6.14) under the assumption of the Tate conjecture (5) for surfaces. The analogous result is also valid for an imperfect field k under the additional assumption of “resolution of singularities” over k.

Using a suggestion of M. Marcolli, we provide an unconditional construction (6.13) of J^{n}(−)⊗ℤ[1/p] (n ≥ 0) and L^{n}(−)⊗ℤ[1/p] (2 ≥ n ≥ 0) of one-motives (up to p-isogeny) for arbitrary varieties over a perfect field k. The J^{n}(V) ⊗ ℤ[1/p] are good substitutes for the (as yet conditional) L^{n}(V) ⊗ ℤ[1/p]; for instance, W_{−1}J^{n}(V) = W_{−1}L^{n}(V).

In particular, we generalize Carlson’s results [6] on L^{2} of a complex projective surface to any variety over a perfect field (and up to p-isogeny in characteristic p > 0).

2V. Voevodsky [51] works over perfect fields and neglects p-torsion in characteristic p > 0.

In Section 7, we use these new invariants L^{n} and J^{n} to provide affirmative answers to special cases of questions [33, 36, 47] in the motivic folklore. These concern “independence of ℓ” of ℓ-adic étale cohomology of arbitrary varieties over number fields and finite fields.

Since the circulation of this manuscript (circa 1998), other authors [2] have independently obtained some of the results presented here; [42] is a leisurely introduction to our results.

Notation. We work over S := Spec k; here k is a field of characteristic p (except in §1, 6, k is perfect unless indicated otherwise); in sections §2, 3, 4, and 5 we assume p to be zero. In §6, we assume p > 0.

We fix an algebraic closure k̄ of k; S̄ := Spec k̄ and 𝔾 := Gal(k̄/k). For any scheme X over S, we set X̄ := X ×_{S} S̄. All schemes will be supposed to be separated and locally noetherian. A variety is a geometrically integral scheme of finite type over S. For any set B, ℤ(B) is the free abelian group generated by the elements of B. π_{0}(X) := the set (with a 𝔾-action) of connected components of X̄. D_{X} := the étale group scheme corresponding to ℤ(π_{0}(X̄)). T_{X} := Hom(D_{X}, 𝔾_{m}), the algebraic torus associated with D_{X}. w(X) := the set of irreducible components of X̄. W_{X} := the étale group scheme corresponding to the 𝔾-module ℤ(w(X)). For any group scheme G, π_{0}(G) is a group with an action of 𝔾; we shall also use π_{0}(G) for the corresponding étale group scheme.

For a variety V over ℂ, V (ℂ) (resp. V^{an}) denotes the associated topological space with the classical topology (resp. analytic variety). Given X over S and an imbedding ι : k ↪ ℂ, we denote by X_{ι} the scheme over ℂ obtained by base change. MHS := the abelian category of (ℤ)-mixed Hodge structures. M_{F} := the additive category of one-motives over a field F. S_{fppf} (resp. S_{fpqc}) is the big site over S with the fppf (resp. fpqc) topology [5, pp. 200-201]. ℤ̂ is the profinite completion \underleftarrow{lim}_{r}ℤ/rℤ of ℤ. 𝔸 = ℤ̂ ⊗ ℚ is the ring of finite adèles of ℚ. ℤ^{p} := \underleftarrow{lim}_{r}ℤ/rℤ with r coprime to p and 𝔸^{p} := ℤ^{p} ⊗ ℚ.

We refer to section x.y by §x.y and to specific results by Theorem x.y or Remark x.y or simply (x.y). We use □ to denote the end of a remark or a proof.

1. Preliminaries

Let us begin by reviewing well-known results, which will be of use in the paper.

One-motives. [14, §10]

A one-motive M := [B ⟶^{u} G] over S (or over k) is a two-term complex consisting of a semi-abelian variety G over k (i.e., G is an extension of an abelian variety A by a torus T), a finitely generated torsion-free abelian group B with a structure of a discrete 𝔾-module, and a homomorphism u : B → G(k̄) of 𝔾-modules. In particular, if k is algebraically closed, then u is a homomorphism of abelian groups. It is convenient to regard B as an étale group scheme (locally constant) on S. A morphism of one-motives is a morphism of complexes. From the category M_{k} of one-motives over k, there are “realization” functors (Hodge) T_{ℤ} — for each ι : k ↪ ℂ — to (the category of) torsionfree ℤ-mixed Hodge structures of type (∗), (étale) T_{ℓ} — for each ℓ ≠ p — to ℤ_{ℓ}-modules with an action of 𝔾, and (de Rham) T_{DR} — if p = 0 — to k-vector spaces.

A morphism ϕ : M_{1} → M_{2} is called an isogeny if ϕ_{B} : B_{1} → B_{2} is injective with finite cokernel and ϕ_{G} : G_{1} → G_{2} is surjective with finite kernel. The (additive) category M_{k} of one-motives over k enjoys Cartier duality [14, 10.2.11]. The dual of an isogeny is also an isogeny. A p-isogeny is an isogeny ϕ such that the orders of Coker(ϕ_{B}) and Ker(ϕ_{G}) are powers of p.

The ℚ-linear abelian category M_{k} ⊗ ℚ of isogeny one-motives over k is obtained from M_{k} by inverting isogenies; M_{k} ⊗ ℚ inherits realization functors (Hodge) T_{ℤ} to ℚ-mixed Hodge structures, (étale) T_{ℓ} to ℚ_{ℓ}-vector spaces, (de Rham) T_{DR} to k-vector spaces, weight filtration W, and Cartier duality from M_{k}. Every one-motive M defines an isogeny one-motive M ⊗ ℚ. The weight filtration W on [B ⟶^{u} G] ⊗ ℚ is W_{−3} = 0,

W_{−2} = [0 → T] ⊗ ℚ, W_{−1} = [0 → G] ⊗ ℚ, and W_{0} = [B ⟶^{u} G] ⊗ ℚ.

Finally, if p = 0, the functors T_{ℓ} can be combined to a functor M ↦ T M = M ⊗ℤ̂ := ∏ℓ T_{ℓ}M; here T M is a ℤ̂-module with a 𝔾-action. The adèlic realization functor M ↦ T M ⊗ ℚ from M_{k} to 𝔸-modules with a 𝔾-action factorizes via M_{k} ⊗ ℚ; this gives the adèlic realization functor of an isogeny one-motive: M ⊗ ℚ ↦ M ⊗ 𝔸.

If p > 0, then the ℤ[1/p]-linear category M_{k} ⊗ ℤ[1/p] of one-motives up to p-isogeny over k is obtained from M_{k} by inverting p-isogenies. The realization functor M ↦ T^{p}M = _{∏ℓ≠p} T_{ℓ}M from M_{k} to the category of ℤ^{p}-modules with a 𝔾-action extends to the category M_{k} ⊗ ℤ[1/p]. Relative representability.

As indicated in [5, pp.200-201], representability issues are best treated in the fppftopology. The following simple lemma will be used often. Lemma 1.1. (i) Let F be a representable contravariant functor from the category of schemes over S to sets. Then F is a sheaf with respect to the fpqc-topology and, hence, with respect to the fppf, étale, and Zariski topologies.

(ii) Let 0 → F → G → H → 0 be an exact sequence of sheaves of abelian groups on S_{fppf}. Suppose F, H are representable, and that F → S is an affine morphism (i.e. the scheme representing F is affine over S). Then G is representable, necessarily by a commutative group scheme.

Proof. (i) [5, Prop. 1, p.200]. (ii) The proof of [41, Prop. 17.4] for S_{fpqc} also works for S_{fppf}. □

Picard functor.

Let f : X → S be a smooth proper scheme. Proposition 1.2. The sheaves f_{∗}O, f_{∗}O^{∗}, and R^{1}f_{∗}O^{∗} on S_{fppf} are representable. The scheme T_{X} := Hom(D_{X}, 𝔾_{m}) represents f_{∗}O^{∗}.

Proof. The representability of f_{∗}O and f_{∗}O^{∗} is rather elementary [5, Cor. 8, Lem. 10, pp.207-208]. Each character of D_{X} provides a non-zero function, constant (since X is proper) on each connected component of X, i.e., on each irreducible component of X (since X is smooth). Thus T_{X} represents f_{∗}O^{∗}. The representability of R^{1}f_{∗}O^{∗} is due to Murre-Oort [5, Thm. 3, p.211]. □

The scheme representing R^{1}f_{∗}O^{∗} is the Picard scheme P ic_{X} of X. It is reduced in characteristic zero but it may not be so in positive characteristic. Its reduced neutral (= identity) component P ic_{X}^{0,red} is the classical Picard variety P ic(X). The Néron-Severi group scheme NS_{X} is the étale group scheme corresponding to the 𝔾-module π_{0}(P ic_{X}^{red}); we often write NS(X) for NS_{X}(S). Proposition 1.3. (p = 0) The S_{fppf}-sheaves R^{i}f_{∗}Ω^{j} and H_{DR}^{i}(X) = R^{i}f_{∗}Ω given by Hodge and de Rham cohomology (Ω is the de Rham complex on X) as well as the sheaf R^{1}f_{∗}Ω^{∗} corresponding to the multiplicative de Rham complex [37, 3.1.7, p.31] on X

Ω^{∗} := [O^{∗} ⟶^{d log} Ω^{1} → Ω^{2} ⋯]

are all representable. The first two are representable by vector group schemes.

Proof. The sheaves R^{i}f_{∗}Ω^{j}, H_{DR}^{i}(X) are coherent, free, and commute with arbitrary base change [34, 1.4.1.8], [25, p.309-310]. For any t : S′ → S, we have R^{i}f_{∗}Ω^{j}(S′) = t^{∗}H^{i}(X, Ω^{j}); similarly for H_{DR}^{i}(X). Any locally free O_{T} -module L on a scheme T gives rise to a sheaf on T_{fppf} which is representable by a vector group scheme [37, p.1]. Thus, R^{i}f_{∗}Ω^{j}, H_{DR}^{i}(X) are representable by vector group schemes. Similarly, the sheaves R^{i}f_{∗}C — here C is [Ω^{1} → Ω^{2} ⋯] — are also representable by vector group schemes. In the exact sequence^{3} (cf. (2.13))

0 → f_{∗}C → R^{1}f_{∗}Ω^{∗} → R^{1}f_{∗}O^{∗} → R^{1}f_{∗}C,

representability is already known for all the sheaves other than R^{1}f_{∗}Ω^{∗}; and f_{∗}C is representable by an affine scheme. The representability of R^{1}f_{∗}Ω^{∗} follows from (1.1).

□

The scheme P ic_{X}^{♮} representing R^{1}f_{∗}Ω^{∗} classifies [38, 2.5] [34, 7.2.1] isomorphism classes of line bundles (=invertible sheaves) on X endowed with an integrable connection; cf. (3.6). Proposition 1.4. Assume that k is of characteristic zero.

(i) The neutral component E^{♮} of P ic_{X}^{♮} is the universal additive (= vectorial) extension of P ic_{X}^{0} = P ic_{X}^{0,red}.

(ii) The additive group scheme Lie E^{♮} represents the sheaf H_{DR}^{1}(X);

Lie E^{♮} = Lie P ic_{X}^{♮} ⟶^{∼} H_{DR}^{1}(X) := ℍ^{1}(X, Ω) ⟵^{∼} ℍ^{1}(X, [O → Ω^{1}]).

Proof. A proof of (i) for X an abelian variety is in [38, 2.1, 2.7, 2.8]; it can also be obtained by combining propositions 2.6.7, 3.2.3, and 4.2.1 of [37, Chapter I].

For a general smooth proper X over S, let g : E^{∗} → E^{♮} be the map induced from the universal additive extension E^{∗} of P ic_{X}^{0}. We need to show that g is an isomorphism. Since both E^{∗} and E^{♮} are compatible with base change, we may assume X(S) ≠ ∅; this provides an Albanese map u : X → Alb(X). We now argue as in [38, 3.0] using the

3The natural map f_{∗}O^{∗} → f_{∗}C is zero: the former is represented by a torus and the latter by a vector group scheme.

standard isomorphisms u^{∗} : H^{0}(Alb(X), Ω^{1}) ⟶^{∼} H^{0}(X, Ω^{1}) and u^{∗} : P ic_{Alb(X)}^{0} ⟶^{∼} P ic_{X}^{0}.

Part (ii) follows from (i) by [38, Lem. 2.6.9]. □

Divisors on a smooth proper variety.

We recall the classical properties of P ic(X) and NS(X) = NS_{X}(S) for f : X → S smooth proper. Remark 1.5. One has

(i) (ℓ ≠ p) an isomorphism H_{et}^{1}(X̄, ℤ_{ℓ}(1)) ⟶^{∼} T_{ℓ}P ic(X) [39, p.125] of 𝔾-modules provided by the Kummer sequence (37).

(ii) (p = 0) Lie P ic(X) ⟶^{∼} H^{1}(X, O) [5, Thm. 1, p.231].

(iii) (p = 0) an exact sequence (1.4)

0 → H^{0}(X, Ω^{1}) → Lie E^{♮} → H^{1}(X, O) → 0.

(iv) (k = ℂ) an isomorphism of pure Hodge structures [24, pp.156-158]:

(1)  H^{1}(X(ℂ), ℤ(1)) ⟶^{∼} H_{1}(P ic(X), ℤ)

provided by the exponential sequence

(2)  0 → ℤ(1) → O ⟶^{exp} O^{∗} → 1;

a commutative diagram [19, Thm. 1.4, p. 17] — vertical maps are [45]:

0 ⟶ H^{0}(X, Ω^{1}) ⟶ H_{DR}^{1}(X) ⟶ H^{1}(X, O) ⟶ 0
≀↓  ^{≀}↓  ^{≀}↓
0 ⟶ H^{0}(X^{an}, Ω^{1}) ⟶ H^{1}(X(ℂ), ℂ) ⟶ H^{1}(X^{an}, O) ⟶ 0.

(v) (ℓ ≠ p) the “cycle class map” [39, VI §9] furnishes a 𝔾-equivariant inclusion [39, 3.2.9 (d), p.216]:

(3)  NS(X̄) ⊗_{ℤ} ℤ_{ℓ} ↪ H_{et}^{2}(X̄, ℤ_{ℓ}(1));

numerical and homological equivalence coincide for divisors (with ℚ_{ℓ}-coefficients).

(vi) (k = ℂ) Lefschetz’s (1, 1)-theorem = the integral Hodge conjecture for divisors [18, p.143] [24, p. 156]:

(4)  NS(X) ⟶^{∼} Hom_{MHS}(ℤ, H^{2}(X(ℂ), ℤ(1))).

(vii) (k finitely generated) the Tate conjecture [49, p.72] [31, 5.1] for divisors asks if

(5)  NS(X) ⊗_{ℤ} ℚ_{ℓ} ⟶^{∼} H_{et}^{2}(X × k^{sep}, ℚ_{ℓ}(1))^{𝔾_{sep}}?

Here k^{sep} is a separable algebraic closure of k, 𝔾_{sep} the associated Galois group, and M^{𝔾_{sep}} denotes the invariants of a 𝔾_{sep}-module M. See [49, §5] for the known cases of (5).

(viii) (p = 0) the Chern class map [12, 2.2.4] gives an injection

(6)  c_{X} : NS(X) ⊗ k ↪ H^{1}(X, Ω^{1}).

(The map d log : O^{∗} → Ω^{1} [12, 2.2.4] induces a map R^{1}f_{∗}O^{∗} → R^{1}f_{∗}Ω^{1} of representable S_{fppf}-sheaves; since the second is affine and the identity component of the first is an abelian scheme, one gets the map c_{X}.) □

Remark 1.6. Let us consider (0.1) for X. By the purity of H^{n}(X, ℤ(1)) (k = ℂ), both t^{n}(X) and I^{n}(X) are zero for n > 2. Parts (i)-(iv) of (1.5) identify [0 → P ic(X)] as the one-motive L^{1}(X/k) whereas parts (v), (vi), and (viii) identify [NS_{X}/torsion → 0] as the one-motive L^{2}(X/k). Conjecture 0.1 is known for smooth proper varieties. □

From (4), one obtains the Theorem 1.7. Let X be a smooth proper scheme over S = Spec k. The dimension of the ℚ-vector space

H_{ℚ}^{1,1}(X_{ι}) := Hom_{MHS}(ℚ(−1), H^{2}(X_{ι}, ℚ))

is independent of the map ι : k ↪ ℂ. □ Smooth varieties.

Let U be the open complement of a strict divisor Y (1.10) with normal crossings in a smooth projective complex variety X. The normalization Ỹ of Y is a smooth projective scheme. Consider the map W_{Y} → P ic_{X} which sends a divisor E to the class of the invertible sheaf O(E) on X. Let N be the cokernel of the induced map λ : W_{Y} → NS_{X}. Proposition 1.8. The (0, 0)-part of H^{2}(U, ℚ(1)) is N ⊗ ℚ. So t^{2}(U) ⊗ ℚ ⟶^{∼} N ⊗ ℚ.

Proof. Let j : U ↪ X denote the inclusion. The cycle class map [12, 2.2.4-5] and the Gysin sequence [15, 3.3] provide the following commutative diagram

W_{Y}  ⟶^{λ} NS(X) ⟶  N  ⟶ 0
∥  ↓  ↓
H^{0}(Ỹ , ℤ) ⟶ H^{2}(X, ℤ(1)) ⟶^{j^{∗}} H^{2}(U, ℤ(1))  .

By [12, 3.2.17], we know that W_{0}H^{2}(U, ℚ(1)) = j^{∗}(H^{2}(X, ℚ(1))). This implies that the (0, 0)-part of H^{2}(U, ℚ(1)) is the image under j^{∗} of the (0, 0)-part of H^{2}(X, ℚ(1)). From (4), NS(X) is the (0, 0)-part of H^{2}(X, ℤ(1)). This proves the first claim. The second follows immediately because t^{2}(U) ⊗ ℚ is the (0, 0)-part of H^{2}(U, ℚ(1)): by [12, 3.2.15 (ii)], W_{−1}H^{2}(U, ℚ(1)) = 0. □

Remark 1.9. Since W_{m−1}H^{m}(U, ℚ) = 0 [12, 3.2.15] for any m ≥ 0, only t^{1}(U) and t^{2}(U) can be nontrivial. Conjecture 0.1 for the case of t^{1}(U) is classically known (3.7); L^{1}(U/k) can be identified as the Picard one-motive [Ker(λ) → P ic(X)] of [43], the Cartier dual of the generalized Albanese variety of U [48]. The result (1.8), an analog of (4) for smooth varieties, identifies [N → 0]⊗ℚ as L^{2}(U/k)⊗ℚ. Thus, (0.1) is known (up to isogeny) for the smooth variety U. □ Simplicial objects. [8]

A simplicial object B_{•} in a category C is a sequence of objects

B_{•} = {B_{0}, B_{1}, B_{2}, ..., B_{n}, ...},

together with morphisms (face) d_{i} : B_{n} → B_{n−1}, (degeneracies) s_{i} : B_{n} → B_{n+1} (0 ≤ i ≤ n) satisfying the simplicial identities of which we need only (the degeneracy maps will not play a role in our discussions)

(7)  d_{i}d_{j} = d_{j−1}d_{i}  i < j.

A truncated simplicial object B_{≥m} is the subsequence of objects {B_{m}, B_{m+1}, B_{m+2}, ...}, together with all the maps d_{i} and s_{j} between them.

Given a simplicial (resp. cosimplicial) commutative group scheme A_{•}, then the pair (A_{•}, δ) — here δ_{n} = Σ_{i=0}^{i=n}(−1)^{i}d_{i} [8, 3.4] — becomes a chain complex of commutative group schemes: it follows from (7) that δ_{n}δ_{n+1} = 0 (resp. δ_{n+1}δ_{n} = 0). Simplicial pairs.

Let a : X → S be a smooth projective morphism. Let Y be a strict divisor with normal crossings on X as in [9, 2.4]; in particular, this means that Y is reduced and its irreducible components Y^{i}(i ∈ I) are regular schemes, and of codimension one in X. Let U be the open subscheme of X corresponding to the complement of Y. The normalization Ỹ of Y is the disjoint sum of the Y^{i}’s. Definition 1.10. A simplicial pair (X_{•}, Y_{•}) consists of the data of

(i) a simplicial scheme X_{•} smooth and projective over S.

(ii) a strict divisor with normal crossings Y_{•} of X_{•}; in particular, each Y_{m} is a strict divisor (as defined above) of X_{m}.

(These conditions are (α), (β) of [9, p. 51]; it follows from [9] that any variety over S (k perfect) admits a proper hypercovering (§5) corresponding to a simplicial pair.)

In particular, (X_{•}, Y_{•}) is a simplicial object in the category of pairs. The schemes U_{m} := X_{m}−Y_{m} give a smooth simplicial subscheme U_{•} of X_{•} [14, 6.2.6]; let j : U_{•} → X_{•} be the natural map.

A simplicial pair (X_{•}, Y_{•}) gives, for each l ≥ 0, a truncated simplicial pair (X_{≥l}, Y_{≥l}) = (X_{•}, Y_{•})_{≥l} which consists of the schemes (X_{m}, Y_{m}) (for m ≥ l) and the maps d, s.

One has an evident notion [14, 6.2.8] [44, p. 75] of a morphism θ : (X_{•}, Y_{•}) → (Z_{•}, J_{•}) of simplicial pairs; θ satisfies [30, 1.10] θ(Y_{m}) ⊂ J_{m} and θ(X_{m} − Y_{m}) ⊂ (Z_{m} − J_{m}). Remark 1.11. Fix a simplicial pair (X_{•}, Y_{•}). The simplicial abelian group scheme D_{•} — here D_{i} = D_{X_{i}} — gives upon normalization a chain complex still denoted D_{•}. Similarly, one obtains the chain complexes of group schemes: T_{•} P ic_{•}, NS_{•}, P ic_{•}^{0}, W_{•}. The map λ_{i} : W_{i} = W_{Y_{i}} → NS_{X_{i}} = NS_{i} which sends a divisor E of X_{i} supported on Y_{i} to the class [O(E)] of the invertible sheaf O(E) gives a map λ : W_{•} → P ic_{•} → NS_{•}. Exact and spectral sequences.

We summarize some results from [14, 5.1-5.3] about cohomology of sheaves on a simplicial scheme Z_{•}. For any sheaf (or complex of sheaves) F on Z_{•}, there is a spectral sequence [14, 5.2.3.2, 5.1.12.2] ^{4}

(8)  E_{1}^{p,q} = H^{q}(Z_{p}, F) ⇒ H^{p+q}(Z_{•}, F)

with associated low-degree exact sequence

0 → E_{2}^{1,0}(F) → H^{1}(Z_{•}, F) → E_{2}^{0,1}(F) → E_{2}^{2,0}(F).

The “filtration bête” σ of Deligne [12, 1.4.7] is:

(9)  σ_{≥m}H^{∗}(Z_{•}, F) := Im(H^{∗}(Z_{•}, σ_{≥m}F) → H^{∗}(Z_{•}, F)).

4Note that the map d_{i} of [14, 5.2.3.2, 5.3.3.2] is denoted here by δ_{i} and vice-versa.

The methods used to deduce (8) in [14, 5.2.3, 5.2.7] also work for truncated simplicial schemes. Given a complex of sheaves C^{•} on Z_{≥m}, there is a spectral sequence

(10)  E_{1}^{p+m,q}(C^{•}) = H^{q}(Z_{p+m}, C^{•}) ⇒ H^{p+q}(Z_{≥m}, C^{•})

with associated low-degree exact sequence

(11)  0 → E_{2}^{1+m,0}(C^{•}) → ℍ^{1}(Z_{≥m}, C^{•}) → E_{2}^{m,1}(C^{•}) → E_{2}^{2+m,0}(C^{•}).

Let g_{r} : Z_{r} → S and f : Z_{≥m} → S be the structure maps. Analogs of (10) and (11) for the associated S_{fppf}-sheaves hold as well:

(12)  E_{1}^{p+m,q}(C^{•}) = R^{q}g_{p+m ∗}(C^{•}) ⇒ R^{p+q}f_{∗}(C^{•})
(13)  0 → E_{2}^{1+m,0}(C^{•}) → R^{1}f_{∗}(C^{•}) → E_{2}^{m,1}(C^{•}) → E_{2}^{2+m,0}(C^{•}).

2. Construction of L^{n} for a simplicial pair

Throughout this section, k will denote a field of characteristic zero.

For each simplicial pair (X_{•}, Y_{•}) and each non-negative integer n, we shall construct one-motives

L^{n} = L^{n}(X_{•}, Y_{•}) = [B_{n} ⟶^{ϕ_{n}} P̃_{n}]

and J^{n}(X_{•}, Y_{•}); these are contravariant functorial. Their realizations will be analyzed in subsequent sections. Our construction was obtained by a careful analysis of [14, 8.1.19.1].

Fix a simplicial pair (X_{•}, Y_{•}); we have a diagram U_{•} ↪^{j} X_{•} ↩^{i} Y_{•}. Let a : X_{•} → S, a_{m} : X_{m} → S, and f : X_{≥n−1} → S be the structure morphisms.^{5} We often write O_{m}^{∗} for O_{X_{m}}^{∗}, P ic_{m} for P ic_{X_{m}}, etc.

Construction of P_{n}.

We begin with the construction of a semi-abelian variety P_{n} which is isogenous to the required P̃_{n}; our method is similar to [43, §3.1]. Proposition 2.1. (i) The sheaf R^{1}f_{∗}O^{∗} = R^{n}a_{∗}σ_{≥n−1}O_{•}^{∗} is representable by a locally algebraic group scheme G′_{n} with neutral component G_{n}.

(ii) Lie G_{n} ⟶^{∼} H^{1}(X_{≥n−1}, O).

(iii) (k = ℂ) H^{i}(X_{≥n−1}, F) ⟶^{∼} H^{i}(X_{≥n−1}^{an}, F), F = O, O^{∗} and i = 0, 1.

Proof. (i) By (13), the sheaf R^{1}f_{∗}O^{∗} sits in an exact sequence

0 → E_{2}^{n,0} → R^{1}f_{∗}O^{∗} ⟶^{π} E_{2}^{n−1,1} ⟶^{ψ} E_{2}^{n+1,0}.

By (1.2), the sheaf a_{m ∗}O_{m}^{∗} is representable by the torus T_{X_{m}}. The group (of multiplicative type) Hom(H^{m}(D_{•}), 𝔾_{m}) dual to the homology H^{m}(D_{•}) (1.11) of

D_{X_{m+1}} ⟶^{δ_{m}} D_{X_{m}} ⟶^{δ_{m−1}} D_{X_{m−1}}

represents E_{2}^{m,0}. Since P ic_{X_{m}} represents R^{1}a_{m ∗}O_{m}^{∗}, the scheme R′ := Ker(δ_{n−1}^{∗} : P ic_{X_{n−1}} → P ic_{X_{n}}) represents E_{2}^{n−1,1}. Since Ker(ψ) is representable and the scheme representing E_{2}^{n,0} is affine, we can apply (1.1).

5We adopt the notation: X_{≥−1} = X_{•}.

(ii) It follows from (i) by [38, Lem. 2.6.9].

(iii) Use the GAGA isomorphisms H^{i}(X_{m}, F) ⟶^{∼} H^{i}(X_{m}^{an}, F) (i = 0, 1) [45, Prop. 17, 18] for F = O, O^{∗} and (10), (11) for X_{≥n−1} and X_{≥n−1}^{an}. □

We call T′ (resp. Q′) the group scheme representing E_{2}^{n,0} (resp. E_{2}^{n+1,0}); we denote its neutral component by T (resp. Q).

The neutral component R of R′ is an abelian scheme (it is a subscheme of P ic_{X_{n−1}}^{0})

and Q is affine; so ψ(R) = 0. Hence R is the neutral component of Ker(R′ ⟶^{ψ} Q′). Thus we have an exact sequence

(14)  0 → T′ → G_{n} ⟶^{π} R → 0.

Remark 2.2. (i) By definition [23, 1.1], an invertible sheaf L on a simplicial scheme Z_{•} is an invertible sheaf L_{m} on each Z_{m} such that: for each morphism τ : Z_{m} → Z_{r} which is a composition of s_{i}’s and d_{j}’s, the map τ^{∗}L_{r} → L_{m} is an isomorphism. In (loc. cit), it is shown that L is determined entirely by the data of L_{0} and the isomorphism α : d_{0}^{∗}L_{0} ⟶^{∼} d_{1}^{∗}L_{0} satisfying d_{2}^{∗}(α) ∘ d_{0}^{∗}(α) = d_{1}^{∗}(α); (2.1), for n = 1, proves the representability of the Picard functor of Z_{•} which is smooth and proper over S. The scheme G′_{n} classifies isomorphism classes of pairs (L, α) where

(a) L is an invertible sheaf on X_{n−1} such that δ_{n−1}^{∗}L is isomorphic to O_{X_{n}} (by (7), δ_{n}^{∗}δ_{n−1}^{∗}L = O_{X_{n+1}});

(b) α is a trivialization δ_{n−1}^{∗}L ⟶^{∼} O_{X_{n}} on X_{n} satisfying a cocycle condition: δ_{n}^{∗}α = 1; namely, δ_{n}^{∗}α is the identity isomorphism of O_{X_{n+1}}.

(ii) For n = 1, we can think of α as an isomorphism d_{0}^{∗}L ⟶^{∼} d_{1}^{∗}L. The cocycle condition d_{2}^{∗}(α) ∘ d_{0}^{∗}(α) = d_{1}^{∗}(α) (⟺ δ_{1}^{∗}α = 0) becomes the commutativity of

d_{0}^{∗}d_{0}^{∗}L ⟶^{d_{0}^{∗}(α)} d_{0}^{∗}d_{1}^{∗}L  d_{2}^{∗}d_{0}^{∗}L
∥  ^{d_{2}^{∗}(α)}↓
d_{1}^{∗}d_{0}^{∗}L ⟶^{d_{1}^{∗}(α)} d_{1}^{∗}d_{1}^{∗}L  d_{2}^{∗}d_{1}^{∗}L

(using the identities d_{0}d_{1} = d_{0}d_{0}, d_{0}d_{2} = d_{1}d_{0}, d_{1}d_{2} = d_{1}d_{1}). □ Remark 2.3. Suppose given a simplicial scheme Z_{•}, an invertible sheaf F on Z_{m} and a nowhere vanishing section s of F. The identity (7) implies that (i) the sheaf δ_{m+1}^{∗}δ_{m}^{∗}F is naturally isomorphic to O_{Z_{m+2}}; (ii) the section δ_{m+1}^{∗}δ_{m}^{∗}s of δ_{m+1}^{∗}δ_{m}^{∗}F = O_{Z_{m+2}} corresponds to the identity section of O_{Z_{m+2}}. □ Definition of B_{n} and ϕ_{n}.

We now turn to the construction of the map ϕ_{n}. This will take several steps.

Let λ′_{m} : W_{m} → P ic_{X_{m}} be the map E ↦ O(E). In general, there is no lifting

of λ′_{n−1} : W_{Y_{n−1}} → P ic_{X_{n−1}} to a map W_{Y_{n−1}} → G′_{n} ⟶^{π} R′ → P ic_{X_{n−1}}. In other

words, the invertible sheaf O(E) on X_{n−1} corresponding to E ∈ W_{n−1} may not always satisfy conditions (a), (b) of (2.2(i)). But a natural lifting does exist on the subgroup K := Ker(δ_{n−1}^{∗} : W_{Y_{n−1}} → W_{Y_{n}}) of W_{Y_{n−1}}.

Lemma 2.4. There exists a canonical and functorial map ϑ′ : K → G′_{n} which fits into a commutative diagram:

K ⟶^{ϑ′} G′_{n}
↓  ↓
W_{Y_{n−1}} ⟶^{λ′_{n−1}} P ic_{X_{n−1}}.

Proof. The scheme G′_{n} classifies pairs (L, α) where L is an invertible sheaf on X_{n−1} and α is a trivialization δ_{n−1}^{∗}L ⟶^{∼} O_{X_{n}} satisfying the cocycle condition δ_{n}^{∗}α = 1. For any E ∈ K, pick a rational section s_{E} of O(E) such that the divisor div(s_{E}) = E; the set of such sections forms a torsor over H^{0}(X_{n−1}, O^{∗}). Since Y_{•} is a simplicial divisor, the pull-back δ_{n−1}^{∗}s_{E} is a rational section of δ^{∗}O(E) with divisor δ_{n−1}^{∗}E. As E lies in K, we have δ_{n−1}^{∗}E = 0. So δ_{n−1}^{∗}s_{E} provides a trivialization α_{E} : δ_{n−1}^{∗}O(E) ⟶^{∼} O on X_{n}. The trivialization δ_{n}^{∗}α_{E} of δ_{n}^{∗}δ_{n−1}^{∗}O(E) on X_{n+1} is provided by the nowhere vanishing regular section t_{E} := δ_{n}^{∗}δ_{n−1}^{∗}s_{E}. Now the rational section s_{E} is a nowhere vanishing section of O(E) on the open subscheme U_{n−1}. Applying (2.3) to U_{•}. we see that t_{E} is the identity section of the sheaf O_{U_{n+1}}. Since t_{E} is regular on X_{n+1}, we deduce that t_{E} is the identity section of O_{X_{n+1}}. The map ϑ′ is defined by E ↦ (O(E), α_{E}). It is clear that modifying s_{E} by an element of H^{0}(X_{n−1}, O^{∗}) does not affect the isomorphism class of the pair (O(E), α_{E}). □

The maps d_{i} : X_{≥n−1} → X_{n−2} collectively provide the morphism

(15)  δ^{∗} : P ic_{X_{n−2}} → G′_{n}.

More explicitly, given an invertible sheaf L on X_{n−2}, the pull-back L′ := δ_{n−2}^{∗}L is an invertible sheaf on X_{n−1}. As in (2.3), (7) implies that there is a canonical trivialization, call it β_{L′}, of δ_{n−1}^{∗}L′ on X_{n} satisfying δ_{n}^{∗}β_{L′} = 1. The map δ^{∗} sends L to the (isomorphism class of the) pair (L′, β_{L′}). Definition 2.5. P′_{n} is the quotient of G′_{n} by δ^{∗}(P ic_{X_{n−2}}^{0}); its neutral component is P_{n}.

Note π_{0}(G′_{n}) ⟶^{∼} π_{0}(P′_{n}).

Lemma 2.6. (k = ℂ) Lie P_{n} ⟶^{∼} \frac{H^{1}(X_{≥n−1},O)}{δ^{∗}H^{1}(X_{n−2},O)} ⟶^{∼} \frac{H^{1}(X_{≥n−1}^{an},O)}{δ^{∗}H^{1}(X_{n−2}^{an},O)}.

Proof. Combine (2.1) (iii) with (1.5 (ii)). □

The map (15), in turn, induces a map NS_{n−2} ⟶^{μ} P′_{n}. Combining this with ϑ : K ⟶^{ϑ′} G′_{n} → P′_{n}, we get the map

ρ : K ⊕ NS_{X_{n−2}} → P′_{n} ρ(a, b) = ϑ(a) + μ(b).

We now turn to the introduction of several group schemes relevant for the definition of B_{n}. The mapping cone complex of λ : W_{•} → NS_{•} (1.11) is:

W_{Y_{n−2}} ⊕ NS_{X_{n−3}} ⟶^{γ_{n−3}} W_{Y_{n−1}} ⊕ NS_{X_{n−2}} ⟶^{γ_{n−2}} W_{Y_{n}} ⊕ NS_{X_{n−1}} (16)

γ_{m} : (E, β) ↦ (δ_{m+1}^{∗}E, δ_{m}^{∗}β − λ_{m+1}(E)). (17)

Since W_{•} is a complex, the image of γ_{n−3} is contained in K ⊕ NS_{X_{n−2}}. One checks that the composite map ρ ∘ γ_{n−3} is zero.

Definition 2.7. (i) C′_{n} denotes \frac{K⊕NS_{X_{n−2}}}{γ_{n−3}(W_{n−2})}; and C_{n} is the kernel of the composite map

ρ′ : C′_{n} ⟶^{ρ} P′_{n} → π_{0}(P′_{n}).

(ii) A_{n} is the kernel of the composite map C′_{n} ⟶^{ρ′} π_{0}(P′_{n}) ⟶^{∼} π_{0}(G′_{n}) → NS_{X_{n−1}}; this map — the middle isomorphism is the inverse of π_{0}(P′_{n}) ⟵^{∼} π_{0}(G′_{n}) — is the map C′_{n} → NS_{X_{n−1}} induced by γ_{n−2}. Note C_{n} is a subgroup of A_{n}.

(iii) B′_{n} is the kernel of the map

(18)  \frac{K ⊕ NS_{X_{n−2}}}{γ_{n−3}(W_{n−2} ⊕ NS_{X_{n−3}})} ⟶^{ρ} P′_{n} → π_{0}(P′_{n}),

which is analogous to the cycle class map; π_{0}(P′_{n}) plays the role of a relative Néron-Severi group.

We have

(19)  \frac{C_{n}}{NS_{X_{n−3}}} ⟶^{∼} B′_{n}.

Restricting ρ to B′_{n}, we obtain a morphism of group schemes B′_{n} ⟶^{ρ} P_{n}. Let τ_{n} be the torsion subgroup of B′_{n}. Replacing B′_{n} by B_{n} := B′_{n}/τ_{n} and P_{n} by P̃_{n} := P_{n}/ρ(τ_{n}), the map ρ induces a map ϕ_{n} : B_{n} → P̃_{n}. Definition 2.8. L^{n}(X_{•}, Y_{•}) is the one-motive

(20)  L^{n} = L^{n}(X_{•}, Y_{•}) := [B_{n} ⟶^{ϕ_{n}} P̃_{n}].

Remark 2.9. (i) The one-motive L^{n} depends only on the schemes X_{m}, Y_{m} for m = n − 3, ⋯, n + 1 (and the maps between them).

(ii) Let k′ be an extension of k and write S′ = Spec k′. It is clear from the definitions that L^{n}(X_{•} ×_{S} S′, Y′_{•} ×_{S} S′) = L^{n} ×_{S} S′.

(iii) By (i) and (ii), there is no loss of generality in assuming that k is finitely generated over ℚ. Such a field always admits an embedding into ℂ.

(iv) L^{n} is a contravariant functor for morphisms of simplicial pairs. □ The one-motive J^{n}.

Definition 2.10. V_{m} (resp. N_{m}) is the kernel (resp. cokernel) of W_{m} ⟶^{λ_{m}} NS_{m}; and K^{0} (resp. M) is the kernel (resp. cokernel) of the composite map K ⟶^{ϑ} P′_{n} → π_{0}(P′_{n}).

The one-motive J^{n} is

(21)  J^{n} = J^{n}(X_{•}, Y_{•}) := [(\frac{K^{0}/V_{n−2}}{torsion} ⟶^{ϕ_{n}} P̃_{n}].

The natural morphism J^{n} → L^{n} is an isomorphism for n ≤ 1; J^{n} is contravariant functorial for morphisms of simplicial pairs.

The remainder of this section is devoted to results which will be used in §3. Lemma 2.11. One has an exact sequence

0 → \frac{V_{n−1} ∩ K}{V_{n−2}} → A_{n} → Ker(δ_{n−2}^{∗} : N_{n−2} → N_{n−1}) → 0.

Proof. Taking the quotient of 0 → K → K ⊕ NS_{n−2} → NS_{n−2} → 0 by the exact sequence 0 → V_{n−2} → W_{Y_{n−2}} → Im(λ_{n−2}) → 0, we get the exact sequence in the first row of the diagram

0 ⟶ \frac{K}{V_{n−2}} ⟶ \frac{K⊕NS_{n−2}}{γ_{n−3}(W_{n−2})} ⟶ N_{n−2} ⟶ 0
↓  ↓  ↓
0 ⟶ Im(λ_{n−1}) ⟶ NS_{n−1} ⟶ N_{n−1} ⟶ 0;

the required sequence is an easy consequence. □

The de Rham cohomology sheaves.

We shall use the following complexes on a (simplicial) smooth projective scheme; the scheme could be either X_{•}, X_{m} or X_{≥m}. We often omit the subscript from the complex if it is clear from the context which scheme it is on.

Ω := [O ⟶^{d} Ω^{1} ⟶^{d} Ω^{2} …]
Ω(log Y) := [O ⟶^{d} Ω^{1}(log Y) ⟶^{d} ...]
Γ_{0} := [O ⟶^{d} Ω^{1}],  Γ := [O ⟶^{d} Ω^{1}(log Y)]
Γ_{0}^{∗} := [O^{∗} ⟶^{d log} Ω^{1}],  Γ^{∗} := [O^{∗} ⟶^{d log} Ω^{1}(log Y)]

Let Ω(log Y) be the logarithmic de Rham complex [14, 3.1.3, 6.2.7] of U_{•} on X_{•}. Lemma 2.12. The map dl : G′_{n} = R^{1}f_{∗}O^{∗} → R^{1}f_{∗}Ω^{1} induced by d log : O^{∗} → Ω^{1} [12, 2.2.4] yields a map dlog : π_{0}(G′_{n}) → H^{1}(X_{≥n−1}, Ω^{1}).

Proof. Since G_{n} is a semi-abelian variety and R^{1}f_{∗}Ω^{1} is a vector scheme, the map dl restricted to G_{n} is zero. □

Proposition 2.13. (i) The sheaf R^{1}f_{∗}Γ^{∗} is representable by a group scheme G_{n}^{♢} with ℍ^{1}(X_{≥n−1}, Γ^{∗}) as its group of k-rational points.

(ii) Lie G_{n}^{♢} ⟶^{∼} ℍ^{1}(X_{≥n−1}, Γ) ⟶^{∼} H_{DR}^{1}(U_{≥n−1}).

(iii) The group G_{n}^{♢} is an extension of a subgroup (containing G_{n}) of G′_{n} by f_{∗}Ω^{1}(log Y).

Proof. (i) In the exact sequence, given by the map Γ^{∗} → O^{∗},

f_{∗}Γ^{∗} ⟶^{β} f_{∗}O^{∗} → f_{∗}Ω^{1}(log Y) ⟶^{h} R^{1}f_{∗}Γ^{∗} → R^{1}f_{∗}O^{∗} → R^{1}f_{∗}Ω^{1}(log Y),

β is clearly an isomorphism: ℍ^{0}(X_{≥n−1}, Γ^{∗}) = H^{0}(X_{≥n−1}, O^{∗}). So h is injective. The sheaves R^{i}f_{∗}Ω^{j}(log Y) are locally free and commute with base change [34, 1.4.1.8]; so they are representable by vector group schemes (cf. the proof of (1.3)). Lemma 1.1 now provides the representability of R^{1}f_{∗}Γ^{∗}.

(ii) the first isomorphism follows from (i) [38, Lemma 2.6.9] and the second isomorphism from [34, 1.0.3.7], [12, 3.1.8]; cf. also [31, 3.4].

(iii) It suffices to show that the map Lie G_{n}^{♢} → Lie G_{n} is onto. Under the identifications, Lie G_{n}^{♢} ⟶^{∼} ℍ^{1}(X_{≥n−1}, Γ) and Lie G_{n} ⟶^{∼} H^{1}(X_{≥n−1}, O), we need the surjectivity of the forgetful map ℍ^{1}(X_{≥n−1}, Γ) → H^{1}(X_{≥n−1}, O). We can use (11) for Γ and O on X_{≥n−1}; since ℍ^{0}(X_{m}, Γ) = H^{0}(X_{m}, O_{X_{m}}), we have E_{2}^{n,0}(Γ) ⟶^{∼} E_{2}^{n,0}(O). Thus we need the surjectivity of E_{2}^{n−1,1}(Γ) → E_{2}^{n−1,1}(O). Let us show that even the composite map π_{0} : E_{2}^{n−1,1}(Γ_{0}) → E_{2}^{n−1,1}(Γ) → E_{2}^{n−1,1}(O) is onto.

Write E_{m}^{♮} for the universal additive extension of P ic_{X_{m}}^{0}. We have

E_{2}^{n−1,1}(Γ_{0}) ⟵^{∼} Ker(δ_{n−1}^{∗} : Lie E_{n−1}^{♮} → Lie E_{n}^{♮})

E_{2}^{n−1,1}(O) ⟵^{∼} Ker(δ_{n−1}^{∗} : Lie P ic_{X_{n−1}}^{0} → Lie P ic_{X_{n}}^{0}) = Lie R.

For any abelian scheme A, the Lie algebra Lie A^{♮} of the universal additive (= vectorial) extension is canonically isomorphic to H_{DR}^{1}(A^{∗}) of the dual abelian scheme A^{∗} [37, 4.1.7, p. 48] (1.4). So the functor ♮ : A ↦ Lie A^{♮} is exact. Thus E_{2}^{n−1,1}(Γ_{0}) ⟵^{∼} Lie R^{♮}. Hence π_{0} is the natural (surjective) projection Lie R^{♮} → Lie R. □

Remark 2.14. (i) For any semi-abelian variety G (with associated abelian variety A and torus T), the universal additive (= vectorial) extension G^{♮} of G is the pullback A^{♮} ×_{A} G [14, 10.1.7c] of the universal additive extension A^{♮} of A. The proof of (2.13) shows that (a) R^{1}f_{∗}Γ_{0}^{∗} is a vectorial extension of a subgroup (containing G_{n}) of G′_{n}. (b) this vectorial extension restricted to G_{n} is the universal extension G_{n}^{♮} = R^{♮} ×_{R} G_{n} of G_{n}. The universal additive extension P_{n}^{♮} of P_{n} is the quotient of G_{n}^{♮} by δ^{∗}E_{n−2}^{♮}, defined as in the proof of (2.13 (iii)).

(ii) The analog of (2.13) is also true for a smooth proper variety Z and a strict divisor V of Z. For the choice of the strict divisor Y_{m} of X_{m}, we obtain the existence of a group scheme P ic_{X_{m}}^{♢} with ℍ^{1}(X_{m}, Γ^{∗}) as its group of k-rational points; one has Lie P ic_{X_{m}}^{♢} ⟶^{∼} ℍ^{1}(X_{m}, Γ) ⟶^{∼} H_{DR}^{1}(U_{m}), the last isomorphism due to [12, 3.1.8], [34, 1.0.3.7]. The cokernel P_{n}^{♢} of δ^{∗} : P ic_{X_{n−2}}^{♢} → G_{n}^{♢} is a vectorial extension of a subgroup (containing P_{n}) of P′_{n}. □ Lemma 2.15. (a) (k = ℂ) Lie P_{n}^{♢} ⟶^{∼} σ_{≥n−1}H^{n}(U_{•}, ℂ).

(b) (k = ℂ) Lie P_{n}^{♮} ⟶^{∼} σ_{≥n−1}H^{n}(X_{•}, ℂ) ⟶^{∼} W_{−1}H^{n}(U_{•}, ℂ).

(c) Lie P_{n}^{♢} ⟶^{∼} σ_{≥n−1}H_{DR}^{n}(U_{•}).

Proof. Combine H^{1}(U_{∗}, ℂ) ⟵^{∼} H^{1}(X_{∗}^{an}, j_{∗}ℂ_{U}) ⟶^{∼} ℍ^{1}(X_{∗}^{an}, Ω(log Y)) (logarithmic Poincaré lemma [12, 3.2.2]), ℍ^{1}(X_{∗}^{an}, Γ) ⟶^{∼} ℍ^{1}(X_{∗}^{an}, Ω(log Y)) (X_{∗} = X_{m} or X_{≥n−1}) with (2.13), (2.14). Part (a) follows from the next isomorphism (the injectivity is a consequence of the degeneration of [14, 8.1.19.1] at E_{2})

(22)  \frac{H^{1}(U_{≥n−1}, ℂ)}{δ_{n−2}^{∗}H^{1}(U_{n−2}, ℂ)} ⟶^{∼} σ_{≥n−1}H^{n}(U_{•}, ℂ);

the surjectivity is by definition of σ. The first (resp. second) isomorphism in (b) is a special case of (a), i.e., for Y = ∅ (resp. from (3.5)). Part (c) is proved similarly using (2.13), (2.14). □

3. Hodge and de Rham realizations of L^{n}

We retain the notations of the previous section but we now take k = ℂ. We have a diagram U_{•} ⟶^{j} X_{•} ⟵^{i} Y_{•} corresponding to our simplicial pair (X_{•}, Y_{•}). As before, f : X_{≥n−1} → S is the structure map.

The main results (3.2) (3.18) of this section prove the Hodge and de Rham part of the conjecture (0.1) for U_{•}; the étale realization will be treated in §4. Statement of the theorem. Lemma 3.1. The mixed Hodge structures H^{∗}(U_{•}, ℤ) are polarizable.

Proof. Since Gr_{r}^{W} H^{m}(U_{•}, ℚ) is a direct sum of subquotients of the cohomology of smooth projective varieties [14, 8.1.19.1] and the cohomology H^{∗}(V, ℚ) of a smooth projective complex variety V is polarizable [12, 2.2.6], this is clear. □

Denote the largest sub-mixed Hodge structure of type (∗) of H^{n}(U_{•}, ℤ(1))/torsion by t^{n}(U_{•}); by the previous lemma, Gr_{−1}^{W}t^{n}(U_{•})⊗ℚ = ℚ(1)⊗Gr_{1}^{W} H^{n}(U_{•}, ℚ) is polarizable. By [14, 10.1.3], the mixed Hodge structure t^{n}(U_{•}) corresponds to a one-motive I^{n}(U_{•}) over ℂ. Theorem 3.2. There is a canonical and functorial isogeny of one-motives

I^{n}(U_{•}) → L^{n}(X_{•}, Y_{•})

over ℂ, i.e., there is a canonical functorial isomorphism of ℚ-mixed Hodge structures:

t^{n}(U_{•}) ⊗ ℚ ⟶^{∼} T_{ℤ}(L^{n}) ⊗ ℚ.

Corollary 3.3. L^{n}(X_{•}, Y_{•}) ⊗ ℚ depends only upon U_{•}.

Proof. Clear. □

We follow, for the most part, Deligne’s arguments in [14, 10.3]. Proof of the W_{−1}-part of Theorem 3.2.

We begin with the isogeny one-motive W_{−1}L^{n} ⊗ ℚ = [0 → P̃_{n}] ⊗ ℚ = [0 → P_{n}] ⊗ ℚ.

Proposition 3.4. W_{−1}t^{n}(U_{•}) ⊗ ℚ ⟶^{∼} T_{ℤ}(W_{−1}L^{n}) ⊗ ℚ.

Since

P_{n} = \frac{G_{n}}{P ic(X_{n−2})}, H^{1}(X_{n−2}, ℤ(1)) ⟶^{∼}_{(1)} H_{1}(P ic(X_{n−2}), ℤ)

and W_{−1}t^{n}(U_{•}) ⊗ ℚ = W_{−1}H^{n}(U_{•}, ℚ(1)), this follows from the next Proposition 3.5. (i) H′ := H^{1}(X_{≥n−1}, ℤ(1)) ⟶^{∼} T_{ℤ}([0 → G_{n}]) = H_{1}(G_{n}, ℤ).

Note t^{1}(X_{≥n−1}) = H^{1}(X_{≥n−1}, ℤ(1)).

(ii) \frac{H^{1}(X_{≥n−1},ℚ(1))}{H^{1}(X_{n−2},ℚ(1))} ⟶^{∼} σ_{≥n−1}H^{n}(X_{•}, ℚ(1)) = W_{−1}H^{n}(X_{•}, ℚ(1)) ⟶^{∼} W_{−1}H^{n}(U_{•}, ℚ(1)).

(The last isomorphism is an analog of a theorem of Grothendieck-Deligne [12, 3.2.16-17], [26, 9.1-9.4].)

Proof. (ii) It follows from the definition of the weight filtration [21, p.55] that the image of H^{n}(X_{•}, σ_{≥n−1}, ℚ(1)) in H^{n}(X_{•}, ℚ(1)) is W_{−1}H^{n}(X_{•}, ℚ(1)). This proves the equality. The rest of (ii) follows from an inspection of the spectral sequence [14, 8.1.19.1] and the fact (ibid. 8.1.20 (ii)) that it degenerates at E_{2}. The relevant E_{1}-terms of [14, 8.1.19.1] are those with b = 1 and −a = n − 1 (and r = 0, p = 1, q = n − 1) since W_{−1}H^{n}(U_{•}, ℚ(1)) is the Tate twist of W_{1}H^{n}(U_{•}, ℚ).

(i) From (2), we get the exact sequence

H^{0}(X_{≥n−1}^{an}, O) ⟶^{exp} H^{0}(X_{≥n−1}^{an}, O^{∗}) → H′ ⟶^{ξ} H^{1}(X_{≥n−1}^{an}, O) ⟶^{exp} H^{1}(X_{≥n−1}^{an}, O^{∗}).

Since the first map is surjective (exp : ℂ → ℂ^{∗} is surjective), ξ is injective. Proposition 2.1 (iii) now gives the required isomorphism

⊞ : H′ = H^{1}(X_{≥n−1}, ℤ(1)) ⟶^{∼} T_{ℤ}([0 → G_{n}]) := H_{1}(G_{n}, ℤ).

Let us show that ⊞ is compatible with the weight and Hodge filtrations. In the sequences (10), (11) on X_{≥n−1}, we have

• isomorphisms of Hodge structures

η_{1} : E_{2}^{n,0}(ℚ(1)) ⟶^{∼} H_{1}(T′, ℚ), η_{2} : E_{2}^{n−1,1}(ℚ(1)) ⟶^{∼} H_{1}(R, ℚ).

For η_{1}, use H^{0}(X_{m}^{an}, ℤ(1)) ⟶^{∼} H_{1}(T_{X_{m}}, ℤ), a consequence of (2). And η_{2} follows from (1.5 (ii),(iv)).

• surjectivity of the map π in the commutative diagram

0 ⟶ E_{2}^{n,0}(ℚ(1)) ⟶_{τ} H′ ⊗ ℚ ⟶_{π} E_{2}^{n−1,1}(ℚ(1)) ⟶ 0
η_{1}↓  ↓_{⊞}  ^{η_{2}}↓
0 ⟶ H_{1}(T , ℚ) ⟶ H_{1}(G_{n}, ℚ) ⟶ H_{1}(R, ℚ) ⟶ 0.

This follows from the degeneration [14, 8.1.20] at E_{2} of (10) for ℚ(1).

The bottom row of the diagram is the the Hodge realization of the exact sequence (14) of isogeny one-motives. The image of τ is W_{−2}H′ ⊗ ℚ [21, p.55]. Since W_{−2}[0 → G_{n}] ⊗ ℚ = [0 → T ] ⊗ ℚ [14, 10.1.4], we find that ⊞ is compatible with the weight filtration.

Since H′ is of type (∗), there is only one nontrivial step in the Hodge filtration, viz.,

F^{0}; and, F^{0}(H′ ⊗ ℂ) ⟶^{∼} F^{0}((H′/W_{−2}H′) ⊗ ℂ) since F^{0} ∩ W_{−2}H′ ⊗ ℂ = 0. Thus,

to show that ⊞ is compatible with F, it suffices to show that η_{2} is a map of Hodge structures. This we have done. □

This finishes the proof of the W_{−1}-part of Theorem 3.2. Interpretation of H^{1} and its applications. Observation 3.6. (Interpretation of H^{1}) Let d : F → G be a morphism of abelian sheaves on a space Z. In [14, 10.3.10], Deligne notes that ℍ^{1}(Z, [F ⟶^{d} G]) can be identified with the set of isomorphism classes of pairs (L, α) where L is a F -torsor and α is a trivialization of the G-torsor dL. This identification is based upon the sequence

H^{0}(Z, F) → H^{0}(Z, G) → ℍ^{1}(Z, [F ⟶^{d} G]) → H^{1}(Z, F) ⟶^{d} H^{1}(Z, G) □

Let j_{∗}^{m}O_{U}^{∗} [14, 10.3.9(i)] be the subsheaf of meromorphic sections of j_{∗}O_{U}^{∗} on X_{•}^{an}.

Remark 3.7. One can prove (3.5(i)) using (3.6); namely, the group H^{1}(X_{≥n−1}, ℤ(1)) ⟶^{∼} ℍ^{1}(X_{≥n−1}, [O ⟶^{exp} O^{∗}]) is the set of pairs (L, α) where L is an O_{X_{≥n−1}}-torsor and α a trivialization of exp(L). Since Aut(L) → Aut(exp(L)) is onto, H^{1}(X_{≥n−1}, ℤ(1)) is the set of isomorphism classes L of O_{X_{≥n−1}}-torsors (= elements of Lie G_{n}) with exp(L) = 0

in G_{n}; thus, H^{1}(X_{≥n−1}, ℤ(1)) ⟶^{∼} H_{1}(G_{n}, ℤ).

Similar results, based on (3.6), are as follows.

(a) the mixed Hodge structure H^{1}(U_{m}, ℤ(1)) is isomorphic to T_{ℤ}[V_{m} → P ic(X_{m})] (1.9).

(b) the mixed Hodge structure H^{1}(U_{≥n−1}, ℤ(1)) is isomorphic to T_{ℤ}[K^{0} → G_{n}]; note that [K^{0} → G_{1}] (n = 1) is the Picard one-motive [43] of U_{•}.

We refer to [43] for the details; a sketch of the proof of (a) is as follows: H^{1}(U_{m}, ℤ(1)) = ℍ^{1}(X_{m}, [O_{X} ⟶^{exp} j_{∗}^{m}O_{U}^{∗}]) can be identified — as in [14, 10.3.10c] — with the set of isomorphism classes of pairs (L, α) where L is an O_{X_{m}}-torsor and α is an isomorphism of the invertible sheaf exp(L) with O_{X_{m}}(E) (E is a divisor supported on Y_{m}). Since Aut(L) → Aut(exp(L)) is surjective, H^{1}(U_{m}, ℤ(1)) is the set of pairs (p, d) where p is an isomorphism class of an O_{X_{m}}-torsor, i.e., an element of Lie P ic(X_{m}), and d ∈ V_{m} with exp(p) as image in P ic(X_{m}). This gives an isomorphism (as abelian groups)

H^{1}(U_{m}, ℤ(1)) ⟶^{∼} T_{ℤ}[V_{m} → P ic(X_{m})].

(Combining (a) and (b) yields — see (21) for the definition of J^{n} —

(23)  T_{ℤ}J^{n} ⊗ ℂ ⟶^{∼} \frac{H^{1}(U_{≥n−1}, ℂ)}{H^{1}(U_{n−2}, ℂ)} ⟶^{∼}_{(22)} σ_{≥n−1}H^{n}(U_{•}, ℂ) ⟵^{∼}_{(2.15)} LieP_{n}^{♢}.)

(c) G_{n}^{♢}(ℂ) ⟶^{∼} ℍ^{1}(X_{≥n−1}, Γ^{∗}) (2.13) is the group of isomorphism classes of pairs (L, ω) with L an invertible sheaf on X_{≥n−1} and ω ∈ H^{0}(X_{≥n−1}, Ω^{1}(log Y)). To relate to [14, 10.3.10a], one uses the “connections θ on invertible sheaves = one-forms ω” dictionary [38, 2.5] [22, 1.5, p.47]. Thus, ℍ^{1}(X_{≥n−1}, Γ^{∗}) (cf. [34, 7.2.1]) is the set of isomorphism classes of pairs (L, θ) where L is as before and θ a connection on L, holomorphic on U_{≥n−1} and allowed to have simple poles along Y_{≥n−1}.

(d) Lie G_{n}^{♢}(ℂ) ⟶^{∼} ℍ^{1}(X_{≥n−1}, Γ) ⟶^{∼} H_{DR}^{1}(U_{≥n−1}) (2.13) is the set of isomorphism classes of pairs (L, θ) where L is an O_{X_{≥n−1}}-torsor and θ a connection on L as in (c).

(e) Similar results hold for Lie P ic_{X_{m}}^{♢}(ℂ) ⟶^{∼} ℍ^{1}(X_{m}, Γ) ⟶^{∼} H_{DR}^{1}(U_{m}) (2.14) and

P ic_{X_{m}}^{♢}(ℂ) ⟶^{∼} ℍ^{1}(X_{m}, Γ^{∗}) (2.14).

(f) Parts (d) and (e) yield an interpretation of Lie P_{n}^{♢}⟶^{∼}σ_{≥n−1}ℍ^{n}(U_{•}, Γ) (2.15). □

While W_{−1}t^{n}(U_{•})⊗ℚ is a quotient of H^{1}(X_{≥n−1}, ℚ(1))), t^{n}(U_{•})⊗ℚ is a subquotient of H^{2}(U_{≥n−2}, ℚ(1)). Thus, more work is necessary to complete the proof of (3.2).

Proof of Theorem 3.2.

This will be accomplished in the following three steps.

• Step 1. Construction of a certain mixed Hodge structure h_{X}^{2}.

• Step 2. Relating h_{X}^{2} and t^{n}(U_{•}).

• Step 3. Interpretation of h_{X}^{2} using (3.6).

Step 1. Construction of h_{X}^{2}.

This uses the truncated complex τ_{≤1}Rj_{∗}ℤ(1)_{U} [12, 1.4.6]. As before, let q : Ỹ_{m} → Y_{m} → X_{m} denote the natural map from the normalization Ỹ_{m} of Y_{m}. Since, on each X_{m}, R^{1}j_{m,∗}ℤ(1) = q_{∗}ℤỸ_{m} [12, 3.1.9], we get the triangle in the derived category of sheaves on X_{m}^{an}

ℤ(1)_{X} → τ_{≤1}Rj_{m,∗}ℤ(1) → q_{∗}ℤ[−1] → ℤ_{X}(1)[1] → ⋯;

the truncated complexes τ_{≤1}Rj_{m,∗}ℤ(1) for each m combine to give a complex on X_{≥n−2} which we denote by τ_{≤1}Rj_{∗}ℤ(1). Thus, in the exact sequence on X_{≥n−2}

0 → σ_{≥n−1}ℤ(1)_{X} → τ_{≤1}Rj_{∗}ℤ(1) → F → 0,

F on X_{m} is (quasi-isomorphic to) the complex q_{∗}ℤ[−1] for m > n − 2 and q_{∗}ℤ[−1] → ℤ_{X_{n−2}}(1)[1] for m = n − 2. The associated cohomology sequence (here we use that H^{i}(X_{≥n−2}, σ_{≥n−1}ℤ(1)) = H^{i−1}(X_{≥n−1}, ℤ(1))) (24)

\frac{H^{1}(X_{≥n−1}, ℤ(1))}{H^{1}(X_{n−2}, ℤ(1))} ↪ ℍ^{2}(X_{≥n−2}, τ_{≤1}Rj_{∗}ℤ(1)) → ℍ^{2}(X_{≥n−2}, F) ⟶^{δ^{∗}} H^{2}(X_{≥n−1}, ℤ(1))

fits into the commutative diagram (defining h_{X}^{2} by pullback via ν)

\frac{t^{1}(X_{≥n−1})}{t^{1}(X_{n−2})}  ⟶  h_{X}^{2}  ⟶  \frac{K⊕NS_{n−2}}{W_{n−2}}
≀↓^{(3.5)}  ↓  ^{ν}↓
\frac{H^{1}(X_{≥n−1},ℤ(1))}{H^{1}(X_{n−2},ℤ(1))}  ⟶ ℍ^{2}(X_{≥n−2}, τ_{≤1}Rj_{∗}ℤ(1)) ⟶ \frac{H^{0}(Ỹ_{≥n−1},ℤ)⊕H^{2}(X_{n−2},ℤ(1))}{H^{0}(Ỹ_{n−2},ℤ)}
↓  ↓  ↓
\frac{ℍ^{2}(X_{≥n−2},[σ_{≥n−1}O⟶^{exp}j_{∗}^{m}O_{U}^{∗}])}{H^{1}(X_{n−2},O)} ⟶ ℍ^{2}(X_{≥n−2}, [O ⟶^{exp} j_{∗}^{m}O_{U}^{∗}]) ⟶  H^{2}(X_{n−2}, O)
↓  ^{a}↓  ∥
\frac{ℍ^{2}(X_{≥n−2},K)}{H^{1}(X_{n−2},O)}  ⟶  ℍ^{2}(X_{≥n−2}, Γ)  ⟶  H^{2}(X_{n−2}, O)
b↓
\frac{H^{1}(X_{≥n−1},O)}{H^{1}(X_{n−2},O)}  ⟵_{(2.6)}^{∼}  Lie P_{n}  (∗∗)

Remark 3.8. (i) The map ν, induced by (4) and the isomorphism H^{0}(Ỹ_{i}, ℤ) ≅ W_{Y_{i}}, is injective. (ii) h_{X}^{2} is a mixed Hodge structure of type (∗); note W_{−1}h_{X}^{2} = \frac{H^{1}(X_{≥n−1},ℤ(1))}{H^{1}(X_{n−2},ℤ(1))}.

(An argument similar to that in the proof of (3.10(i)) shows that the natural map h_{X}^{2} ⊗ ℚ → H^{2}(U_{≥n−2}, ℚ(1)) is injective with image t^{2}(U_{≥n−2}) ⊗ ℚ.) (iii) The map a is induced by the (first) morphism of complexes

O_{X} ⟶^{exp} j_{∗}^{m}O_{U}^{∗} ⟶  0
∥  ↓^{d log}
O_{X} ⟶^{d} Ω^{1}(log Y) ⟶  0
∥  ∥
O_{X} ⟶^{d} Ω^{1}(log Y) ⟶^{d} Ω^{2}(log Y) → ⋯;

it is compatible with the natural map

ℍ^{2}(X_{≥n−2}, Rj_{∗}ℤ(1)) ⟶^{∼} H^{2}(U_{≥n−2}, ℤ(1)) ↪ ℍ^{2}(X_{≥n−2}, Ω(log Y)) ⟶^{∼} H^{2}(U_{≥n−2}, ℂ).

The last isomorphism is the logarithmic Poincaré lemma [12, 3.2.2]. (iv) On X_{≥n−1}, K = Γ and, on X_{n−2}, K = [0 → Ω^{1}(log Y)].

(v) Since C′_{n} = \frac{K⊕NS_{n−2}}{W_{n−2}} is of type (0, 0), the map h_{X}^{2} → H^{2}(X_{n−2}, O) in (∗∗) is zero.

We deduce an injection h_{X}^{2} ↪ \frac{ℍ^{2}(X_{≥n−2},[σ_{≥n−1}O⟶^{exp}j_{∗}^{m}O_{U}^{∗}])}{H^{1}(X_{n−2},O)} (with finite cokernel — see

(30) below) and a commutative diagram

h_{X}^{2}  ⟶ h_{X}^{2} ⊗ ℂ ⟵ W_{−1}h_{X}^{2} ⊗ ℂ
↓  ↓  ^{≀}↓^{(ii)}
\frac{ℍ^{2}(X_{≥n−2},[σ_{≥n−1}O⟶^{exp}j_{∗}^{m}O_{U}^{∗}])}{H^{1}(X_{n−2},O)} ⟶^{a} \frac{ℍ^{2}(X_{≥n−2},K)}{H^{1}(X_{n−2},O)} ⟵ \frac{ℍ^{1}(X_{≥n−1},Γ_{0})}{ℍ^{1}(X_{n−2},Γ_{0})}
↓  ↓_{b}  ^{(2.15)}↓^{≀}
\frac{H^{1}(X_{≥n−1},O)}{H^{1}(X_{n−2},O)}  ⟵_{(2.6)}^{∼} Lie P_{n} ⟵ Lie P_{n}^{♮}.

(vi) The Hodge filtration [12, 3.2.2], restricted to K, is F^{0}K = K and F^{1}K = [0 → Ω^{1}(log Y)] on X_{≥n−2}. It induces the map b — see (∗∗) — thereby defining the Hodge filtration on h_{X}^{2} ⊗ ℂ. Because h_{X}^{2} is defined using cohomology with ℤ(1)-coefficients, F^{1} on K corresponds to F^{0} on h_{X}^{2} ⊗ ℂ. □ Lemma 3.9. One has a commutative diagram

C′_{n}  ⟶^{ρ}  P′_{n}  ⟶ π_{0}(P′_{n})
ν↓  ^{≀}↑
\frac{H^{0}(Ỹ_{≥n−1},ℤ)⊕H^{2}(X_{n−2},ℤ(1))}{H^{0}(Ỹ_{n−2},ℤ)} ⟶_{(24)}^{δ^{∗}} H^{2}(X_{≥n−1}, ℤ(1)) ⟵ π_{0}(G′_{n}).

Proof. Straightforward. □

Combining (3.9), (3.4) and (3.5), we obtain the exact sequence

(25)  0 → H_{1}(P_{n}, ℚ) → h_{X}^{2} ⊗ ℚ → C_{n} ⊗ ℚ → 0.

Step 2. Relating h_{X}^{2} and t^{n}(U_{•}).

Using the next lemma, we shall show that t^{n}(U_{•}) is a quotient of h_{X}^{2} by NS_{n−3}.

Lemma 3.10. (i) One has an isomorphism ν : B_{n} ⊗ ℚ ⟶^{∼} Gr_{0}^{W} t^{n}(U_{•}) ⊗ ℚ. (ii) C_{n} has finite index in A_{n}; cf. (2.7).

(iii) K^{0} := Ker(K ⟶^{ϑ} π_{0}(G′_{n})) has finite index in K ∩ V_{n−1} := Ker(K ⟶^{λ_{n−1}} NS_{n−1}).

Proof. (i) By [14, 8.1.19.1], Gr_{b}^{W} H^{n}(U_{•}, ℚ) is a subquotient of E_{1}^{n−b,b}. Since Gr_{0}^{W} t^{n}(U_{•})⊗ ℚ is a Tate twist of the (1, 1)-part of Gr_{2}^{W} H^{n}(U_{•}, ℚ), the relevant E_{1}-terms correspond to b = 2 and −a = n − 2 (with p, q, r satisfying p + 2r = 2 and q − r = n − 2). If written explicitly (as is done immediately after 8.1.19.1 in [14])^{6}, we obtain — using the degeneration [14, 8.1.20 (ii)] at E_{2} of [14, 8.1.19.1] — that Gr_{2}^{W} H^{n}(U_{•}, ℚ) is (the coefficients in (26) are ℚ)

(26)  \frac{Ker(H^{0}(Ỹ_{n−1})(−1) ⊕ H^{2}(X_{n−2}) ⟶^{t_{n−1}} H^{0}(Ỹ_{n})(−1) ⊕ H^{2}(X_{n−1}))}{Im(H^{0}(Ỹ_{n−2})(−1) ⊕ H^{2}(X_{n−3}) ⟶^{t_{n−2}} H^{0}(Ỹ_{n−1})(−1) ⊕ H^{2}(X_{n−2}))};

here t_{m}(a, b) = (δ_{m}^{∗}a, δ_{m−1}^{∗}b − λ_{m}(a)). This is easily compared with (16).

By [14, 8.1.20 (ii)], the group in (26) is unchanged if one replaces H^{2}(X_{n−1}) by H^{2}(X_{≥n−1}) and uses the map δ^{∗} of (24). The (1, 1)-part of Gr_{2}^{W} H^{n}(U_{•}, ℚ) can be identified using (4). The lemma now follows from π_{0}(P′_{n}) ≅ π_{0}(G′_{n}), π_{0}(G′_{n}) ↪ H^{2}(X_{≥n−1}, ℤ(1)) — a consequence of (2), and (3.9).

(ii) and (iii) also follow from the degeneration [14, 8.1.20 (ii)] at E_{2} of [14, 8.1.19.1].

□

Using (3.10(i)) and (3.4), we can rewrite

(27)  0 → W_{−1}t^{n}(U_{•}) → t^{n}(U_{•}) → Gr_{0}^{W} t^{n}(U_{•}) → 0

as the sequence (exact modulo finite groups) of mixed Hodge structures

(28)  0 → H_{1}(P_{n}, ℤ) → t^{n}(U_{•}) → B_{n} → 0.

Proposition 3.11. We have an isomorphism of mixed Hodge structures

(29)  \frac{h_{X}^{2} ⊗ ℚ}{δ^{∗}NS(X_{n−3}) ⊗ ℚ} ⟶^{∼} t^{n}(U_{•}) ⊗ ℚ.

Proof. Since h_{X}^{2} is of type (∗), the natural map

H^{2}(X_{≥n−2}, τ_{≤1}Rj_{∗}ℤ(1)) → H^{2}(U_{≥n−2}, ℤ(1)) → H^{n}(U_{•}, ℤ(1))

gives a map h_{X}^{2} → t^{n}(U_{•}). The proposition follows from (28), (19), (3.10), and (25). □

Step 3. Interpretation of h_{X}^{2} using (3.6).

Theorem 3.12. One has a natural isomorphism

\frac{ℍ^{2}(X_{≥n−2}, [σ_{≥n−1}O ⟶^{exp} j_{∗}^{m}O_{U}^{∗}])}{H^{1}(X_{n−2}, O)} ⟶^{∼} T_{ℤ}[C_{n} ⟶^{ρ} P_{n}].

Remark 3.13. Every j_{∗}^{m}O_{U_{r}}^{∗}-torsor P gives an invertible sheaf on U_{r} and this extends to an invertible sheaf P′ on X_{r}; given any two such extensions P′ and P′′, one has an isomorphism of invertible sheaves P′′ ⟶^{∼} P′ ⊗ O(v) (v ∈ W_{Y_{r}}) on X_{r}. □

Proof. The group ℍ^{2}(X_{≥n−2}, [σ_{≥n−1}O ⟶^{exp} j_{∗}^{m}O_{U}^{∗}]) is actually a ℍ^{1} in disguise (this shift occurs for reasons of degree for j_{∗}^{m}O_{U}^{∗} — this is in degree one — and of truncation for O): it sits in an exact sequence

→ ℍ^{2}(X_{≥n−2}, [σ_{≥n−1}O ⟶^{exp} j_{∗}^{m}O_{U}^{∗}]) → H^{1}(X_{≥n−1}, O_{X}) ⊕ H^{1}(X_{n−2}, j_{∗}^{m}O_{U}^{∗}) →

→ H^{1}(X_{≥n−1}, j_{∗}^{m}O_{U}^{∗}) → Via (3.6), we can interpret ℍ^{2}(X_{≥n−2}, [σ_{≥n−1}O ⟶^{exp} j_{∗}^{m}O_{U}^{∗}]) as the set of isomorphism classes of triples (L, P, α) where L is an O_{X_{≥n−1}}-torsor and P is a j_{∗}^{m}O_{U_{n−2}}^{∗}-torsor and α is an isomorphism^{7} exp_{m}(L) ⟶^{∼} δ^{∗}P of j_{∗}^{m}O_{U_{≥n−1}}^{∗}-torsors. Via (3.13), α can also

6We note a harmless typo in (loc. cit): Gysin maps go from H^{b−2r} to H^{b−2(r−1)} and not H^{b−2(r−2)}.

be thought of as an isomorphism (as in [14, 10.3.10c]) exp(L) ⟶^{∼} δ^{∗}P′ ⊗ O(D′) of invertible sheaves on X_{≥n−1} with D′ ∈ K. If P′′ is another invertible sheaf on X_{n−2} corresponding to P, then we have α′′ : exp(L) ⟶^{∼} δ^{∗}P′′ ⊗ O(D′′) and P′′ ⟶^{∼} P′ ⊗ O(v) (v ∈ W_{n−2}); so one can rewrite α as exp(L) ⟶^{∼} δ^{∗}(O(−v) ⊗ P′′) ⊗ O(D′′ + δ^{∗}v). As (P′, D′) − (P′′, D′′) = γ_{n−3}(v) ∈ NS_{n−2} ⊕ K, the element w = (P′, D′) ∈ C′_{n} depends only on the isomorphism class of the triple (L, P, α). Thus, we may associate the pair (w, [L]) ∈ C′_{n} ⊕ Lie G_{n} with the isomorphism class of (L, P, α).

The map δ^{∗} : H^{1}(X_{n−2}, O) → ℍ^{2}(X_{≥n−2}, [σ_{≥n−1}O ⟶^{exp} j_{∗}^{m}O_{U}^{∗}]) can be described as

follows. The class [I] ∈ H^{1}(X_{n−2}, O) of an O_{X_{n−2}}-torsor I is mapped to the class of the triple (δ^{∗}I, P = exp_{m}(I), α_{I}) with α_{I} : exp_{m}(δ^{∗}I) ⟶^{∼} δ^{∗}(exp_{m}(I)) the tautological isomorphism; associated with this triple is the invertible sheaf P′ = exp(I) (3.13), the element D = 0 of K and the isomorphism α_{I} : exp(δ^{∗}I) ⟶^{∼} δ^{∗}(exp(I)). If P′′ is another invertible sheaf corresponding to the j_{∗}^{m}O_{U}^{∗}-torsor exp_{m}(I), then as before P′′ ⟶^{∼} P′ ⊗ O(v) (v ∈ W_{n−2}); thus, to the triple (δ^{∗}I, exp_{m}(I), α_{I}), we may attach the pair (0, [δ^{∗}I]) ∈ C′_{n} ⊕ Lie G_{n}.

Taking into account the surjectivity of H^{0}(X_{≥n−1}, O) ⟶^{exp} H^{0}(X_{≥n−1}, O^{∗}), we find that elements of

\frac{ℍ^{2}(X_{≥n−2}, [σ_{≥n−1}O ⟶^{exp} j_{∗}^{m}O_{U}^{∗}])}{H^{1}(X_{n−2}, O)}

can be identified with pairs (w, L) where L is an element of Lie P_{n} and (as before) w ∈ C′_{n} with exp(L) = ρ(w) in P_{n}. This last equality forces, by (3.9), w to be an element of C_{n}. Recalling the definition of T_{ℤ} [14, 10.1.3.1], we deduce the required isomorphism. □

Consider the composite inclusion

(30)  h_{X}^{2} ↪ \frac{ℍ^{2}(X_{≥n−2}, [σ_{≥n−1}O ⟶^{exp} j_{∗}^{m}O_{U}^{∗}])}{H^{1}(X_{n−2}, O)} ⟶^{∼} T_{ℤ}[C_{n} ⟶^{ρ} P_{n}];

h_{X}^{2} ⊗ ℚ (resp. T_{ℤ}[C_{n} ⟶^{ρ} P_{n}] ⊗ ℚ) is an extension (25) (resp. [14, 10.1.3.1]) of C_{n} ⊗ ℚ by H_{1}(P_{n}, ℚ). Thus, we obtain an isomorphism of abelian groups

h_{X}^{2} ⟶^{∼} T_{ℤ}[C_{n} ⟶^{ρ} P_{n}] (modulo finite groups).

From (19) and (29), we deduce an isomorphism of abelian groups

Λ : t^{n}(U_{•}) ⟶^{∼} T_{ℤ}L^{n} (modulo finite groups).

Compatibilities of Λ.

• Weight filtration: Since NS(X_{n−3}) ⊗ ℚ = t^{2}(X_{n−3}) ⊗ ℚ is of type (0, 0), from (29) we have W_{−1}h_{X}^{2} ⊗ ℚ ⟶^{∼} W_{−1}t^{n}(U_{•}) ⊗ ℚ. The elements of W_{−1}h_{X}^{2} are characterized, by (27), (28) and (3.7), as those corresponding to pairs (L, β) with exp(L) = 0 = β. This proves the compatibility of Λ with the weight filtration.

7We write exp_{m}(L) for the j_{∗}^{m}O_{U_{≥n−1}}^{∗}-torsor and exp(L) for the O_{X}^{∗}-torsor associated with L.

• Hodge filtration: From [14, 10.1.3.1], the map α : T_{ℤ}L^{n} → Lie P_{n} (used to construct T_{ℤ}L^{n}) gives the Hodge filtration. Namely,

F^{0}(T_{ℤ}L^{n} ⊗ ℂ) = Ker(α_{ℂ} : (T_{ℤ}L^{n}) ⊗ ℂ → Lie P_{n}).

Therefore, we obtain

α_{ℂ} : \frac{(T_{ℤ}L^{n}) ⊗ ℂ}{F^{0}} ⟶^{∼} Lie P_{n}.

Since t^{n}(U_{•}) is of type (∗), we get [14, 10.1.3.3] the isomorphism

\frac{t^{n}(U_{•}) ⊗ ℂ}{F^{0}} ⟵^{∼} \frac{W_{−1}t^{n}(U_{•}) ⊗ ℂ}{F^{0} ∩ W_{−1}}.

Now (2.15) and (3.4) together imply that

W_{−1}t^{n}(U_{•}) ⊗ ℂ ⟶^{∼} \frac{ℍ^{1}(X_{≥n−1}, Γ_{0})}{ℍ^{1}(X_{n−2}, Γ_{0})} ⟵^{∼} Lie P_{n}^{♮}.

Noting that the Hodge filtration on H^{∗}(U_{•}, ℂ) is induced by the filtration ([12, 3.2.2], [14, 8.1.8, 8.1.12])

F^{i}Ω(log Y) := 0 → 0 → ⋯ → Ω^{i}(log Y) ⟶^{d} Ω^{i+1}(log Y) → ,

we obtain — cf. (3.8 (v), (vi)) — the commutativity of the following diagram

W_{−1}t^{n}(U_{•}) ⊗ ℂ ⟵^{∼} Lie P_{n}^{♮} ⟶_{(2.15)}^{∼} \frac{ℍ^{1}(X_{≥n−1},Γ_{0})}{ℍ^{1}(X_{n−2},Γ_{0})}
↓  ↓  ↓
\frac{t^{n}(U_{•})⊗ℂ}{F^{0}} ⟵^{∼} \frac{W_{−1}t^{n}(U_{•})⊗ℂ}{F^{0}∩W_{−1}}  ⟵^{∼} Lie P_{n} ⟶_{(2.6)}^{∼} \frac{H^{1}(X_{≥n−1},O)}{H^{1}(X_{n−2},O)},

which fits into a larger commutative diagram

H^{n}(U_{•}, ℤ(1))/torsion ⟵ t^{n}(U_{•}) ⟶^{Λ} T_{ℤ}L^{n} ⟶^{α} Lie P_{n}
↓  ↓  ∥
H^{n}(U_{•}, ℂ)  ⟵ t^{n}(U_{•}) ⊗ ℂ ⟶ \frac{t^{n}(U_{•})⊗ℂ}{F^{0}} ⟶ Lie P_{n}.

This diagram shows that Λ is compatible with the Hodge filtration thereby finishing the proof of Theorem 3.2. □

Theorem 3.2 partly proves Conjecture 0.1 (up to isogeny) for the simplicial scheme U_{•}; it remains to prove the statements concerning the de Rham and étale realizations. We first treat the de Rham realization. The étale realization is dealt with in §4. The de Rham realization of L^{n}.

Using T_{DR}L^{n} ⟶^{∼} T_{ℂ}L^{n} [14, 10.1.8], T_{ℂ}L^{n} ⟶^{∼} t^{n}(U_{•}) ⊗ ℂ (3.2), and t^{n}(U_{•}) ⊗ ℂ ↪

H^{n}(U_{•}, ℂ) ⟶^{∼} H_{DR}^{n}(U_{•}) [12, 3.1.8, 3.2.2] gives us a map

T_{DR}L^{n} → H_{DR}^{n}(U_{•});

our next task is to show that this map can be constructed purely algebraically (3.18) and this, over any field of characteristic zero.

In the remainder of this section, k denotes a field of characteristic zero. Our next main result (3.18) of this section requires us to construct a group scheme Ũ, a map ψ̃ : B_{n} → Ũ such that, when k = ℂ, ψ̃ lifts to a compatible homomorphism ψ : T_{ℤ}L^{n} → Lie Ũ. Once these are acquired, the criterion [14, 10.1.9] may be applied to deduce that Lie Ũ is the de Rham realization T_{DR}L^{n} of L^{n}. Step 1. Construction of a map ψ_{1} : K → G_{n}^{♢}.

Recall the map q : Ỹ_{m} → Y_{m} → X_{m} from the normalization Ỹ_{m} of Y_{m}. The Poincaré residue sequence [12, 3.1.5.2] on X_{m} and X_{≥n−1}

0 → Ω^{1} → Ω^{1}(log Y) ⟶^{Res} q_{∗}OỸ → 0

gives the exact sequences

(31)  H^{0}(Ỹ_{m}, O) → H^{1}(X_{m}, Ω^{1}) → H^{1}(X_{m}, Ω^{1}(log Y)),
(32)  H^{0}(Ỹ_{≥n−1}, O) → H^{1}(X_{≥n−1}, Ω^{1}) → H^{1}(X_{≥n−1}, Ω^{1}(log Y)).

Since the first map of (31) is the composition of

W_{Y_{m}}(S) ⊗ k ⟶^{∼} H^{0}(Ỹ_{m}, O) ⟶^{λ_{m}} NS_{X_{m}}(S) ⊗ k ⟶^{c_{Xm}}_{(6)} H^{1}(X_{m}, Ω^{1}),

there is a natural injection — recall N_{m} := Coker(λ_{m}) (2.10) —

(33)  κ_{m} : N_{m}(S) ⊗ k ↪ H^{1}(X_{≥n−1}, Ω^{1}(log Y)).

The boundary map of (32) is induced by the composite map

K(S) ⟶^{ϑ′} H^{1}(X_{≥n−1}, O^{∗}) = G′_{n}(S) → π_{0}(G′_{n}) ⟶^{dlog}_{(2.12)} H^{1}(X_{≥n−1}, Ω^{1})

via K(S) ⊗ k ⟶^{∼} H^{0}(Ỹ_{≥n−1}, O). Thus, in the exact sequence

ℍ^{1}(X_{≥n−1}, [O^{∗} ⟶^{d log} Ω^{1}(log Y)]) → H^{1}(X_{≥n−1}, O^{∗}) → H^{1}(X_{≥n−1}, Ω^{1}(log Y)),

one finds, by the exactness of (32), that the map ϑ′ admits a lifting to ℍ^{1}(X_{≥n−1}, Γ^{∗}).

In fact, one can easily construct a natural lifting ψ_{1} : K(S) → ℍ^{1}(X_{≥n−1}, Γ^{∗}) = G_{n}^{♢}(S) of ϑ′ using Čech cohomology; let C^{∗}(G) denote Čech cochains of X_{≥n−1} with coefficients in G (relative to a suitable open cover {U_{i}}) and ∂ the Čech differential. By (3.7c), for each element E ∈ K, we have to construct an invertible sheaf L on X_{≥n−1} and a connection ∇ with at most logarithmic poles along Y_{≥n−1}. In terms of cocycles [38, 2.5] [34, 7.2], using C^{1}(Γ^{∗}) = C^{1}(O^{∗})⊕C^{0}(Ω^{1}(log Y)), if {s_{ij}} ∈ C^{1}(O^{∗}) represents L and {ω_{i}} ∈ C^{0}(Ω^{1}(log Y)) represents ∇, then these satisfy ds_{ij}/s_{ij} = ω_{j} − ω_{i}; ∇ is integrable if and only if ω_{i} is closed.

Let E ∈ K be an effective divisor. If {f_{i}} are local equations for E, consider the cochain (s_{ij}, df_{i}/f_{i}) ∈ C^{1}(O^{∗}) ⊕ C^{0}(Ω^{1}(log Y)) where f_{j} = f_{i}s_{ij}; clearly, df_{j}/f_{j} − df_{i}/f_{i} = ds_{ij}/s_{ij}. Thus, the pair (s_{ij}, df_{i}/f_{i}) is a cocycle; and, it represents an element of ℍ^{1}(X_{≥n−1}, Γ^{∗}). If {g_{i}} are different local equations for E, then one gets an element (t_{ij}, dg_{i}/g_{i}) ∈ C^{1}(O^{∗}) ⊕ C^{0}(Ω^{1}(log Y)) with g_{j} = g_{i}t_{ij}. There exist u_{i} ∈ C^{0}(O^{∗}) with f_{i} = u_{i}g_{i}. Since s_{ij}/t_{ij} = u_{j}/u_{i} and ds_{ij}/s_{ij}−dt_{ij}/t_{ij} = du_{j}/u_{j}−du_{i}/u_{i}, i.e., the cocycle (t_{ij}, dg_{i}/g_{i}) − (s_{ij}, df_{i}/f_{i}) is a coboundary (∂u), the element ψ_{1}(E) = (s_{ij}, df_{i}/f_{i}) of ℍ^{1}(X_{≥n−1}, Γ^{∗}) depends only on E but not on the choice of the local defining equations. The association E ↦ ψ_{1}(E) defined for effective E easily extends to a homomorphism ψ_{1} : K → G_{n}^{♢}.

Step 2. Construction of the scheme Ũ.

Let g be the structure map X_{≥n−2} → S; recall a_{m} is the structure map X_{m} → S.

Lemma 3.14. The S_{fppf}-sheaf R^{2}g_{∗}K^{∗} associated with the complex K^{∗} := [σ_{≥n−1}O^{∗} ⟶^{d log} Ω^{1}(log Y)] on X_{≥n−2} is representable. One has ξ_{∗} : π_{0}(R^{2}g_{∗}K^{∗}) ↪ π_{0}(G′_{n}); here R^{2}g_{∗}K^{∗} denotes the associated representing group scheme.

Proof. One has an exact sequence

H^{0}(X_{≥n−1}, O^{∗}) → H^{1}(X_{≥n−2}, Ω^{1}(log Y)) → ℍ^{2}(X_{≥n−2}, K^{∗}) ⟶^{ξ}

→ H^{1}(X_{≥n−1}, O^{∗}) → H^{2}(X_{≥n−2}, Ω^{1}(log Y));

the sheaf R^{1}f_{∗}O^{∗} is representable by G′_{n}, f_{∗}O^{∗} by a torus, and R^{i}g_{∗}Ω^{1}(log Y) are representable by vector group schemes. Since Hom(f_{∗}O^{∗}, R^{1}g_{∗}Ω^{1}(log Y)) = 0, one can now invoke (2.1) to obtain the representability of R^{2}g_{∗}K^{∗}.

Since Ker(ξ) is connected, the map ξ_{∗} is injective. □

The inclusion σ_{≥n−1}Γ^{∗} ↪ K^{∗} induces an exact sequence

(34)  0 → \frac{G_{n}^{♢}}{a_{n−2 ∗}Ω^{1}(log Y)} → R^{2}g_{∗}K^{∗} ⟶^{υ} R^{1}a_{n−2 ∗}Ω^{1}(log Y).

Consider the commutative diagram

H^{1}(X_{n−3}, Ω^{1}(log Y)) ⟶^{δ^{∗}} H^{1}(X_{≥n−2}, Ω^{1}(log Y))
↓^{δ_{n−3}^{∗}}  ↓
H^{1}(X_{n−2}, Ω^{1}(log Y)) ⟵^{υ}  ℍ^{2}(X_{≥n−2}, K^{∗}),

and the map

P ic_{X_{n−2}}^{0}(S) ↪ H^{1}(X_{n−2}, O^{∗}) → ℍ^{2}(X_{≥n−2}, K^{∗})

induced by the inclusion K^{∗} ↪ Γ^{∗} on X_{≥n−2}. Taking the quotient of ℍ^{2}(X_{≥n−2}, K^{∗}) by the images of H^{1}(X_{n−3}, Ω^{1}(log Y)) and P ic_{X_{n−2}}^{0}(S) under these maps, and pulling back via (cf. (34) (33))

(35) \frac{Ker(δ_{n−2}^{∗} : N_{n−2} → N_{n−1})(S) ⊗ k}{NS_{n−3}(S) ⊗ k} → \frac{N_{n−2}(S) ⊗ k}{NS_{n−3}(S) ⊗ k} → \frac{H^{1}(X_{n−2}, Ω^{1}(log Y))}{H^{1}(X_{n−3}, Ω^{1}(log Y))}

(actually we do this at the level of the group schemes which represent the associated sheaves) gives us a group scheme U′. Its identity component is denoted U. Lemma 3.15. The map ξ_{∗} : π_{0}(U′) ↪ π_{0}(P′_{n}) is injective.

Proof. It is clear that the previous operations of quotient and pullback do not affect π_{0} and thus π_{0}(R^{2}g_{∗}K^{∗}) ≅ π_{0}(U′). The natural map ξ : R^{2}g_{∗}K^{∗} → G′_{n} provides a map ξ : U′ → P′_{n}; by (3.14), one has an injection ξ_{∗} : π_{0}(U′) ↪ π_{0}(P′_{n}). □

Lemma 3.16. One has an exact sequence

(36)  0 → Lie P_{n}^{♢} → Lie U ⟶^{υ} \frac{Ker(δ_{n−2}^{∗} : N_{n−2} → N_{n−1})(S) ⊗ k}{NS_{n−3}(S) ⊗ k} .

Proof. By (2.14), Lie P_{n}^{♢} is a quotient of Lie G_{n}^{♢} by Lie P ic_{X_{n−2}}^{♢}. Since Lie P ic_{X_{n−2}}^{♢} is an extension of Lie P ic_{X_{n−2}}^{0} by a_{n−2 ∗}Ω^{1}(log Y), we obtain (36) noting (34). One just has to observe that the image of P ic_{X_{n−2}}^{0} in R^{2}g_{∗}K^{∗} is equal to that of the neutral

component of P ic_{X_{n−2}}^{♢} under the natural map P ic_{X_{n−2}}^{♢} ⟶^{δ^{∗}} G_{n}^{♢} → R^{2}g_{∗}K^{∗}. □

Since the composite map

H^{1}(X_{n−2}, O^{∗}) → ℍ^{2}(X_{≥n−2}, K^{∗}) ⟶^{υ} H^{1}(X_{n−2}, Ω^{1}(log Y))

is the map induced by d log : O^{∗} → Ω^{1} → Ω^{1}(log Y), the natural map NS_{n−2} → \frac{R^{2}g_{∗}K^{∗}}{P ic_{n−2}^{0}} induces a map

ψ_{2} : NS_{n−2} → U′.

By definition (35) of U′, Im(δ_{n−3}^{∗} : NS_{n−3} → NS_{n−2}) is contained in Ker(ψ_{2}). Lemma 3.17. The map ψ′ : K ⊕ NS_{n−2} → U′ defined by (u, v) ↦ ψ_{1}(u) + ψ_{2}(v) provides a map ψ′ : B′_{n} → U.

Proof. Since ψ′ = ψ_{2} on the subgroup γ_{n−3}(NS_{n−3}) = δ_{n−3}^{∗}(NS_{n−3}), NS_{n−3} is contained in Ker(ψ′). It is straightforward to check that γ_{n−3}(W_{Y_{n−2}}) is in Ker(ψ′). This gives a map ψ′ : B′_{n} → U′. Note that the composite map

K ⊕ NS_{n−2} ⟶^{ψ′} U′ ⟶^{ξ} P′_{n}

is ρ. By (3.15), the image under ψ′ of B′_{n} is contained in U. □

Put Ũ = U/ψ′(τ_{n}); here, τ_{n} is the torsion subgroup of B′_{n}. The map ψ′ induces a map ψ̃ : B_{n} → Ũ. Theorem 3.18. (i) (k = ℂ) There is a natural commutative diagram

B_{n} ⟶^{ψ̃} Ũ ⟶^{ξ} P̃_{n}
↑  ↑  ↑
T_{ℤ}L^{n} ⟶^{ψ} Lie Ũ ⟶^{ξ} Lie P̃_{n}

whose exterior square is [14, 10.1.3.1] for L^{n}; further, ψ induces an isomorphism ψ_{ℂ} :

T_{ℂ}L^{n} ⟶^{∼} Lie Ũ.

(ii) T_{DR}L^{n} ⟶^{∼} Lie Ũ.

(iii) There is a natural map Lie Ũ → H_{DR}^{n}(U_{•}).

Proof. (i) (k = ℂ) From (2.11), (3.10) and (21), we obtain an exact sequence of isogeny one-motives over ℂ

0 → \frac{[K^{0} → G_{n}]}{[V_{n−2} → P ic_{n−2}^{0}]}⊗ℚ → [C_{n} ⟶^{ρ} P_{n}]⊗ℚ → [Ker(δ_{n−2}^{∗} : N_{n−2} → N_{n−1}) → 0]⊗ℚ → 0.

The proof (in Step 3.) of Theorem 3.2 also shows that (30) is an isomorphism (modulo finite groups) of mixed Hodge structures. Combining this with (23), we obtain that h_{X}^{2} ⊗ ℂ ⟶^{∼} T_{ℂ}[C_{n} ⟶^{ρ} P_{n}] sits in an exact sequence

0 → Lie P_{n}^{♢} → h_{X}^{2} ⊗ ℂ → Ker(δ_{n−2}^{∗} : N_{n−2} → N_{n−1}) ⊗ ℂ → 0.

Recall that there is a map h_{X}^{2} ⊗ ℂ → ℍ^{2}(X_{≥n−2}, K) = Lie R^{2}g_{∗}K^{∗} and that the latter sits in an exact sequence

0 → \frac{ℍ^{1}(X_{≥n−1}, Γ)}{H^{0}(X_{n−2}, Ω^{1}(log Y))} → ℍ^{2}(X_{≥n−2}, K) → H^{1}(X_{n−2}, Ω^{1}(log Y)) → ℍ^{2}(X_{≥n−1}, Γ).

Comparing with (36), the map h_{X}^{2} ⊗ ℂ → Lie R^{2}g_{∗}K^{∗} gives a map

ψ_{ℂ} : T_{ℂ}L^{n} ⟵^{∼}_{(29),(3.2)} \frac{h_{X}^{2} ⊗ ℂ}{NS(X_{n−3}) ⊗ ℂ} → Lie Ũ,

whose composition with T_{ℤ}L^{n} ↪ T_{ℂ}L^{n} gives us the map ψ. Since the composite map

K ⊕ NS_{n−2} ⟶^{ψ′} U′ ⟶^{ξ} P′_{n} is ρ — see (18), it follows easily that the composite map ξ ∘ ψ̃ : B_{n} → P̃_{n} is ϕ_{n}. Thus, the exterior square is the one that intervenes in the definition [14, 10.1.3.1] of T_{ℤ}L^{n}.

We leave it to the reader to check that ψ_{ℂ} is an isomorphism and that it is compatible with the weight and Hodge filtrations.

(ii) This follows from (i) by [14, 10.1.9].

(iii) Compose the natural maps

Lie Ũ → ℍ^{2}(X_{≥n−2}, K) → ℍ^{n}(X_{•}, Ω(log Y)) ⟶^{∼} H_{DR}^{n}(U_{•}),

the last being a consequence of [12, 3.1.8], [34, 1.0.3.7], [31, 6.11.4]. □

4. Etale realization of L^{n}

Fix a simplicial pair (X_{•}, Y_{•}) over a field k of characteristic zero. When k = ℂ, we have a map (see below for notations)

T L^{n} ⊗ ℚ ⟶^{∼} (T_{ℤ}L^{n}) ⊗_{ℤ} 𝔸 ⟶^{∼}_{(3.2)} T_{ℤ}I^{n}(U_{•}) ⊗_{ℤ} 𝔸 ↪ H^{n}(U_{•}, ℤ(1)) ⊗_{ℤ} 𝔸 ⟶^{∼}_{(41)} H_{et}^{n}(U_{•}, 𝔸(1));

the first isomorphism is from [14, 10.1.6]. We shall show that this map can be constructed purely algebraically for all k; combined with (3.2), (3.18), this will prove (0.1) up to isogeny for U_{•}. By (2.9), we may and do assume that k is a finitely generated extension of ℚ.

We adopt the following notations: r is a positive integer, ℓ is a positive prime integer, H_{et}^{i}(V̄ , ℤ̂(1)) is \underleftarrow{lim}_{r}H_{et}^{i}(V̄ , μμ_{r}) and H_{et}^{i}(V̄ , 𝔸(1)) is H_{et}^{i}(V̄ , ℤ̂(1)) ⊗ ℚ. For any one-

motive M, recall the finite ℤ/rℤ-module T_{ℤ/rℤ}M := H^{0}(M ⊗^{L} ℤ/rℤ) [14, 10.1.5]; note T_{ℓ}M := \underleftarrow{lim}_{d}T_{ℤ/ℓ^{d}ℤ}M, M ⊗ ℤ̂ = T M = \underleftarrow{lim}_{r}T_{ℤ/rℤ}M, and M ⊗ 𝔸 := T M ⊗ ℚ. For any commutative group (scheme) A, _{r}A is the Ker(A ⟶^{r} A)(S̄) and A_{tor} the torsion subgroup (scheme). Let ℤ/rℤ denote the constant group. All maps in this section are 𝔾-equivariant.

Recall the Kummer sequence of étale sheaves on a (simplicial) scheme V

(37)  0 → μμ_{r} → 𝔾_{m} ⟶^{r} 𝔾_{m} → 0.

For proper V, this gives, by the divisibility of H_{et}^{0}(V̄ , 𝔾_{m}),

(38)  H_{et}^{1}(V̄ , μμ_{r}) ⟶^{∼} _{r}H_{et}^{1}(V̄ , 𝔾_{m});

for V smooth and proper, we also obtain

(39)  κ_{V} : NS(V̄) ⊗ ℤ/rℤ ↪ H_{et}^{2}(V̄ , μμ_{r}).

Similarly, using the divisibility of G_{n}(S̄), one obtains an injection

(40)  π_{0}(G′_{n})(S̄) ⊗ ℤ/rℤ ↪ H_{et}^{2}(X̄_{≥n−1}, μμ_{r}).

Remark 4.1. The spectral sequence [14, 8.1.19.1] has an étale analogue E_{1}^{et} [9, Introduction] [11, §6] [15, §14]; it calculates the étale cohomology of Ū_{•} with ℚ_{ℓ}-coefficients.

The degeneration of the étale analogue at E_{2} (i.e., that E_{2}^{et} = E_{∞}^{et}) and the definition of the weight filtration W on H_{et}^{∗}(Ū, ℚ_{ℓ}(1)) is a consequence of [13]; see [11, §6,7], [15, §14] for these and the fact that the weight filtrations in mixed Hodge theory are compatible with those in étale cohomology, compared via (41).

Recall the Artin-Grothendieck comparison theorem [1, §4] [36, p.23]: for any variety V over ℂ, one has canonical isomorphisms

(41)  H_{et}^{i}(V, ℤ/rℤ) ⟶^{∼} H^{i}(V (ℂ), ℤ/rℤ)

between the étale and classical (singular) cohomology of V. Any imbedding ι : k ↪ ℂ gives an isomorphism of k̄ with the algebraic closure \overline{ι(k)} of ι(k) in ℂ; this, in turn,

provides an isomorphism H_{et}^{∗}(U_{m} ×_{k} k̄, ℤ/rℤ) ⟶^{∼} H_{et}^{∗}(U_{m} ×_{k} \overline{ι(k)}, ℤ/rℤ) ⟶^{∼} H_{et}^{∗}(U_{m} ×_{ι}

ℂ, ℤ/rℤ); the last isomorphism is from [39, VI 2.6]. Now (41) and the degeneration [14, 8.1.20 (ii)] of [14, 8.1.19.1] at E_{2} together provide another proof that E_{2}^{et} = E_{∞}^{et}. □ Lemma 4.2. One has (i)

\frac{H_{et}^{1}(X̄_{≥n−1}, 𝔸(1))}{H_{et}^{1}(X̄_{n−2}, 𝔸(1))} ⟶^{∼} σ_{≥n−1}H_{et}^{n}(X̄_{•}, 𝔸(1)) = W_{−1}H_{et}^{n}(X̄_{•}, 𝔸(1)) ⟶^{∼} W_{−1}H_{et}^{n}(Ū_{•}, 𝔸(1)).

(ii) H_{et}^{1}(X̄_{≥n−1}, ℤ̂(1)) ⟶^{∼} T [0 → G_{n}] and T [0 → P_{n}] ⊗ ℚ ⟵^{∼} W_{−1}H_{et}^{n}(Ū_{•}, 𝔸(1)).

Proof. (i) As in (3.5), this follows from the degeneration at E_{2} of the étale analogue of [14, 8.1.19.1] for ℚ_{ℓ}(1) — see (4.1).

(ii) The first isomorphism follows from (38) for V = X_{≥n−1} by taking the inverse limit over r (note π_{0}(G′)_{tor} is finite). Similarly, one has (1.5 (i)) H_{et}^{1}(X̄_{n−2}, ℤ_{ℓ}(1)) ⟶^{∼} T_{ℓ}P ic(X_{n−2}). Combining this with (i) gives the second isomorphism. □

Proposition 4.3. (i) H^{0}([W_{Y_{m}} ⟶^{λ_{m}} P ic_{X_{m}}] ⊗^{L} ℤ/rℤ)) ⟵^{∼} H_{et}^{1}(Ū_{m}, μμ_{r}).

(ii) H^{0}([K ⟶^{ϑ′} G′_{n}] ⊗^{L} ℤ/rℤ) ⟵^{∼} H_{et}^{1}(Ū_{≥n−1}, μμ_{r}).

Proof. It is exactly identical to [14, 10.3.6]; we repeat the proof of (i) here for the convenience of the reader.

By (3.6), H_{et}^{1}(Ū_{m}, μμ_{r}) ⟶^{∼} ℍ_{et}^{1}(Ū_{m}, [𝔾_{m} ⟶^{r} 𝔾_{m}]) is the set of isomorphism classes of pairs (L, α) where L is an invertible sheaf on Ū_{m} together with an isomorphism α : L^{⊗r} ⟶^{∼} O. Let (L, α) be such a pair. The invertible sheaf L extends to an invertible sheaf L̃ on X̄_{m}, and there exists a divisor E of X̄_{m} with support in Ȳ_{m} such that α extends to an isomorphism α′ : L̃^{⊗r} ⟶^{∼} O(E). If there exists an isomorphism β of L̃^{⊗r} with O(E), this isomorphism is uniquely determined up to multiplication by an element of the divisible group H_{et}^{0}(X̄_{m}, 𝔾_{m}). One deduces that the pair (L̃, E) determines (L, α) up to isomorphism. For a pair (L̃, E) to come from a suitable (L, α), it is necessary and sufficient that r[L̃] = [O(E)] in P icX̄_{m}. It comes from (OX̄_{m}, 0) if and only if it is of the form (OX̄_{m}(E), rE). This identifies H_{et}^{1}(Ū_{m}, μμ_{r}) with H^{0} of the

complex which is the tensor product of [W_{Y_{m}}(S̄) ⟶^{λ_{m}} P ic(X̄_{m})] (degrees 0 and 1) and [ℤ ⟶^{r} ℤ] (degrees −1 and 0):

W_{Y_{m}}(S̄) ⟶^{λ_{m}} P ic(X̄_{m})
↑^{r}  ↑^{−r}
W_{Y_{m}}(S̄) ⟶^{λ_{m}} P ic(X̄_{m})

This proves the first isomorphism. □

Lemma 4.4. (i) The maps [B_{n} ⟶^{ϕ_{n}} P̃_{n}] ⟵^{b} [B′_{n} ⟶^{ρ} P_{n}] ⟶^{a} [\frac{C′_{n}}{NS_{n−3}} ⟶^{ρ} P′_{n}] of complexes (concentrated in degrees 0 and 1) induce isomorphisms (modulo finite groups)

(42) T L^{n} ⟵^{∼} \underleftarrow{lim}_{r}H^{0}([B′_{n} ⟶^{ρ} P_{n}] ⊗^{L} ℤ/rℤ) ⟶^{∼} \underleftarrow{lim}_{r}H^{0}([\frac{C′_{n}}{NS_{n−3}} ⟶^{ρ} P′_{n}] ⊗^{L} ℤ/rℤ).

(ii) The map [C_{n} ⟶^{ρ} P_{n}] to [C′_{n} ⟶^{ρ} P′_{n}] induces an isomorphism (modulo finite groups)

(43)  \underleftarrow{lim}_{r}H^{0}([C_{n} ⟶^{ρ} P_{n}] ⊗^{L} ℤ/rℤ) ⟶^{∼} \underleftarrow{lim}_{r}H^{0}([C′_{n} ⟶^{ρ} P′_{n}] ⊗^{L} ℤ/rℤ).

Proof. (cf. [14, 10.3.5 ii]) (i) The first isomorphism is clear because b is an isogeny; the cokernel [Im(ρ′) → π_{0}(P′_{n})] of the map a is quasi-isomorphic to [0 → Coker(ρ′)] and Coker(ρ′)_{tor} is finite. This yields the second isomorphism.

(ii) The argument is similar to that of (i). □

The map [σ_{≥n−1}𝔾_{m} ⟶^{r} 𝔾_{m}] → [𝔾_{m} ⟶^{r} 𝔾_{m}] of étale complexes on U_{≥n−2} gives the exact sequence

(44) \frac{ℍ_{et}^{2}(Ū_{≥n−2}, [σ_{≥n−1}𝔾_{m} ⟶^{r} 𝔾_{m}])}{H_{et}^{1}(Ū_{n−2}, 𝔾_{m})} → ℍ_{et}^{2}(Ū_{≥n−2}, [𝔾_{m} ⟶^{r} 𝔾_{m}]) → H_{et}^{2}(Ū_{n−2}, 𝔾_{m})

Theorem 4.5. (i) One has a canonical isomorphism

Π_{r} : \frac{ℍ_{et}^{2}(Ū_{≥n−2}, [σ_{≥n−1}𝔾_{m} ⟶^{r} 𝔾_{m}])}{H_{et}^{1}(Ū_{n−2}, 𝔾_{m})} ⟶^{∼} H^{0}([C′_{n} ⟶^{ρ} P′_{n}] ⊗^{L} ℤ/rℤ).

This gives an inclusion Λ′ : T [C_{n} ⟶^{ρ} P_{n}] ↪ H_{et}^{2}(Ū_{≥n−2}, ℤ̂(1)).

(ii) The map Λ′ of (i) induces an inclusion Λ_{et} : T L^{n} ⊗ ℚ ↪ H_{et}^{n}(Ū_{•}, 𝔸(1)).

(iii) One has Λ_{et}(T (W_{j}L^{n} ⊗ ℚ)) ⊆ W_{j}H_{et}^{n}(Ū_{•}, 𝔸(1)) with equality for j = −2, −1.

Proof. (i) The proof follows that of (4.3). Via (3.6), ℍ_{et}^{2}(Ū_{≥n−2}, [σ_{≥n−1}𝔾_{m} ⟶^{r} 𝔾_{m}]) is the group of isomorphism classes of triples (I, L, α) where I is a 𝔾_{m}-torsor on Ū_{n−2} and L is a 𝔾_{m}-torsor on Ū_{≥n−1} and α : δ^{∗}I ⟶^{∼} L^{⊗r}. The class [L] ∈ H_{et}^{1}(Ū_{n−2}, 𝔾_{m}) of a 𝔾_{m}-torsor L gets mapped to the class δ^{∗}[L] of the triple (L^{⊗r}, δ^{∗}L, α_{L}) where α_{L} is the canonical isomorphism δ^{∗}(L^{⊗r}) ⟶^{∼} (δ^{∗}L)^{⊗r}.

Fix the class f of a triple (I, L, α). The torsors I and L extend to torsors Ĩ and L̃ on X̄_{n−2} and X̄_{≥n−1} respectively and there exists a divisor E of X̄_{≥n−1} supported on Ȳ_{≥n−1} such that α extends to an isomorphism α̃ : δ^{∗}Ĩ⊗ O(E) ⟶^{∼} L̃^{⊗r}. This says that the elements w = (E, [Ĩ]) ∈ K ⊕ NS_{n−2} and [L̃] ∈ P′_{n} satisfy ρ(w) = r[L̃]; in other

words, the pair (w, [L̃]) defines an element Π_{r}(f) of H^{0}([C′_{n} ⟶^{ρ} P′_{n}] ⊗^{L} ℤ/rℤ).

This element Π_{r}(f) is clearly zero if our triple is isomorphic to δ^{∗}[L]: if L extends to a 𝔾_{m}-torsor L̃ on X̄_{n−2}, then Π_{r}(f) is given by w = (0, r[L̃]) and the class of ρ([L̃]) ∈ P′_{n}. If Ĩ′ is another extension of I, then there is an isomorphism Ĩ ⟶^{∼} Ĩ′ ⊗O(d) (d ∈ W_{n−2}). Using Ĩ′ instead of Ĩ modifies w by γ_{n−3}(d). If L̃′ is another extension of L, then there is an isomorphism L̃ ⟶^{∼} L̃′ ⊗ O(F) (F ∈ K). Using L̃′ instead of L̃ modifies w by ±rF. This shows that the map Π_{r} is well-defined; on the image of the

map \frac{H_{et}^{1}(Ū_{≥n−1},μμ_{r})}{H_{et}^{1}(Ū_{n−2},μμ_{r})} → ℍ_{et}^{2}(Ū_{≥n−2}, [σ_{≥n−1}𝔾_{m} ⟶^{r} 𝔾_{m}]), the map Π_{r} reduces to the maps

(isomorphisms) of (4.3).

Given an element g of H^{0}([C′_{n} ⟶^{ρ} P′_{n}] ⊗^{L} ℤ/rℤ), pick representatives u ∈ K, J_{1} ∈

P ic_{n−2}, and J ∈ G′_{n} for g. By definition, O(u) ⊗ δ^{∗}J_{1} ⟶^{∼} J^{⊗r} ⊗ J_{2} where J_{2} is

an element of P ic_{n−2}^{0}; by the divisibility of P ic_{n−2}^{0}(S̄), the relation can be rewritten as O(u) ⊗ δ^{∗}J_{1} ⟶^{∼} J^{⊗r} for an appropriate element of G′_{n}. Restricting to Ū_{≥n−1}, we obtain an isomorphism β : δ^{∗}J_{1} ⟶^{∼} J^{⊗r}; the triple (J_{1}, J , β) defines an element of ℍ_{et}^{2}(Ū_{≥n−2}, [σ_{≥n−1}𝔾_{m} ⟶^{r} 𝔾_{m}]). One easily verifies that this is an inverse to the map Π_{r} and that Π_{r} is an isomorphism. The map Λ′ is obtained by composing the inverse of \underleftarrow{lim}_{r}Π_{r} with the maps in (44) and (43). (ii) Since δ^{∗}H_{et}^{2}(Ū_{n−3}, 𝔸(1)) is contained in Ker(H_{et}^{2}(Ū_{≥n−2}, 𝔸(1)) → H_{et}^{n}(Ū_{•}, 𝔸(1))), the following diagram gives us Λ_{et}:

H_{et}^{2}(Ū_{n−3}, 𝔸(1)) ⟶^{δ^{∗}} H_{et}^{2}(Ū_{≥n−2}, 𝔸(1))⟶ H_{et}^{n}(Ū_{•}, 𝔸(1))
(39)↑  ↑^{(i)}
NS(X̄_{n−3}) ⊗ 𝔸 ⟶^{δ^{∗}} T [C_{n} ⟶^{ρ} P_{n}] ⊗ ℚ;

note NS(X̄_{n−3}) ⊗ 𝔸 ⟶^{∼} T [NS(X̄_{n−3}) → 0] ⊗ ℚ. Since W_{−1}H_{et}^{2}(Ū_{n−3}, 𝔸(1)) = 0 [15,

7.2] [12, 3.2.15], the intersection

δ^{∗}H_{et}^{2}(Ū_{n−3}, 𝔸(1)) ∩ W_{−1}H_{et}^{2}(Ū_{≥n−2}, 𝔸(1))

is zero. Now (4.2 (ii)) shows that Λ_{et} is injective on W_{−1}T L^{n} ⊗ ℚ. It remains to show the injectivity of either

Λ_{et} : B_{n}(S) ⊗ 𝔸 → Gr_{0}^{W} H_{et}^{n}(Ū_{•}, 𝔸(1))

or its ℚ_{ℓ}-analogue. The base change of this map via any ι : k ↪ ℂ can be identified, via (41) and the compatibility of the étale and classical cycle class maps [31, 5.3], with the injective map ν of (3.10(i)):

B_{n}(ℂ)⊗ℚ_{ℓ} ⟶^{∼}_{ν⊗id} Gr_{0}^{W} t^{n}(U_{•,ι})⊗ℚ_{ℓ} ↪ Gr_{0}^{W} H^{n}(U_{•,ι}, ℚ(1))⊗ℚ_{ℓ} ⟵^{∼}_{(41)} Gr_{0}^{W} H_{et}^{n}(U_{•,ι}, ℚ_{ℓ}(1)).

(iii) Follows from the definition of the filtration W ([11], [13], [15, §14]) and (4.2). □

Definition 4.6. t_{ℓ}^{n}(U_{•}) := Λ_{et}(T_{ℓ}L^{n}⊗ℚ) is a 𝔾-invariant ℚ_{ℓ}-subspace of H_{et}^{n}(Ū_{•}; ℚ_{ℓ}(1)). Similarly, t_{𝔸}^{n}(U_{•}) := Λ_{et}(T L^{n} ⊗ ℚ) is a 𝔾-invariant 𝔸-submodule of H_{et}^{n}(Ū_{•}; 𝔸(1)). Remark 4.7. Over k = ℂ, one has a commutative diagram

T L^{n} ⊗ ℚ ⟶^{Λ_{et}} H_{et}^{n}(U_{•}, 𝔸(1))
≀↓  ^{≀}↓^{(41)}
T_{ℤ}L^{n} ⊗ 𝔸 ⟶_{(3.2)}^{Λ} H^{n}(U_{•}(ℂ), 𝔸(1));

the left vertical map is from [14, 10.1.6]. This diagram, together with (3.2), also provides the injectivity of Λ_{et} in (4.5(ii)). It also provides canonical isomorphisms

t_{ℓ}^{n}(U_{•}) ⟶^{∼} t^{n}(U_{•}) ⊗ ℚ_{ℓ} and t_{𝔸}^{n}(U_{•}) ⟶^{∼} t^{n}(U_{•}) ⊗ 𝔸. □

5. The one-motives L^{n}(V)

In this section, k is any field of characteristic zero. Throughout the rest of the paper, V will denote a variety (over S).

We shall now translate our results for simplicial schemes into ones for varieties; in particular, we show how to construct one-motives L^{n}(V) associated with any algebraic variety V over S. We will also show that L^{n}(−) is a contravariant functor.

The main result of this section is the proof of Conjecture 0.1 for fields of characteristic zero. Proper hypercoverings.

Let V be a variety over S. By [14, 6.2.8], [44, 5.3], there exists a simplicial pair (X_{•}, Y_{•}) with a proper map α : U_{•} → V which makes U_{•} into a proper hypercovering of V. Namely, we have

(i) for any ι : k ↪ ℂ, an isomorphism of mixed Hodge structures [14, 8.2.2]

(45)  α^{∗} : H^{∗}(V_{ι}, ℤ) ⟶^{∼} H^{∗}(U_{•,ι}, ℤ);

(ii) an isomorphism of 𝔾-modules

(46)  α^{∗} : H_{et}^{∗}(V̄ , ℤ̂(1)) ⟶^{∼} H_{et}^{∗}(Ū_{•}, ℤ̂(1));

(iii) and an isomorphism of k-vector spaces

(47)  α^{∗} : H_{DR}^{∗}(V) ⟶^{∼} H_{DR}^{∗}(U_{•}).

For (ii), one uses the result [44, 4.3.2] that a proper surjective morphism is a map of universal cohomological descent for étale torsion sheaves and proceeds as in [14, 6.2.5]; cf. [4, p.306].

In (iii), the definition of H_{DR}^{∗}(V) (as indicated in [17, p. 89], [31, 6.11.4]) is that of [28] or, by [29, 1.2], as the crystalline cohomology of V; see also [27, Expose III], especially 1.3, 1.13, 1.14. for a cubical variant.

We denote by β_{U} the inverse of any of these isomorphisms α^{∗}. Remark 5.1. Given two such proper hypercoverings of V, one can find another such proper hypercovering which dominates both. More generally, given any morphism h : V → W between two schemes, one can find such proper hypercoverings of V and W and a morphism between them which lifts the morphism h; we refer to [44, 5.1.4], [14, 6.2.8] for details. □

Suppose given two proper hypercoverings of V fitting into a diagram

V ⟵ U_{•} ⟶ X_{•} ⟵ Y_{•}
∥  _{θ}↓  _{θ}↓
V ⟵ ′U_{•} ⟶ ′X_{•} ⟵ ′Y_{•}

which yields a morphism of one-motives

θ_{L}^{∗} : L^{n}(′X_{•},′ Y_{•}) → L^{n}(X_{•}, Y_{•}).

Lemma 5.2. The morphism θ_{L}^{∗} is an isogeny.

Proof. The construction of the one-motives L^{n}(U_{•}) and L^{n}(′U_{•}) relies only on a finite number of schemes and a finite number of associated morphisms between them. Therefore, we may assume that these one-motives are defined over a finitely generated subfield k′ of k. Such a field k′ always admits an embedding into ℂ. For any such embedding ι : k′ ↪ ℂ, one has a commutative diagram

T_{ℤ}L^{n}(′X_{•},′ Y_{•}) ⊗ ℚ ⟶ H^{n}(′U_{•} × ℂ, ℚ(1))
T θ_{L}^{∗}↓  _{θ^{∗}}↓^{≀}
T_{ℤ}L^{n}(X_{•}, Y_{•}) ⊗ ℚ ⟶ H^{n}(U_{•} × ℂ, ℚ(1));

the isomorphism θ^{∗} [14, 8.2.2] of mixed Hodge structures provides an isomorphism t^{n}(′U_{•}) ⟶^{∼} t^{n}(U_{•}). By Theorem 3.2 applied to ′U_{•} × ℂ and U_{•} × ℂ, the map T θ_{L}^{∗} is an isomorphism of ℚ-mixed Hodge structures. By [14, 10.1.3], θ_{L}^{∗} is an isogeny. □

Definition 5.3. The isogeny one-motive L^{n}(V) ⊗ ℚ is the isogeny class of the onemotive L^{n}(X_{•}, Y_{•}) of any simplicial pair (X_{•}, Y_{•}) corresponding to a proper hypercovering U_{•} of V. Definition 5.4. Let t_{ℓ}^{n}(V) be the image of t_{ℓ}^{n}(U_{•}) under the map β_{U}, inverse to the isomorphism α^{∗} : H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) ⟶^{∼} H_{et}^{n}(Ū_{•}, ℚ_{ℓ}(1)) for any proper hypercovering α : U_{•} → V corresponding to a simplicial pair (X_{•}, Y_{•}). Also define t_{𝔸}^{n}(V) = β_{U}(t_{𝔸}^{n}(U_{•})), a 𝔸-submodule of H_{et}^{n}(V̄ , 𝔸(1)). Note that t_{𝔸}^{n}(V) and t_{ℓ}^{n}(V) are independent of the choice of the proper hypercovering U_{•} of V.

By (5.1), any two proper hypercoverings of V are dominated by a third one; thus, as in [14, §8.2], the isogeny one-motive L^{n}(V) ⊗ ℚ is well-defined by [44, 5.1.4], [14, 6.2.8]. Lemma 5.5. L^{n}(−) ⊗ ℚ, t_{ℓ}^{n}(−) and t_{𝔸}^{n}(−) are contravariant functorial.

Proof. Given a morphism f : V → W, there exists a proper hypercovering (X_{•}, Y_{•}) (resp. (Z_{•}, I_{•})) of V (resp. W) and a morphism θ between them which lifts f [14, 6.2.8]. This induces a morphism θ^{∗} : L^{n}(W) ⊗ ℚ → L^{n}(V) ⊗ ℚ. One has to check that this induced morphism does not depend upon the choices of the auxiliary hypercoverings and for this we refer to (loc. cit). □

The next lemma, inspired by [40], will be used to deduce an integral version (i.e., an actual one-motive L^{n}(V)) of the isogeny one-motive L^{n}(V) ⊗ ℚ. Let C be the category whose objects are pairs (R, L) where R is an isogeny one-motive over k and L is a 𝔾-stable ℤ̂-lattice of the 𝔸-module T R, and morphisms from (R_{1}, L_{1}) to (R_{2}, L_{2}) are those morphisms ϕ : R_{1} → R_{2} of isogeny one-motives such that the induced map ϕ_{∗} : T R_{1} → T R_{2} satisfies ϕ_{∗}(L_{1}) ⊂ L_{2}. Lemma 5.6. The natural functor M ↦ (M ⊗ ℚ, M ⊗ ℤ̂) from the category M_{k} of one-motives over k to C is an equivalence of categories.

Proof. This follows from [40, 2.2]; the main point is as follows: as ℤ̂-lattices of the 𝔸-module T R are compact and open in T R, any two 𝔾-stable ℤ̂-lattices in T R are commensurable. □

Let V be any variety over k. Consider the 𝔾-stable ℤ̂-lattice

t_{𝔸}^{n}(V) ∩ H_{et}^{n}(V̄ , ℤ̂(1))/torsion ⊂ H_{et}^{n}(V̄ , 𝔸(1))

in t_{𝔸}^{n}(V). Since the map Λ_{et} : T L^{n}(V) ⊗ ℚ → H_{et}^{n}(V̄ , 𝔸(1)) is injective with image t_{𝔸}^{n}(V), this defines a 𝔾-stable ℤ̂-lattice of T L^{n}(V) ⊗ ℚ which, by (5.6), determines a one-motive L^{n}(V) =: L^{n}(V/k) over k. Theorem 5.7. Conjecture 0.1 is true for fields of characteristic zero. The one-motives L^{n}(V/k) defined for an arbitrary variety V over a field k of characteristic zero are functorial in V and k. Furthermore, one has

(i) an injection of 𝔾-modules Λ_{et} : T L^{n}(V/k) ↪ H_{et}^{n}(V̄ , ℤ̂(1))/torsion;

(ii) an injection of k-vector spaces T_{DR}L^{n}(V/k) ↪ H_{DR}^{n}(V);

(iii) a canonical and functorial isomorphism L^{n}(V/ℂ) ⟶^{∼} I^{n}(V).

If k is finitely generated over ℚ, then Λ_{et} is compatible with the weight filtration: Λ_{et}(W_{i}L^{n}(V/k) ⊗ 𝔸) ⊂ W_{i}H_{et}^{n}(V̄ , 𝔸(1)) for i = −2, −1, 0 (equality for i = −2, −1).

Proof. The contravariant functoriality of L^{n}(V) is proved as in (5.5) and compatibility with base change is clear (2.9).

(i) follows from the definition of L^{n}(V/k), (5.2), (4.5), (46); and (ii) follows from (3.18), (47). (iii) follows from the definition of L^{n}(V/k), (3.2), (45), and (41).

The last statement follows from (4.5). □

Remark 5.8. Let N be the dimension of V. The one-motives L^{n}(V) vanish for n > N + 1; this follows from weight considerations [15, 7.3]. If V is smooth, then L^{n}(V) is zero for n > 2 (1.9). □

6. Positive characteristic

In this section, k is a field of characteristic p > 0, k^{perf} its perfection, k^{sep} a separable closure (𝔾_{sep} is the associated Galois group), k̄ (an algebraic closure of k) the compositum of k^{perf} and k^{sep}; ℓ is a prime distinct from p.

Even though we do not assume k to be perfect, all our results involve passage to k^{perf}. For any variety V over a perfect field k and any integer n, we use “neat hypercoverings” to construct L^{n}(V)⊗ℤ[1/p] (0 ≤ n ≤ 2), J^{n}(V)⊗ℤ[1/p] (n ≥ 0) — one-motives up to pisogeny, i.e., objects of M_{k} ⊗ℤ[1/p] — which are contravariant functorial; these provide a partial proof of (0.1) for k. Finally, we reduce Conjecture 0.1 (up to p-isogeny), if k is perfect, to the validity of (5) for surfaces; an analogous result (6.16) holds, even if k is not perfect, under the additional assumption of “resolution of singularities”.

Generalized one-motives.

Let (X_{•}, Y_{•}) be a simplicial pair over S and f be the structure map of X_{≥n−1}. The sheaf R^{1}f_{∗}O^{∗} on S_{fppf} is representable (2.1). The (reduced) neutral component of the corresponding group scheme is not guaranteed to be a semi-abelian variety unless k is perfect. Theorem 6.1. [20, Corollary 2.3, p.288] Let G be a locally algebraic group scheme over a field F. If F is perfect, then the reduced scheme G^{red} is a smooth group scheme.

This motivates the following definition.

Definition 6.2. A generalized one-motive over k is a two-term complex M = [B ⟶^{u} G] of group schemes over S such that, after base change to Spec k^{perf}, [B ⟶^{u} G^{red}] is a one-motive over Spec k^{perf}.

The category M̃_{k} of generalized one-motives over k is an additive category. The functors M ↦ T_{ℓ}M (ℓ ≠ p), M ↦ T^{p}M on M_{k} extend to the category M̃_{k}.

Our constructions in Section 2 provide generalized one-motives L^{n}, J^{n} over k (and one-motives L^{n}, J^{n} over k^{perf}) associated with the simplicial pair (X_{•}, Y_{•}) over S; these are contravariant functorial and they are compatible with base change.

Relating L^{n} to étale cohomology.

For any (simplicial) variety V, set H_{et}^{∗}(V̄ , ℤ^{p}(1)) := \underleftarrow{lim}_{r}H_{et}^{∗}(V̄ , μμ_{r}) with (r, p) = 1, H_{et}^{∗}(V̄ , 𝔸^{p}(1)) := H_{et}^{∗}(V̄ , ℤ^{p}(1)) ⊗ ℚ.

Using the generalized one-motives L^{n}, one checks that the proofs of (4.3), (4.4), (4.5(i)) are valid with 𝔸^{p} instead of 𝔸. Also, (4.2) is valid: the étale analogue E_{∗}^{et} of [14, 8.1.19.1] calculating the étale cohomology of Ū_{•} with ℚ_{ℓ}-coefficients degenerates at E_{2} (4.1). This is a consequence of the weight filtration W [16, 5.3.6] on H^{i}(V̄ , ℚ_{ℓ}) for any scheme V over S. Compatibility with W forces the vanishing of the differentials d_{r} of E_{∗}^{et} for r > 1; cf. [16, 5.3.7] for another application. The point is that given any finite set of (smooth) schemes X_{i}, Ỹ_{j}, there exist an integral scheme V of finite type over Spec 𝔽_{p}, a dominant morphism η : S → V, smooth schemes X_{i}, Ỹ_{j} over Spec 𝔽_{p} with smooth proper morphisms X_{i} → V, Ỹ_{j} → V such that X_{i} ×_{η} S = X_{i}, Ỹ_{j} ×_{η} S = Ỹ_{j}. The terms of the spectral sequence E_{∗}^{et} are pull-backs via η of pure lisse sheaves on V (purity follows from [16, 6.2.6]); cf. [31, p.89, pp.115-117].

Relations with the Tate conjecture.

The proof of the injectivity of the map Λ_{et} : T_{ℓ}L^{n} ⊗ ℚ → H_{et}^{n}(Ū_{•}, ℚ_{ℓ}(1)) of (4.5(ii)) uses (3.10(i)) which, in turn, is based on (4); the injectivity is necessary for a proof of (5.2) in this context. Thus we need a proof (valid in positive characteristic) of the injectivity of

(48)  ν_{et} : B_{n}(S) ⊗ ℚ_{ℓ} → Gr_{0}^{W} H_{et}^{n}(Ū_{•}, ℚ_{ℓ}(1))

where, by the degeneration (4.1) of E_{∗}^{et} at E_{2}, we have — (3.10) —

Gr_{0}^{W} H_{et}^{n}(Ū_{•}, ℚ_{ℓ}(1)) = \frac{Ker(H^{0}(Ỹ_{n−1}) ⊕ H^{2}(X̄_{n−2}) ⟶^{t_{n−1}} H^{0}(Ỹ_{n}) ⊕ H^{2}(X̄_{n−1}))}{Im(H^{0}(Ỹ_{n−2}) ⊕ H^{2}(X̄_{n−3}) ⟶^{t_{n−2}} H^{0}(Ỹ_{n−1}) ⊕ H^{2}(X̄_{n−2}))};

here H^{0}(Ỹ_{m}) = H_{et}^{0}(Ỹ_{m} × S̄, ℚ_{ℓ}) and H^{2}(X̄_{m}) = H_{et}^{2}(X̄_{m}, ℚ_{ℓ}(1)) and the maps t_{m} are

as in (26). By [39, 2.6, p. 224], H_{et}^{2}(X_{m} × k^{sep}, ℚ_{ℓ}(1))) ⟶^{∼} H_{et}^{2}(X̄_{m}, ℚ_{ℓ}(1))).

Lemma 6.3. Let k be a finitely generated extension of the prime field 𝔽_{p}. The map ν_{et} of (48) is injective if either

(i) δ_{n−3}^{∗}H_{et}^{2}(X_{n−3} × k^{sep}, ℚ_{ℓ}(1))^{𝔾_{sep}} is equal to the image of the map

δ_{n−3}^{∗}NS_{n−3}(S) ⊗ ℚ_{ℓ} ↪ NS_{n−2}(S) ⊗ ℚ_{ℓ} ↪ H_{et}^{2}(X_{n−2} × k^{sep}, ℚ_{ℓ}(1)).

or

(ii) the Tate conjecture (5) is true for either X_{n−3} or X_{n−2}. In particular, ν_{et} is injective when n ≤ 2.

Proof. It is straightforward to check that (i) implies the desired injectivity. Let us show that (ii) implies (i). The exact sequences of 𝔾_{sep}-modules

0 → Ker(δ_{n−3}^{∗}) → H_{et}^{2}(X_{n−3} × k^{sep}, ℚ_{ℓ}(1)) → δ_{n−3}^{∗}H_{et}^{2}(X_{n−3} × k^{sep}, ℚ_{ℓ}(1)) → 0,

0 → δ_{n−3}^{∗}H_{et}^{2}(X_{n−3} × k^{sep}, ℚ_{ℓ}(1)) → H_{et}^{2}(X_{n−2} × k^{sep}, ℚ_{ℓ}(1)) → Coker(δ_{n−3}^{∗}) → 0

are split: this follows from [49, 2.10] — see the proof of [49, 5.2(b)]. Thus, as in (loc. cit), these exact sequences remain exact after taking 𝔾_{sep}-invariants. This, by (1.5(v)), suffices for the implication (ii) ⇒ (i). □

Since the map \frac{K^{0}(S)}{W_{n−2}(S)} ⊗ℚ_{ℓ} → B_{n}(S)⊗ℚ_{ℓ} ⟶^{ν_{et}} Gr_{0}^{W} H_{et}^{n}(Ū_{•}, ℚ_{ℓ}(1)) is clearly injective,

we obtain the injectivity of

(49)  Λ_{et} : T_{ℓ}J^{n}(X_{•}, Y_{•}) ⊗ ℚ ↪ H_{et}^{n}(Ū_{•}, ℚ_{ℓ}(1));

its image is denoted s_{ℓ}^{n}(X_{•}, Y_{•}). A similar definition gives s_{𝔸^{p}}^{n}(X_{•}, Y_{•}). Remark 6.4. (i) Without a condition such as the injectivity of ν_{et}, it is hard to show that the Galois representations Λ_{et}(T_{ℓ}L^{n} ⊗ ℚ) are “independent of ℓ”; this is unclear already for their dimensions [36, pp.27-29].

(ii) If n ≤ 2, then ν_{et} is injective; set t_{ℓ}^{n}(X_{•}, Y_{•}) := Im(Λ_{et}), a 𝔾-invariant subspace of H_{et}^{n}(Ū_{•}, ℚ_{ℓ}(1)). A similar definition gives t_{𝔸^{p}}^{n}(X_{•}, Y_{•}). □

Varieties over perfect fields.

From now on, we assume that k is perfect. Let V be an arbitrary variety over S. The results of [9] show [9, §1] [4, 6.3] the existence of a simplicial pair (X_{•}, Y_{•}) over S with a morphism α : U_{•} → V which makes U_{•} a proper hypercovering of V. One has (46) α^{∗} : H_{et}^{∗}(V̄ , ℤ^{p}(1))⟶^{∼}H_{et}^{∗}(Ū_{•}, ℤ^{p}(1)); as before, β_{U} denotes the inverse of α^{∗}. The methods of [14, 6.2] imply, using [9], the abundance of smooth hypercoverings so that (5.1) is valid as well. Definition 6.5. Let^{8} s_{ℓ}^{n}(V) be the direct limit of β_{U}(s_{ℓ}^{n}(X_{•}, Y_{•})) over all proper hypercoverings α : U_{•} → V. It is a ℚ_{ℓ}-subspace of H_{et}^{n}(V̄ , ℚ_{ℓ}(1)), with an action of 𝔾. A similar definition holds for t_{ℓ}^{n}(V) (n ≤ 2).

Since H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) [36, p. 24] is a finite dimensional vector space over ℚ_{ℓ}, there is actually a (by no means unique) proper hypercovering U_{•} of V such that β_{U}(s_{ℓ}^{n}(X_{•}, Y_{•})) = s_{ℓ}^{n}(V); let us call such hypercoverings “neat”. Proposition 6.6. The notion of “neat” does not depend on the auxiliary prime ℓ ≠ p.

Proof. Clear: the map Λ_{et} of (49) is injective which means that the dimension of s_{ℓ}^{n}(X_{•}, Y_{•}) is independent of ℓ; similarly, the dimension of s_{ℓ}^{n}(V) is independent of ℓ. □

Lemma 6.7. (i) Any proper hypercovering of V which dominates a “neat” proper hypercovering of V is “neat”.

(ii) Any two “neat” proper hypercoverings are dominated by a “neat” proper hypercovering.

(iii) More generally, given a pair of proper hypercoverings one of which is “neat”, there is a proper hypercovering (automatically “neat”) dominating them.

(iv) Any proper hypercovering is dominated by a “neat” proper hypercovering. □ The advantage of “neat” hypercoverings is the

Lemma 6.8. For any morphism θ between two “neat” proper hypercoverings U_{•} and ′U_{•} of V, the induced map θ^{∗} : J^{n}(′X_{•},′ Y_{•}) ⊗ ℚ → J^{n}(X_{•}, Y_{•}) ⊗ ℚ is an isomorphism.

Here J^{n}(−) ⊗ ℚ denotes the isogeny one-motive obtained from the generalized onemotive — see (6.2).

Proof. It is enough to show that the map θ^{∗} : T_{ℓ}J^{n}(′X_{•},′ Y_{•}) ⊗ ℚ → T_{ℓ}J^{n}(X_{•}, Y_{•}) ⊗ ℚ is an isomorphism. This map is always injective (for arbitrary proper hypercoverings). But if the hypercoverings are “neat”, then both terms are actually equal to s_{ℓ}^{n}(V). □

Our definition of “neat” depends upon the integer n. But since H_{et}^{i}(V̄ , 𝔸^{p}(1)) = 0 for i > 2 dim V [36, pp. 23-24], there exist, by (6.7), “neat” hypercoverings such that β_{U}(s_{ℓ}^{n}(X_{•}, Y_{•})) = s_{ℓ}^{n}(V) holds for all n. From now on, let us consider only such “neat” hypercoverings. Definition 6.9. We define J^{n}(V) ⊗ ℚ to be the isogeny one-motive J^{n}(X_{•}, Y_{•}) ⊗ ℚ of any “neat” proper hypercovering U_{•} of V.

The following theorem is a trivial consequence of the previous definition; the weight filtration on étale cohomology is given by [16, 5.3.6].

8This follows a suggestion of M. Marcolli.

Theorem 6.10. One has 𝔾-equivariant injections

Λ_{et} : T_{ℓ}W_{i}J^{n}(V) ⊗ ℚ ↪ W_{i}H_{et}^{n}(V̄ , ℚ_{ℓ}(1));

these are isomorphisms for i = −2, −1. Similar statement holds for Λ_{et} : T^{p}J^{n}(V) ⊗ ℚ ↪ H_{et}^{n}(V̄ , 𝔸^{p}(1)). □ Remark 6.11. Note s_{ℓ}^{n}(V) is a subspace (possibly proper) of H_{et}^{n}(V̄ , ℚ_{ℓ}(1)); the allowed weights on s_{ℓ}^{n}(V) are −2, −1 and 0; the allowed weights on H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) lie between −2 and 2n − 2. A similar statement is true for s_{𝔸^{p}}^{n}(V) := Λ_{et}(T^{p}J^{n}(V) ⊗ ℚ) ⊂ H_{et}^{n}(V̄ , 𝔸^{p}(1)). □ Remark 6.12. (Functoriality) Given any morphism f : Z → V, there is an induced morphism f^{∗} : J^{n}(V)⊗ℚ → J^{n}(Z)⊗ℚ: given f, pick a “neat” proper hypercovering U_{•} of V. One can find a proper hypercovering E_{•} of Z and a morphism F : E_{•} → U_{•} lifting f. By (6.7), “neat” proper hypercoverings of Z are cofinal among proper hypercoverings of Z. So we may choose E_{•} to be “neat”. This yields the functoriality of J^{n}(−) ⊗ ℚ, s_{ℓ}^{n}(−), and s_{𝔸^{p}}^{n}(−). □ Remark 6.13. (i) A variant of (5.6) shows that the 𝔾-invariant ℤ^{p}-lattice

(s_{𝔸^{p}}^{n}(V) ∩ H_{et}^{n}(V̄ , ℤ^{p}(1))/torsion) ⊂ H_{et}^{n}(V̄ , 𝔸^{p}(1))

provides a ℤ[1/p]-integral structure on J^{n}(V) ⊗ ℚ, i.e., determines J^{n}(V) ⊗ ℤ[1/p], a one-motive up to p-isogeny, defined over k. Controlling p would require an integral p-adic cohomology for arbitrary varieties.

(ii) It is probable that the p-adic/crystalline realization of J^{n}(V) ⊗ ℚ is related to the rigid cohomology of V [50]. □ The one-motives up to p-isogeny L^{n}(V) ⊗ ℤ[1/p].

Our methods for the construction of J^{n}(V) ⊗ ℤ[1/p] show the following Theorem 6.14. If (5) is true for any surface over any finitely generated (over 𝔽_{p}) subfield of a perfect field k, then (0.1) (up to p-isogeny) is true for k.

Proof. Using the injectivity of ν_{et} of (48) assured by (6.3), we can apply the previous methods to define L^{n}(V) ⊗ ℚ (using a variant of “neat” hypercoverings) and refine it, as in (6.13(i)), to L^{n}(V) ⊗ ℤ[1/p]. The fact that the Tate conjecture (5) for divisors reduces to the case of surfaces is well-known; this is proved along the ideas of the proof of (6.3) using the weak Lefschetz theorem [16, 4.1.6]. □

Remark 6.15. (i) By (6.3), ν_{et} of (48) is injective for n ≤ 2. In this case, our methods provide L^{n}(V) ⊗ ℤ[1/p] (for n ≤ 2), one-motives up to p-isogeny, associated with V which are contravariant functorial. It follows from the definitions (20), (21) that J^{n}(V) ⊗ ℤ[1/p] = L^{n}(V) ⊗ ℤ[1/p] for n ≤ 1.

(ii) If the field k in (6.14) is not taken to be perfect, then one obtains that L^{n}(V) ⊗ ℤ[1/p] (attached to a variety V over k) is defined over k^{perf}. □ Theorem 6.16. Let F be a field of characteristic p > 0; write F^{perf} for its perfection. Assume that “resolution of singularities” holds over F and that (5) holds for any surface over any finitely generated (over 𝔽_{p}) subfield of F. Then, (0.1) (up to p-isogeny) is true for F.

Proof. Straightforward variant of (6.14). □

7. Applications and related results

Related work.

For any curve C, Deligne [14, 10.3] constructed a one-motive H_{m}^{1}(C)(1) (isomorphic to our L^{1}(C)) and used it to prove Theorem 3.2 for the H^{1} of a curve over ℂ. The one-motive L^{2}(V) of a projective complex surface V was already obtained by J. Carlson [6]. Carlson mentions in [7] that he has constructed other one-motives for a special class of varieties (over ℂ) but these results remain unpublished (email, 3 Dec 2001). The one-motive L^{1}(X_{•}, Y_{•}) is the Picard one-motive P ic^{+} of [3] and M^{1} of [43]. Finally, [2] contains independent proofs of some of our results. Motivic principles: illustrations. [33, 1.7, 2.5]

Let V be a variety over a field k of characteristic zero; by (2.9), we may assume k to be finitely generated over ℚ without loss of generality. Proposition 7.1. The rank of

H_{ℚ}^{1,1}(V_{ι})^{n} := Hom_{MHS}(ℚ(0), Gr_{0}^{W} H^{n}(V_{ι}, ℚ(1))),

(the so-called (1, 1)-part) is independent of the imbedding ι : k ↪ ℂ.

Proof. This follows from Theorem 3.2 since the dimension of the left hand side is the rank of B_{n}(ℂ) of the one-motive L^{n}(V). □

Remark 7.2. Put k = ℂ; (7.1) yields an algebraic characterization of the (1, 1)-part of H^{n}(V, ℚ) (1.7). The analogous statement for the (m, m)-part of H^{n}(V, ℚ) is not known (for m > 1) in general; it is part of the (homological) Hodge conjecture [31, 7.2]. □

Consider the Galois representations M_{ℓ} := H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) (these have a weight filtration W by Galois submodules [15, §13, §14]. For each prime ℓ, h_{ℓ}(V) := W_{−1}M_{ℓ} defines an element ζ_{ℓ} ∈ Ext_{𝔾}^{1}(Gr_{−1}^{W}M_{ℓ}, Gr_{−2}^{W}M_{ℓ}). For each ι : k ↪ ℂ, the substructure h_{ι}(V) := W_{−1}M_{ι}(V) of the mixed Hodge structure M_{ι} := H^{n}(V_{ι}, ℚ(1)) defines an element ζ_{ι} ∈ Ext_{MHS}^{1}(Gr_{−1}^{W}M_{ι}, Gr_{−2}^{W}M_{ι}).

The isogeny one-motive W_{−1}L^{n} ⊗ ℚ can be viewed as extension ζ of two isogeny one-motives given by the exact sequence (14):

ζ : 0 → [0 → T ] ⊗ ℚ → [0 → P̃_{n}] ⊗ ℚ → [0 → R] ⊗ ℚ → 0

The element ζ ∈ Ext^{1}(R, T ) ⊗ ℚ is zero if and only if P̃_{n} is isogenous to the direct product R × Q. Corollary 7.3. One has

(a) ζ is zero ⇔ ζ_{ι} is zero for all ι ⇔ ζ_{ι} is zero for one ι : k ↪ ℂ.

(b) ζ is zero ⇔ ζ_{ℓ} is zero for all primes ℓ ⇔ ζ_{ℓ} is zero for one ℓ.

Proof. The relation between ζ, ζ_{ℓ}, and ζ_{ι} is given by (4.5) and (3.4). Statements (a) and (b) follow easily from [33, Thm. 4.3] and [14, 10.1.3] that the ℓ-adic (or Hodge) realizations of an isogeny one-motive is a direct sum of pure objects if and only if the isogeny one-motive is isogenous to a direct product. □

Independence of the prime ℓ in étale cohomology.

We now take k to be a finite field; and let Φ ∈ 𝔾 be the Frobenius automorphism.

The weight filtration W on H_{et}^{n}(V̄ , ℚ_{ℓ}) is defined via the endomorphism Φ_{ℓ}^{∗} of H_{et}^{n}(V̄ , ℚ_{ℓ}) induced by Φ. Let b_{i,ℓ,n}(V) be the dimension of the ℚ_{ℓ}-vector space Gr_{i}^{W} H_{et}^{n}(V̄ , ℚ_{ℓ}); we recall the well-known results: Lemma 7.4. (Deligne) [13] [16, 3.3.9] If V is smooth and proper, then

(1) b_{i,ℓ,n}(V) = 0 if i ≠ n.

(2) b_{n,ℓ,n}(V) is independent of ℓ; thus, we may set b_{n}(V) = b_{n,ℓ,n}(V).

(3) the characteristic polynomial of Φ_{ℓ}^{∗} on H_{et}^{n}(V̄ , ℚ_{ℓ}) has coefficients in ℚ independent of the prime ℓ.

(4) [16, 4.1.5] (V projective) b_{2n+1}(V) is even.^{9} □ Question 7.5. (N. Katz) [36, pp. 27-29] Which parts of (7.4) are valid for b_{i,ℓ,n}(V), Gr_{i}^{W} H_{et}^{n}(V̄ , ℚ_{ℓ}) for general V, i.e., for V possibly singular and non-proper?

This is not answered by a formal application of [13] and resolution of singularities; cf. [9, §1]. Genuinely new ingredients are needed for an answer in general. Katz [35] has proved it for certain smooth varieties; no general results are known for singular varieties. A consequence of (7.7(v)) is: Theorem 7.6. Fix an arbitrary variety V and an integer n ≥ 0. The following systems of Galois representations satisfy (2) and (3) of (7.4):

(i) W_{−2}H_{et}^{n}(V̄ , ℚ_{ℓ}(1)). (ii) Gr_{−1}^{W}H_{et}^{n}(V̄ , ℚ_{ℓ}(1)). (iii) W_{−1}H_{et}^{n}(V̄ , ℚ_{ℓ}(1)). □ Let c_{n}(V) denote the rank (over ℤ) of the Galois module I_{n}(S̄); here I_{n} is the “lattice”

in the one-motive J^{n}(V) = [I_{n} → P_{n}]. Theorem 7.7. (i) The integers b_{−2,ℓ,n}(V) and b_{−1,ℓ,n}(V) are independent of ℓ ≠ p.

(ii) b_{−1,ℓ,n}(V) is an \underline{even} integer; cf. [15, 16.1].

(iii) b_{0,ℓ,n(V)} ≥ c_{n}(V).

(iv) Let f : V → V be any morphism. The characteristic polynomial of the induced endomorphism f_{ℓ}^{∗} : s_{ℓ}^{n}(V) → s_{ℓ}^{n}(V) has ℚ-coefficients which are independent of ℓ.

(v) The characteristic polynomial of the automorphism Φ_{ℓ} on the 𝔾-submodule s_{ℓ}^{n}(V) of H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) has ℚ-coefficients which are independent of ℓ.

(vi) For any morphism g : V → W, we have a commutative diagram

s_{ℓ}^{n}(W)  ⟶^{g_{ℓ}^{∗}}  s_{ℓ}^{n}(V)
↓  ↓
H_{et}^{n}(W̄ , ℚ_{ℓ}(1)) ⟶ H_{et}^{n}(V̄ , ℚ_{ℓ}(1));

the characteristic polynomials of Φ_{ℓ} on the 𝔾-modules Ker(g_{ℓ}^{∗}) and Coker(g_{ℓ}^{∗}) have ℚ-coefficients which are independent of ℓ.

Proof. Write P as an extension of an abelian variety A by a torus T ; it follows from (4.5) that b_{−2,ℓ,n}(V) (resp. b_{−1,ℓ,n}(V)) are the dimensions of T (resp. A). This proves (i) and (ii). (iii) also follows from (4.5) since we know (49) that I(S̄) ⊗_{ℤ} ℚ_{ℓ} injects into Gr_{0}^{W} H_{et}^{n}(V̄ , ℚ_{ℓ}(1)).

The endomorphism algebra End(J^{n}(V) ⊗ ℚ) is a finite dimensional ℚ-algebra. By functoriality of J^{n}(−) ⊗ ℚ, the morphism f induces an element f^{∗} ∈ End(J^{n}(V) ⊗ ℚ).

9Deligne (loc. cit) remarks that this is not yet known for V proper smooth.

The characteristic polynomial of f^{∗} is a polynomial with ℚ-coefficients. Since s_{ℓ}^{n}(V) is the ℓ-adic realization of J^{n}(V) ⊗ ℚ, by functoriality, the map f_{ℓ}^{∗} on s_{ℓ}^{n}(V) has the same characteristic polynomial. This proves (iv). The same argument proves (v): the Frobenius morphism F_{V} of V and the geometric Frobenius Φ^{−1} induce the same endomorphism on H^{∗}(V̄ , ℚ_{ℓ}(1)) [13, 1.15].

The map g : V → W induces a map g^{∗} : J^{n}(W) ⊗ ℚ → J^{n}(V) ⊗ ℚ. Since the category of isogeny one-motives is abelian, we have the isogeny one-motives Ker(g^{∗}) and Coker(g^{∗}). Their ℓ-adic realizations are the Galois modules Ker(g_{ℓ}^{∗}) and Coker(g_{ℓ}^{∗}). Apply the argument in the previous paragraph to End(Ker(g^{∗})) and End(Coker(g^{∗})). This proves (vi). □

Rationality of systems of certain ℓ-adic Galois representations.

Question 7.8. (J.-P. Serre) [46, I-10] [47, 12.5?] For a fixed variety V over a number field k and integer n, is the system of Galois representations H_{et}^{n}(V̄ , ℚ_{ℓ}) “rational”?

The same question for the systems W_{i}H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) clearly refines the above one. For smooth proper V, Deligne’s theorem (Weil conjectures) [16] provides an affirmative answer (see [47] Exemple after 12.5? on page 393). For any imbedding ι : k ↪ ℂ, the weight filtrations on H^{n}(V_{ι}, ℚ(1)) and H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) are compatible [15, §14]. But this does not imply the “rationality” of the system W_{i}H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) in general. However, the combination of (loc. cit) and the proof of (7.7 (v)) yields Theorem 7.9. The system of Galois representations W_{j}H_{et}^{n}(V̄ , ℚ_{ℓ}(1)) is “rational” for j = −2, −1 as is the system t_{ℓ}^{n}(V). □ Acknowledgements. I heartily thank S. Lichtenbaum for his constant encouragement, guidance, and support. I would like to express my gratitude to S. Bloch, P. Deligne, H. Gillet, J. Gordon, M. Marcolli, J. Milne, M. Nori, and B. Totaro. The debt to the work of Deligne and Carlson should be evident. I would like to thank the referees for many constructive remarks and suggestions.

“iyaM visRSTiryata AbabhUva yadi vA dadhe yadi vA na yo asyAdhyakSaH parame vyoman so aNga veda yadi vA naveda”

Nasadiya Sukta (Rigveda X 129).

References

1. M. Artin, Théorème de changement de base par un morphisme lisse, et applications, Théorie des topos et cohomologie étale des schémas. Tome 3, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 305, pp. 206–249. MR 50 #7132

2. L. Barbieri-Viale, A. Rosenschon, and M. Saito, Deligne’s conjecture on 1-motives, (2001), Preprint. Available at http://xxx.lanl.gov/abs/math.AG/0102150.

3. L. Barbieri-Viale and V. Srinivas, Albanese and Picard 1-motives, Mém. Soc. Math. Fr. (N.S.) (2001), no. 87, vi+104. MR 1 891 270

4. P. Berthelot, Altérations de variétés algébriques (d’après A. J. de Jong), Astérisque (1997), no. 241, Exp. No. 815, 5, 273–311, Séminaire Bourbaki, Vol. 1995/96. MR 98m:14021

5. S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer-Verlag, Berlin, 1990. MR 91i:14034 6. J. A. Carlson, The one-motif of an algebraic surface, Compositio Math. 56 (1985), no. 3, 271–314.

MR 87g:14005

7. , The geometry of the extension class of a mixed Hodge structure, Algebraic geometry, (Proc. Sympos. Pure Math., Vol. 46, Part 2, Bowdoin, 1985), Amer. Math. Soc., Providence, RI, 1987, pp. 199–222. MR 89b:32036

8. E. B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107–209 (1971). MR 43 #5529

9. A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, 51–93. MR 98e:14011

10. P. Deligne, Letter to the author. 15 April 1998.

11. , Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 425–430. MR 56 #356

12. , Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57. MR 58 #16653a

13. , La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273–307. MR 49 #5013

14. , Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5–77. MR 58 #16653b

15. , Poids dans la cohomologie des variétés algébriques, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, pp. 79–85. MR 55 #5634

16. , La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137–252. MR 83c:14017

17. , Le groupe fondamental de la droite projective moins trois points, Galois groups over Q (Berkeley, CA, 1987), Springer, New York, 1989, pp. 79–297. MR 90m:14016

18. , À quoi servent les motifs?, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 143–161. MR 95c:14013

19. P. Deligne, J. S. Milne, A. Ogus, and K-y. Shih, Hodge cycles, motives, and Shimura varieties, Springer-Verlag, Berlin, 1982, Lecture Notes in Mathematics, Vol. 900. MR 84m:14046

20. M. Demazure and P. Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland Publishing Co., Amsterdam, 1980. MR 82e:14001

21. F. El Zein, Introduction à la théorie de Hodge mixte, Hermann, Paris, 1991. MR 93c:14013

22. H. Esnault and E. Viehweg, Deligne-Beĭlinson cohomology, Beĭlinson’s conjectures on special values of L-functions, Academic Press, Boston, MA, 1988, pp. 43–91. MR 89k:14008

23. H. Gillet, Universal cycle classes, Compositio Math. 49 (1983), no. 1, 3–49. MR 84j:14016

24. M. Green, J. Murre, and C. Voisin, Algebraic cycles and Hodge theory, Springer-Verlag, Berlin, 1994, Lecture Notes in Mathematics, Vol. 1594. MR 96c:14001

25. A. Grothendieck, Crystals and the de Rham cohomology of schemes, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 306–358. MR 42 #4558

26. , Le groupe de Brauer. III. Exemples et compléments, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 88–188. MR 39 #5586c

27. F. Guillén, V. Navarro Aznar, P. Pascual Gainza, and F. Puerta, Hyperrésolutions cubiques et descente cohomologique, Springer-Verlag, Berlin, 1988, Lecture Notes in Mathematics, Vol. 1335. MR 90a:14024

28. R. Hartshorne, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. (1975), no. 45, 5–99. MR 55 #5633

29. L. Illusie, Report on crystalline cohomology, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R.I., 1975, pp. 459– 478. MR 52 #13845

30. U. Jannsen, Deligne homology, Hodge-D-conjecture, and motives, Beĭlinson’s conjectures on special values of L-functions, Academic Press, Boston, MA, 1988, pp. 305–372. MR 89h:14016 31. , Mixed motives and algebraic K-theory, Springer-Verlag, Berlin, 1990, Lecture Notes in

Mathematics, Vol. 1400. MR 91g:14008

32. , Motivic sheaves and filtrations on Chow groups, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 245–302. MR 95c:14006

33. , Mixed motives, motivic cohomology, and Ext-groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 667–679. MR 97j:14022

34. N. M. Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118. MR 49 #2728

35. , Wild ramification and some problems of “independence of l”, Amer. J. Math. 105 (1983), no. 1, 201–227. MR 85f:14021

36. , Review of l-adic cohomology, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 21–30. MR 95j:14021

37. B. Mazur and W. Messing, Universal extensions and one dimensional crystalline cohomology, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathematics, Vol. 370. MR 51 #10350

38. W. Messing, The universal extension of an abelian variety by a vector group, Symposia Mathematica, Vol. XI (Convegno di Geometria, INDAM, Rome, 1972), Academic Press, London, 1973, pp. 359–372. MR 49 #2760

39. J. S. Milne, Étale cohomology, Princeton University Press, Princeton, N.J., 1980. MR 81j:14002 40. J. S. Milne and N. Ramachandran, Integral motives and special values of zeta functions, (2002),

Preprint. Available at http://xxx.lanl.gov/abs/math.NT/0204065.

41. F. Oort, Commutative group schemes, Springer-Verlag, Berlin, 1966. MR 35 #4229

42. N. Ramachandran, From Jacobians to one-motives: exposition of a conjecture of Deligne, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), Amer. Math. Soc., Providence, RI, 2000, pp. 215–234. MR 2001k:14022

43. , Duality of Albanese and Picard 1-motives, K-Theory 22 (2001), no. 3, 271–301. MR 2002e:14013

44. B. Saint-Donat, Techniques de descente cohomologique, Théorie des topos et cohomologie étale des schémas. Tome 2, Springer-Verlag, Berlin, 1972, (SGA 4), Lecture Notes in Mathematics, Vol. 270, pp. 83–162. MR 50 #7131

45. J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955– 1956), 1–42. MR 18,511a

46. , Abelian l-adic representations and elliptic curves, second ed., Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 91b:11071

47. , Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 377–400. MR 95m:11059

48. , Morphismes universels et différentiels de troisième éspèce, Exposés de Séminaires (1950-1999), Documents Mathématiques 1, Soc. Math. France, Paris, 2001.

49. J. Tate, Conjectures on algebraic cycles in l-adic cohomology, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 71–83. MR 95a:14010

50. N. Tsuzuki, Cohomological descent of rigid cohomology for proper coverings, (2001), Preprint.

51. V. Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238. MR 1 764 202

52. A. Weil, Variétés abéliennes et courbes algébriques, Hermann & Cie., Paris, 1948. MR 10,621d

Department of Mathematics, University of Maryland, College Park MD 20742, USA

Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

E-mail address: atma@math.umd.edu