DUALITY OF ALBANESE AND PICARD 1-MOTIVES

NIRANJAN RAMACHANDRAN

Abstract. We define Albanese and Picard 1-motives of smooth (simplicial) schemes over a perfect field. For smooth proper schemes, these are the classical Albanese and Picard varieties. For a curve, these are the homological 1-motive of Lichtenbaum and the motivic H^{1} of Deligne. This paper proves a conjecture of Deligne about providing an algebraic description, via 1-motives, of the first homology and cohomology groups of a complex algebraic variety. (L. Barbieri-Viale and V. Srinivas have also proved this independently.) It also contains a purely algebraic proof of Lichtenbaum’s conjecture that the Albanese and the Picard 1-motives of a (simplicial) scheme are dual. This gives a new proof of an unpublished theorem of Lichtenbaum that Deligne’s 1-motive of a curve is dual to Lichtenbaum’s 1-motive.

0. Introduction

Let k be a perfect field. Two classical invariants of any smooth projective variety X over k are the abelian varieties Alb(X) (the Albanese variety), and P ic(X) (the Picard variety) [5]. It is a classical observation of W. L. Chow, J. Igusa, and A. Weil that Alb(X) and P ic(X) are naturally dual.

In this paper, we extend this classical result, using 1-motives, to arbitrary schemes over k. P. Deligne’s technique [6] §6.3 of defining a cohomological invariant for arbitrary varieties consists of defining it for simplicial schemes X_{•} (this is the complement of a simplicial divisor with normal crossings E_{•} in a smooth projective simplicial scheme X′_{•}) and using smooth hypercoverings to define it for varieties (see §7). For any such scheme X_{•}, we will define the Albanese 1-motive M_{1}(X_{•}) and the Picard 1-motive M^{1}(X_{•}). Our construction of the covariant functor M_{1} is based on the generalized Albanese variety of Rosenlicht-Lang-Serre. The construction of the contravariant functor M^{1} is based upon the Picard scheme of a simplicial scheme.

Our main result (Theorem 5.1) is that, for any X_{•}, the 1-motives M^{1} and M_{1} are naturally dual. This was conjectured by S. Lichtenbaum. When X_{•} = X′_{•} is a constant scheme corresponding to a smooth projective variety V (i.e., each X_{i} = V), Theorem 5.1 reduces to the classical duality of Alb(V) and P ic(V).

For any curve C, Deligne has defined a 1-motive H_{m}^{1}(C)(1) and obtained, via the realization functors on one-motives, the ℓ-adic cohomology, De Rham cohomology etc., of C from H_{m}^{1}(C)(1). Lichtenbaum [21] showed that a 1motive appears, rather naturally, in the Suslin complex of the curve C. This led him to define h_{1}(C), the homological 1-motive of C. The homological 1-motive of a smooth curve is readily seen to be a generalized Jacobian of M. Rosenlicht [32, 33].

Date: 30 September 2000.

Supported by NSF and the Alfred P. Sloan Doctoral Dissertation Fellowship, 1995-96.

Associated with any curve C is a canonical smooth simplicial scheme Z_{•} (§6; [6] §10.3). We show that M_{1}(Z_{•}) = h_{1}(C) and M^{1}(Z_{•}) = H_{m}^{1}(C)(1). As a consequence of Theorem 5.1, we obtain a new proof of an unpublished theorem of Lichtenbaum (Theorem 6.8): the 1-motive h_{1}(C) is naturally dual to the 1-motive H_{m}^{1}(C)(1).

Let X_{•} be a scheme (as above) over ℂ. We also show that the Hodge realization of M^{1}(X_{•}) is the mixed Hodge structure H^{1}(X_{•}; ℤ(1)). It follows from the duality theorem (Theorem 5.1) that the Hodge realization of M_{1}(X_{•}) is the mixed Hodge structure on H_{1}(X_{•})/torsion. This settles the conjecture of Deligne for I(H^{1}(X_{•})(1)) (§10.4.1 of [6]). Results about the other realizations of these 1-motives will appear elsewhere.

Weak Lefschetz-type theorems are available for varieties which are either non-singular quasi-projective or affine ([14] pp. 23-24); it is also remarked in (loc. cit) that such theorems do not hold in general for arbitrary varieties. Therefore, the study of the motivic H^{1} and H_{1} for general varieties does not seem to reduce to the case of curves. This means that the results obtained here about the motivic H^{1} and H_{1} are perhaps of interest. Related work in this direction includes [27], [12], [1], [2], [11], [22], and [30]. I should mention that [30] generalizes to H^{n} (for n > 1) ^{1} the results obtained here for H^{1}. But the Albanese 1-motive and the duality theorem are not touched there. Recently L. Barbieri-Viale and V. Srinivas [2] (announced in [1]) have independently proved Theorems 3.2 and 4.4. One (the essential) difference between their work and ours is that their Albanese 1-motive is defined to be the dual of the Picard 1-motive. Another is that they define their invariants only in characteristic zero, whereas we do so for an arbitrary perfect field (The restriction to perfect fields is forced by Theorem 3.4).

0.1. Notations. We use the following conventions.

S = Spec k, the spectrum of a perfect field k.

The characteristic of k is denoted by p.

S̄ = Spec k̄, the spectrum of an algebraic closure k̄ of k.

g = Gal(k̄/k), the Galois group of k̄ over k.

We consider only separated schemes X over S. Such schemes are supposed to be of finite type over S unless otherwise mentioned. We use X̄ to denote the S̄-scheme X ×_{S} S̄ defined by the S-scheme X.

X̃ := the normalization of X.

X^{red} := the maximal open and closed (reduced) subscheme of X.

w(X):= the set of connected components of X̃ ×_{S} S̄ = the set of irreducible components of X̄. A variety is a scheme V over S such that V̄ is a integral scheme. We use S and k interchangeably: we say k-points of a scheme to mean its points

1In [30], the conjecture of Deligne is proved up to isogeny for H^{n}(V); here, our theorem proves the integral version of the conjecture for H^{1}.

with values in S.

The group schemes we consider are commutative. For such a group scheme G, let G^{0} denote the connected component of the identity and G^{0,red} denote the connected component of the identity of the reduced scheme G^{red}. For a positive integer r prime to p, we denote by _{r}G the group scheme corresponding to Ker(G ⟶^{r} G), i.e., the r-torsion subgroup scheme of G.

A semiabelian scheme is a commutative group scheme which is an extension of an abelian scheme by a torus.

\underline{ℤ} is the locally algebraic group scheme defined by \underline{ℤ}(S̄) = ℤ with trivial g action. 𝔾′ := the category of group schemes over S. 𝔾 := the category of locally algebraic semiabelian schemes over S. 𝔻 := the abelian category of sheaves (of abelian groups) on S_{F L}. S_{F L} := the (big) flat site of S [26]. Properties of a simplicial scheme X_{•} are properties that hold for each X_{i}: X_{•} is

proper if each X_{i} is proper over S, same for smooth, normal, reduced, etc.

Let W be a set, and denote by ℤ[W] the free abelian group generated by W. We define the abelian group ℤ^{W} to be the group of integer valued functions on W i.e. ℤ^{W} := Maps(W, ℤ). For a finite set W, the groups ℤ(W) and ℤ^{W} are naturally dual as abelian groups.

For any finitely generated abelian group G, we put G_{tor} to denote the torsion subgroup of G. We denote by G/tor the maximal torsion-free quotient of G.

MHS:= the abelian category of mixed Hodge structures (MHS).

MH_{pol}^{1} := the category of torsion-free MHS N of type

{(0, 0), (−1, 0), (0, −1), (−1, −1)}

such that Gr_{−1}^{W}N is polarizable [6] §10.1.

𝔗(M) ∈ MH_{pol}^{1} denotes the Hodge realization ([6] 10.1) of a 1-motive M.

We use H_{ℤ} to denote the underlying ℤ-lattice of the mixed Hodge structure H. We use W_{i}M and W_{i}H to indicate the weight filtration on 1-motives and mixed Hodge structures. The Tate Hodge structure ℤ(1) corresponds to the lattice 2πiℤ ⊂ ℂ; the Hodge structure ℤ(n) (for n > 0) is defined to be ℤ(1)^{⊗n}. For n < 0, we define ℤ(n) to be the Hodge structure dual to ℤ(−n) i.e we have ℤ(n) = Hom_{MHS}(ℤ(−n), ℤ). The Tate Hodge structure ℤ(n) is of weight −2n of type (−n, −n). For any mixed Hodge structure H, the Tate twist H(n) denotes H⊗ℤ(n). For any scheme V over ℂ, we use H^{∗}(V) to indicate the mixed Hodge structure H^{∗}(V (ℂ); ℤ). Similar convention for homology applies as well. Given a sheaf F on V, we write H^{i}(V; F) for H^{i}(V (ℂ); F).

Acknowledgements. It is a pleasure for me to thank S. Lichtenbaum for much encouragement. The results presented here would not have been possible without his guidance; that they appear in published form is due to his insistence. This paper is a greatly revised version of [29]. I would like to thank the mathematics department at Brown, the Alfred P. Sloan foundation, and the National Science Foundation (NSF) for their support. Lastly, I heartily thank the referee for the careful reading, enthusiasm, and numerous suggestions thereby improving the exposition.

1. Albanese Schemes

This section is based on a succinct suggestion of J.-P. Serre [37]. The goal is the definition of the Albanese scheme of a scheme X over S. This is different from the (very interesting) universal regular quotient of [11].

1.1. Locally algebraic group schemes.

Definition 1.1. A locally algebraic group scheme is a group scheme locally of finite type over S.

A (group) scheme of finite type will sometimes be called an algebraic (group) scheme. A locally algebraic group scheme G such that G^{0} = S is said to be discrete or of dimension zero; it is an étale S-scheme. Such a group G is determined (up to isomorphism) by the discrete g-module G(S̄).

Recall from [9] pp. 284-6 that there is a discrete locally algebraic group π_{0}(G) associated with a locally algebraic group scheme G, a scheme-theoretic analog of the group of connected components; it is the universal object for homomorphisms from G to discrete locally algebraic group schemes. One has an exact sequence of locally algebraic group schemes:

(1)  0 → G^{0} → G ⟶^{q_{G}} π_{0}(G) → 0

where q_{G} is a homomorphism of group schemes.

1.2. Construction. Let B be a commutative group scheme (algebraic) over S. Given two B-torsors P_{1} and P_{2} (torsor = principal homogeneous space), the B-torsor obtained by Baer summation is denoted P_{1} ∨_{B} P_{2}. For any B-torsor P, one can define a new torsor P^{−1} by twisting the action of B by the automorphism corresponding to the inverse i : B → B. With each B-torsor P, one can associate a locally algebraic group B_{P} as follows. For n positive, let P^{⊗n} denote the B-torsor P ∨_{B} ⋯ ∨_{B} P (n terms). For n negative, let P^{⊗n} denote the B-torsor P^{−1} ∨_{B} ⋯ ∨_{B} P^{−1} (n terms). Let P^{⊗0} be the trivial torsor B. We define

(2)  B_{P} := ∐_{n∈ℤ}P^{⊗n};

note the canonical morphism a_{P} : B_{P} ↠ \underline{ℤ}. The natural morphisms

(3)  P^{⊗n} ×_{S} P^{⊗m} → P^{⊗n} ∨_{B} P^{⊗m} := P^{⊗(m+n)}

endow B_{P} with the structure of a commutative group scheme; B_{P} is an extension of \underline{ℤ} by the group scheme B. Any scheme T defines a sheaf T of sets on S_{F L}: X ↦ Hom_{S}(X, T). There is a natural inclusion of P ↪ B_{P} of sheaves of sets on S_{F L}; in fact, P is the subsheaf of B_{P} corresponding to the inverse image under a_{P} of the section 1 ∈ \underline{ℤ}.

A semiabelian scheme is an algebraic group scheme which is an extension of an abelian scheme by a torus; it is geometrically connected. We call a locally algebraic group scheme G semiabelian if G^{0} is a semiabelian scheme. If B is a semiabelian scheme and P is a torsor under B, then the group scheme B_{P} is a semiabelian locally algebraic group scheme.

Definition 1.2. A large group is a semiabelian locally algebraic group scheme.

We use this terminology because (1) the name is short and (2) a large group is not a scheme of finite type. We denote by 𝔻 the abelian category of sheaves (of abelian groups) on the (big) flat site S_{F L} of S. The category 𝔾 of large groups embeds as a full subcategory (denoted 𝔾) of 𝔻.

1.3. Morphisms to large group schemes. For any set W, let ℤ[W] be the free abelian group generated by the set W. Let X be a scheme. Define P_{X} to be the presheaf of abelian groups on S_{F L} by T ↦ ℤ[X(T)].

Definition 1.3. Z_{X} is the sheaf on S_{F L} associated with the presheaf P_{X}.

The association X ↦ Z_{X} determines a covariant functor from the category of schemes over S to 𝔻. If we consider X as a sheaf X of sets, then we have a canonical inclusion j : X ↪ Z_{X}. The sheaf Z_{X} has the following universal property: Given any abelian sheaf F on S_{F L} and a map g : X → F of sheaves of sets, there is a unique homomorphism of abelian sheaves g̃ : Z_{X} → F which factors g, i.e., g̃j = g.

It is known that the sheaf Z_{X} is not representable in any reasonable topology [40]. One is naturally led to consider the universal representable quotient of Z_{X}; we restrict ourselves to the situation where the representing scheme is a large group.

It is well known that there is an étale algebraic S-scheme π_{0}(X) and a S-morphism q_{X} : X → π_{0}(X) which is universal for morphisms from X into étale S-schemes ([9], Prop 6.5, p.154).

Consider the sheaf Z_{π_{0}(X)} on S_{F L} defined by the scheme π_{0}(X); as above, q̃_{X} denotes the induced homomorphism from Z_{X} → Z_{π_{0}(X)}. The following lemma is well known (loc. cit).

Lemma 1.4. (a) Z_{π_{0}(X)} is represented by a discrete large group D_{X}.

(b) The pair (Z_{π_{0}(X)}, q̃_{X}) is universal for morphisms from Z_{X} to discrete group schemes.

The group D_{X} of a variety X is isomorphic to \underline{ℤ}.

Definition 1.5. The Albanese sheaf of X is the universal object for morphisms from Z_{X} to objects in the subcategory 𝔾 (i.e. sheaves corresponding to large groups).

We say that X admits a universal morphism if the Albanese sheaf of X exists.

Grant, for the moment, the existence of the Albanese sheaf of X. The sheaf A_{X} comes equipped with a surjective homomorphism u_{X} : Z_{X} → A_{X}. Let A_{X} be the large group corresponding to the sheaf A_{X}. The morphism X ↪ Z_{X} → A_{X} of sheafs of sets on S_{F L} corresponds, by Yoneda’s lemma, to a morphism u_{X} : X → A_{X} of schemes.

We call A_{X} the Albanese scheme of X and u_{X} is the universal morphism associated with X. The pair (A_{X}, u_{X}) is uniquely determined (up to unique isomorphism). An easy consequence of definition 1.5 is a canonical isomorphism

(4)  π_{0}(A_{X}) ⟶^{∼} D_{X}.

The assignment X ↦ A_{X} determines a covariant functor A from the category of algebraic schemes over S to large groups over S.

For a scheme W finite over S (e.g., a finite group scheme), we have canonical isomorphisms

(5)  Z_{W} ⟶^{∼} A_{W} ⟶^{∼} D_{W} .

The next proposition follows from the corresponding properties of the functor P ↦ Z_{P}.

Proposition 1.6. (Properties of the Albanese scheme)

(i) (base change) Let S′ be the spectrum of a perfect field k′ containing k. One has A_{X} ×_{S} S′ = A_{X×_{S}S′}.

(ii) Let X := Y ∐ V be the disjoint union of schemes Y and V. One has A_{X} = A_{Y} ×_{S} A_{V}.

(iii) Let X ⟶^{f} Y be a surjective morphism. The induced map f_{∗} : A_{X} → A_{Y} is a surjection.

Remark 1.7. Serre has shown that every scheme X admits a universal morphism to a torsor P under a semiabelian scheme B: in [34], he shows that every scheme X admits a universal morphism (over k̄) to a semiabelian scheme B (the scheme B is defined over k) and the method of descent as in [36] (pp.102-107) assure the existence of a universal map (defined over k) to a torsor P under the universal semiabelian scheme B. This semiabelian scheme B is the “generalized Albanese” of X.

Remark 1.8. Let T (resp. R) be a torsor under a group scheme G (resp. under H). Consider the group schemes G_{T} and H_{R} together with the canonical surjective homomorphisms a_{T} : G_{T} → \underline{ℤ} and a_{R} : H_{R} → \underline{ℤ}. Specifying a morphism h of torsors between T and R is equivalent to specifying a homomorphism of group schemes h̃ : G_{T} → H_{R} such that a_{R}h̃ = a_{T} (i.e. h̃ induces the identity on the quotient \underline{ℤ}).

Let V be a connected scheme (so that there is a map g_{V} : D_{V} → \underline{ℤ}). Let f : V → P be the universal morphism from V to torsors under semiabelian schemes (as in Remark 1.7). Consider the natural inclusion of P in B_{P}. The map f induces a map f̃: Z_{V} → B_{P}. Define B′′ := B_{P} ×\underline{ℤ} D_{V}, where the fiber product is taken via the maps a_{P} : B_{P} → \underline{ℤ} and g_{V}. The map f̃

factorizes as Z_{V} ⟶^{f′′} B′′ → B_{P} by the universal property of D_{V}. The main difference between B′′ and B_{P} lies in their group of connected components.

Proposition 1.9. The pair (B′′, f′′) is the Albanese scheme of V.

Proof. Consider a homomorphism d′ : Z_{V} → H′ to a large group H′. By universality of D_{V}, the map d′ factorizes uniquely as jd where

(a) d : Z_{V} → H is a homomorphism to a large group H inducing an isomorphism D_{V} ⟶^{∼} π_{0}(H); notice H = H′ ×_{π_{0}(H′)} D_{V}.

and

(b) j : H → H′ is a homomorphism of large groups.

It suffices to show that the map d factorizes via (B′′, f′′). Let H denote the sheaf of abelian groups on S_{F L} corresponding to the group scheme H. The composed map d_{1} : V ↪ Z_{V} ⟶^{d} H of sheaves of sets on S_{F L} gives, by Yoneda’s lemma, a S-morphism d_{1} : V → H of schemes. Since V is connected, its image under d_{1} is contained in a connected component Q of H. The neutral component C of H is a semiabelian scheme. The scheme Q has the structure of a torsor under C such that the pair of inclusions Q ↪ H and C ↪ H combine into a morphism of torsors. In other words, there is a (injective) morphism of group schemes k : C_{Q} → H. Now we use the fact that f is universal for morphisms from V to torsors under semiabelian schemes. We get a homomorphism b : B → C and a morphism of torsors h : P → Q such that d_{1} = hf. By Remark 1.8, this is equivalent to specifying a homomorphism h̃ : B_{P} → C_{Q} compatible with the augmentation to \underline{ℤ}. We have shown that d can be factorized as kh̃f̃. But, as we saw, f̃ can be factorized via f′′. This demonstrates the universal property of the pair (B′′, f′′). □

Corollary 1.10. Let A_{V} be the Albanese scheme of a connected scheme V, as above. The neutral component of A_{V} is the “ generalized Albanese variety” of V defined by Serre (1.7).

Theorem 1.11. Let X be a reduced algebraic scheme over S. There exists a large group A_{X} together with a map u_{X} : Z_{X} → A_{X} which is universal for homomorphisms from Z_{X} to large groups.

Proof. We have just seen that the Albanese scheme exists for connected schemes. Now use Proposition 1.6. □

The identification of Corollary 1.10 yields the

Corollary 1.12. (i) Given a torsor P under a semiabelian scheme B, the natural inclusion of P ⟶^{j} B_{P} induces a homomorphism j̃ : Z_{P} → B_{P}. The large group B_{P}, together with the map j̃, is the Albanese scheme of P.

(ii) If X is proper, then A_{X}^{0} is an abelian scheme.

(iii) If U is an open dense subscheme of X, then the induced map A_{U} → A_{X} is surjective.

(iv) (homotopy invariance) As usual, 𝔸^{n} denotes n-dimensional affine space over S. Let Y := X ×_{S} 𝔸^{n}. The natural projection π : Y → X induces an isomorphism π_{∗} : A_{Y} ⟶^{∼} A_{X}.

(v) For any semiabelian scheme B, one has A_{B} = B ×_{S} \underline{ℤ}.

(vi) (Kunneth) Let X := Y ×_{S} V and let α : D_{X} → D_{Y} ×_{S} D_{V} denote the natural map. One has

A_{X} = (A_{Y} ×_{S} A_{V}) ×_{(D_{Y} ×D_{V} )} D_{X}.

Proof. We prove (iv), (v), (vi) and leave the rest to the reader. For (iv), observe that A_{𝔸^{n}} = \underline{ℤ}: there are no nonconstant morphisms from 𝔸^{n} to semiabelian varieties. Statement (v) is a special case of (i) which follows from Proposition 1.9. As for (vi), it is clear that π_{0}(A_{X}) = D_{X}. Now, use the fact that A_{X}^{0} = A_{Y}^{0} ×_{S} A_{V}^{0} which follows (using Corollary 1.10) from the corresponding property for the “generalized Albanese varieties” of Serre (1.7) proved in [34]. □

Example 1.13. A curve is a scheme of pure dimension one over S. Let X be a normal integral curve. There is a unique normal projective curve X′ containing X as an open dense subscheme; let R be the closed complement of X in X′. The semiabelian variety A_{X}^{0} is the generalized Jacobian of Rosenlicht [32] of X′ corresponding to the modulus R (cf. 6.4 and [35] p. 4).

2. Special schemes

Let X be the open complement of a divisor E in a smooth and projective variety X′ of dimension d. Let j : X ↪ X′ denote the inclusion. Schemes such as X are termed special. We fix these notations for the rest of this section. We now turn to a discussion of the “generalized Albanese variety” of X defined by Serre [35].

Let w(E) denote the set of geometric points of π_{0}(Ẽ) i.e. w(E) is the set of connected components of Ẽ ×_{S} S̄ or, what is the same, the set of irreducible components of Ē. Given a set W, one defines the abelian group ℤ^{W} to be the group of integer valued functions on W i.e. ℤ^{W} := Maps(W, ℤ). For a finite set W, the groups ℤ(W) and ℤ^{W} are naturally dual as abelian groups.

The étale group scheme \underline{ℤ}^{E} is defined by its geometric points: \underline{ℤ}^{E}(S̄) = ℤ^{w(E)}. If P ic(X̄′) := H^{1}(X̄′; O∗ ; X̄′) denotes the absolute Picard group of X̄′, there is a natural g-equivariant homomorphism

(6)  ℤ^{w(E)} ⟶^{b} P ic(X̄′)  d ↦ O(d);

this provides a morphism^{2}:

(7)  b : \underline{ℤ}^{w(E)} → D_{P ic_{X′}^{red}}

2P ic_{X′} is the Picard scheme of X′ (see §3)

whose kernel is denoted B; set I = B(S̄). Under the natural duality between ℤ^{w(E)} and ℤ(w(E)), the group I is dual to the subgroup J of divisors of X′ ×_{S} S̄, supported on E ×_{S} S̄, which are algebraically equivalent to zero. Consequently, I and J have the same rank. The Néron-Severi group NS(X̄′) of X̄′ is the group of geometric points of D_{P ic_{X′}}. There is a natural morphism

B ⟶^{v} P ic_{X′}^{0}.

The abelian scheme A_{X′}^{0} is the “classical” Albanese variety of X′. It is the maximal abelian quotient of the semiabelian scheme A_{X}^{0} (the “generalized Albanese variety” of Serre).

2.1. Serre’s Construction. Let us review the construction of A_{X}^{0}, via the Picard scheme of X′, due to Serre [35]. Denote by v the natural map from B to the neutral component P ic_{X′}^{0} of P ic_{X′}. This defines the Picard 1-motive of X

(8)  M^{1}(X) := [B ⟶^{v} P ic_{X′}^{0,red}].

The Cartier dual of this 1-motive is the Albanese 1-motive of X

(9)  M_{1}(X) := [0 → G];

the group G is an extension of A_{X′}^{0} by the torus T := Hom(B, 𝔾_{m}).

Theorem 2.1. (J.-P. Serre) The semiabelian scheme G is naturally isomorphic to A_{X}^{0} [35].

Corollary 2.2. (a) The kernel of the natural morphism A_{X}^{0} → A_{X′}^{0} is the torus T = Hom(B, 𝔾_{m}).

(b) The dimension of the torus T is the rank of the abelian group I.

(c) the 1-motives (8) and (9) are independent of the compactification X′ of X.

Let u : X′ → L′ be a universal morphism (see Remark 1.7) where L′ is a torsor for A_{X′}^{0}. One has the classical isomorphisms

P ic_{A_{X′}^{0}}^{0} ≅ P ic_{L′}^{0} ≅ P ic_{X′}^{0}, (10)

P := P ic_{X′}^{0,red} P ic_{P}^{0,red} ≅ A_{X′}^{0}. (11)

Using this, one interprets the map v as specifying certain 𝔾_{m}-torsors on A_{X′}^{0} (and on L′). The group scheme G is the geometric scheme corresponding to the total space of the T -bundle (given by v) over A_{X′}^{0}. Likewise, L is the scheme over L′ corresponding to the total space of the T -bundle. Using the universal map u, one may pull this back to X′ to get a T -bundle over X′ with the following property: given any element b ∈ Hom(T, 𝔾_{m}) = B, the 𝔾_{m}-torsor over X′ obtained via b corresponds to the line bundle defined by b, viewed as a Cartier divisor on X′. Observe the trivial fact that X is disjoint from the support (contained in E) of all the elements of B(S̄) viewed as Cartier divisors on X′. Therefore, given any b as above, the 𝔾_{m}-torsor defined by b has a canonical trivialization over X: one has a rational section of the 𝔾_{m}-torsor whose divisor is exactly b, and this is unique up to an element of H^{0}(X′; O_{X′}^{∗}). This section can be used for the trivialization over X. Further, this can be done in an uniform manner to derive a morphism X → L, which is unique up to translations by elements of T. Serre [35] shows that this morphism is universal for maps to torsors under semiabelian schemes. This extends, by universality of the sheaf Z_{X}, to a morphism Z_{X} → G_{L} where G_{L} is the large group scheme defined by the G-torsor L (as in 1.2). One deduces that this last morphism is the universal morphism for X.

The next theorem is true by definition of M_{1}(X) and M^{1}(X). But the proof of the corresponding result for simplicial smooth schemes (proved in §5) is nontrivial.

Theorem 2.3. The 1-motives M_{1}(X) and M^{1}(X) are dual.

Let Y be an open subscheme of X whose complement in X has codimension greater than one. Let B_{Y} be the group dual to the group of divisors of X′ ×_{S} S̄ (supported on (X′ − Y) ×_{S} S̄) which are algebraically equivalent to zero.

Corollary 2.4. The inclusion of Y in X induces an isomorphism of the Albanese schemes A_{Y} ⟶^{∼} A_{X}.

Proof. Observe that B and B_{Y} are all the same. □

2.2. Hodge realization. Just as a harbinger of later results, we prove a result about the Hodge realization of M_{1}(X). Consider an imbedding ι : k ↪ ℂ ^{3}. By base-change via ι, we may and do assume (in this subsection) that all our schemes are over ℂ. Since we are over an algebraically closed field, there is no essential difference between a group scheme and its torsors. Let u : X → A_{X}^{0} be the universal morphism (uniquely determined by the choice of a base point). Recall that d is the dimension of X.

Consider the long exact sequence in cohomology (with coefficients in ℤ) of the pair (X′, E):

→ H^{2d−2}(X′)(d) ⟶^{α} H^{2d−2}(E)(d) → H^{2d−1}(X′, E)(d) → H^{2d−1}(X′)(d) → 0

Poincaré duality gives H^{2d−i}(X′)(d) ≅ H_{i}(X′). By Lefschetz duality, we have that H^{2d−i}(X′, E)(d) ≅ H_{i}(X). The image of α is the dual MHS of the

kernel of (H^{0}(E)(−1) ⟶^{β} H^{2}(X′)) where β sends an element of H^{0}(E)(−1) to its Poincaré dual, an element of H^{2}(X′). The group H^{0}(E) is the free abelian group ℤ^{w(E)} i.e. the group dual to ℤ(w(E)). The map b in (7) induces a map on the geometric points and there is a natural inclusion of D_{P ic_{X′}} (ℂ) in H^{2}(X′), via the exponential sequence. The composed map is the map β. The kernel of β corresponds, under the duality of ℤ^{w(E)} and ℤ(w(E)), to the subgroup J of divisors of X′, supported on E, which are homologically equivalent to zero^{4}. Hence we find that Kerβ = I. If I^{∗} = Hom_{ℤ}(I, ℤ), then Im α ≅ Hom_{ℤ}(I(−1), ℤ) = I^{∗}(1); both are MHS of type (−1, −1). So we obtain the exact sequence

3The variety X is always defined over a finitely generated extension of ℚ. So, we may assume that k is finitely generated over ℚ.

0 → I^{∗}(1) → H_{1}(X) → H_{1}(X′) → 0.

Theorem 2.5. (Hodge realization)

(i) The map

u_{∗} : H_{1}(X; ℤ)/tor → H_{1}(A_{X}^{0}; ℤ) =: 𝔗(M_{1}(X))

of mixed Hodge structures is an isomorphism.

(ii) There is a natural isomorphism

𝔗(M^{1}(X)) ⟶^{u^{∗}} H^{1}(X; ℤ(1))

of mixed Hodge structures.

Proof. Recall from ([36] Chap. VI, §14, Prop. 13, p.128) that u_{∗} is surjective; universal morphisms are maximal in the sense of loc. cit.

Consider the following commutative diagram:

0 ⟶ I^{∗}(1) ⟶ H_{1}(X)/tor ⟶ H_{1}(X′)/tor ⟶ 0
↓  ^{u_{∗}}↓  ^{u′_{∗}}↓
0 ⟶ H_{1}(T) ⟶ H_{1}(A_{X}^{0}) ⟶ H_{1}(A_{X′}^{0}) ⟶ 0

All the vertical maps are maps of MHS because they are induced by morphisms. It is a classical result (see p. 553 of [15]) that u′_{∗} : H_{1}(X′)/tor → H_{1}(A_{X′}^{0}) is an isomorphism. By the snake lemma, I^{∗}(1) → H_{1}(T) is surjective. But the rank of H_{1}(T) is the same as the dimension of T which equals the rank of I , as remarked earlier. Since they both have the same rank, the left vertical map is an isomorphism. Thus, we obtain that the map u_{∗} : H_{1}(X)/tor → H_{1}(A_{X}^{0}) is a isomorphism of MHS. By definition, 𝔗(M_{1}(X)) is the mixed Hodge structure H_{1}(A_{X}^{0}; ℤ) [6], 10.1.3 p. 54. We turn to (ii). The universal coefficient theorem says that the groups H_{1}(X; ℤ)/tor and H^{1}(X; ℤ) are naturally dual. So H^{1}(X)(1) and H_{1}(X)/tor are dual as elements of MH_{pol}^{1}, i.e., H^{1}(X)(1) = Hom_{MHS}(H_{1}(X), ℤ(1)). Since duality of 1-motives is compatible with duality in MH_{pol}^{1} ([6], §10.2), the result follows from the first part. □

Remark 2.6. One cannot expect the previous result to be true for an arbitrary complex algebraic variety (thereby justifying the entrance of 1motives in this context): the weights in the H_{1} of a semiabelian variety are -2 and -1, whereas the weights that can and do occur in the H_{1} of an arbitrary variety are -2, -1 and 0 ([7] 7.1, 8.3). A noncompact nonrational curve with nodes provides an easy example of the last property.

4For divisors, homological equivalence coincides with algebraic equivalence.

3. Picard schemes

In this section, we extend the classical construction of the Picard scheme to a simplicial setting. The arguments are based on spectral sequences.

Let f : V → S be a proper reduced scheme of finite type over S.

We define R^{1}f_{∗}O_{V}^{∗} on S_{F L} to be the sheaf associated with the presheaf T ↦ H^{1}(V ×_{S} T; O_{V ×_{S}T}^{∗}). By a theorem of J. Murre and F. Oort ([3], §8.2, Theorem 3, p. 211), the sheaf R^{1}f_{∗}O_{V}^{∗} is representable by a locally algebraic group scheme P ic_{V}. One refers to P ic_{V} as the Picard scheme of V. One obtains that

H^{1}(V̄; O∗ ; V̄) = P ic_{V} (S̄).

We recall the well known ^{5}

Proposition 3.1. (i) (de Rham realization) Let k be of characteristic zero. There is a canonical isomorphism ([24], [25], [6] §10.1)

𝔗_{dR}([0 → P ic_{V}^{0} ]) ≅ H_{dR}^{1}(V).

(ii)(Hodge realization) Let k = ℂ. The exponential sequence and G.A.G.A. furnish a natural isomorphism

H^{1}(V)(1) ⟶^{∼} 𝔗([0 → P ic_{V}^{0} ]).

3.1. The Picard scheme of proper simplicial schemes. Let Z_{•} be a simplicial scheme. The absolute Picard group P ic(Z_{•}) of Z_{•} is the group H_{Zar}^{1}(Z_{•}; O_{Z_{•}}^{∗}) where O_{Z_{•}}^{∗} is the sheaf of units of the structure sheaf O_{Z_{•}} of Z_{•}. We often use the notation O^{∗} to indicate the sheaf of units of a simplicial scheme (specified by the context).

By the definition of cohomology in the simplicial setting ([6], pp. 12-14, esp. 5.2.3 and 5.2.7, [13], Ex. 1.1, p.7), elements of the group P ic(Z_{•}) correspond to isomorphism classes of pairs (L, α) where

(1) L is an invertible sheaf on Z_{0} such that δ_{0}^{∗}L is isomorphic to δ_{1}^{∗}L;

(2) α is an isomorphism δ_{0}^{∗}L ⟶^{∼} δ_{1}^{∗}L on Z_{1} satisfying a cocycle condition on Z_{2}, i.e., the following diagram is commutative:

δ_{2}^{∗}δ_{0}^{∗}L ⟶^{δ_{2}^{∗}α} δ_{2}^{∗}δ_{1}^{∗}L  δ_{0}^{∗}δ_{0}^{∗}L
∥  ^{δ_{0}^{∗}α}↓
δ_{1}^{∗}δ_{0}^{∗}L ⟶^{δ_{1}^{∗}α} δ_{1}^{∗}δ_{1}^{∗}L  δ_{0}^{∗}δ_{1}^{∗}L

The maps δ_{0}, δ_{1} : Z_{1} → Z_{0} and δ_{0}, δ_{1}, δ_{2} : Z_{2} → Z_{1} form part of the simplicial structure of the scheme Z_{•} ([6] III, 5.3.6, pp. 15-16). We write the cocycle condition as

(12)  δ_{2}^{∗}(α)δ_{0}^{∗}(α) = δ_{1}^{∗}(α).

Let V_{•} be a proper reduced simplicial scheme. The structure morphism of V_{n} → S is denoted by f_{n} and the structure map V_{•} → S is denoted by f.

5The crystalline analog may be found in [18] II 3.11 and [19] pp.46-47.

For any scheme T over S, we get a simplicial scheme V_{•} ×_{S} T. By [6] III, 5.2.3.2, p. 13, there is a spectral sequence

(13)  E_{1,T}^{p,q} = H^{q}(V_{p} ×_{S} T; O_{V_{p}×_{S}T}^{∗}) ⟹ H^{p+q}(V_{•} ×_{S} T; O^{∗}).

We define R^{1}f_{∗}O^{∗} to be the sheaf on S_{F L} associated with the presheaf E^{1}: T ↦ H^{1}(V_{•} ×_{S} T; O^{∗}). The first result that one has about this sheaf is the

Theorem 3.2. The sheaf R^{1}f_{∗}O^{∗} on S_{F L} is representable by a locally algebraic group scheme (denoted P ic_{V_{•}}).

Proof. We define R^{i}f_{n∗}O_{n}^{∗} to be the sheaf on S_{F L} associated with the presheaf:

T ↦ H^{i}(V_{n} ×_{S} T; O_{V_{n}×_{S}T}^{∗})

The sheaves R^{0}f_{n∗}O_{n}^{∗} are representable by tori. By the classical theorem of Murre and Oort ([3], §8.2, Theorem 3, p. 211), the sheaves R^{1}f_{n∗}O_{n}^{∗} are representable by locally algebraic group schemes.

The spectral sequence (13) gives rise to an exact sequence (natural in T):

(14)  0 → E_{2,T}^{1,0} → H^{1}(V_{•} ×_{S} T; O^{∗}) → E_{2,T}^{0,1} → E_{2,T}^{2,0} .

Let E^{p,q} be the presheaf defined by T ↦ E_{2,T}^{p,q} and let E^{p,q} be the sheaf associated with the presheaf E^{p,q}.

One has an exact sequence of sheaves on S_{F L}:

0 → E^{1,0} → R^{1}f_{∗}O^{∗} → E^{0,1} ⟶^{d_{2}} E^{2,0}.

The sheaf E^{1,0} is the homology of the complex of sheaves:

R^{0}f_{0∗}O_{0}^{∗} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} R^{0}f_{1∗}O_{1}^{∗} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}+δ_{2}^{∗}} R^{0}f_{2∗}O_{2}^{∗}.

Each of the sheaves in the above complex is representable by a torus. Therefore the sheaf E^{1,0} is representable by an affine group scheme C′ whose neutral component C is a torus.

- The same argument demonstrates the representability of the sheaf E^{2,0} by an affine group scheme W with W^{0} a torus.

- The sheaf E^{0,1} is the sheaf Ker (R^{1}f_{0∗}O_{0}^{∗} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} R^{1}f_{1∗}O_{1}^{∗}). Since each

of these last sheaves is representable by locally algebraic group schemes, we find that the sheaf E^{0,1} is also representable by a locally algebraic group scheme, denoted K.

- The homomorphism d_{2} corresponds to a morphism of locally algebraic group schemes K ⟶^{d} W. The kernel of d is a locally algebraic group scheme Q′ representing the sheaf Ker(d_{2}). Thus, in the exact sequence

0 → E^{1,0} → R^{1}f_{∗}O^{∗} → Ker(d_{2}) → 0,

the left term is representable by an affine scheme, and the right term is representable. The representability of the sheaf R^{1}f_{∗}O^{∗} follows from the next proposition. □

Proposition 3.3. Let 0 → F → G → H → be an exact sequence of sheaves of abelian groups on S_{F L}. 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. This is Proposition 17.4 on page III.17-6 of [28]. □

The Picard scheme P ic_{V_{•}} of V_{•} is an extension of Q′ by the (reduced) affine group scheme C′. We now recall the ([9] Corollary 2.2, II §5, no. 2, p. 288):

Theorem 3.4. Let G be a locally algebraic group scheme over a field F. If the field F is perfect, then the reduced scheme G^{red} is a smooth group scheme.

This is the main reason why we assume that k is perfect. Other reasons include

(i) the structure theorems for commutative group schemes are particularly nice over a perfect field (Theorems 1 and 2 of §9.2 in [3]). (ii) Over a perfect field, a curve X is smooth if and only if it is normal. These two notions are not equivalent over arbitrary fields (related to the distinction between smoothness and regularity of schemes over imperfect fields; cf. Exercise 10.1 of [17]). (iii) Grothendieck has expressed doubts about the suitability of 1-motives over imperfect fields (see the appendix to [20]). This is based on the fact that the 1-motives associated with a (singular) curve need not be compatible with base change. (iv) our eventual interest is in constructing 1-motives for general varieties by using simplicial smooth varieties. Our construction (§7) is based on the results of [10] which are valid only over perfect fields (page 1 of [10]).

Corollary 3.5. Let V_{•} be a proper reduced simplicial scheme. Assume that V_{0}×_{S} S̄ is normal i.e. V_{0} is geometrically normal. The group scheme P ic_{V_{•}}^{0,red} is a semiabelian scheme.

Proof. The sheaf R^{1}f_{0∗}O_{0}^{∗} is representable by a group scheme whose reduced subscheme is an abelian locally algebraic scheme [16], Theorem 2.1, p.

11. In other words, the neutral component of K^{red} is an abelian scheme. Recall that every morphism from an abelian variety to an affine group scheme is constant; in particular, a homomorphism must necessarily be the zero map. Therefore, the neutral component of K^{red} must be in the kernel of the morphism d. In other words, if Q denotes the neutral component of (Q′)^{red}, the inclusion of Q in K^{0,red} is an equality. We obtain that P ic_{V_{•}}^{0,red} is a semiabelian scheme. □

In particular, Corollary 3.5 applies to a smooth proper simplicial scheme.

Proposition 3.6. Let V_{•} be a proper reduced simplicial scheme. One has the following natural isomorphisms:

(0) H^{1}(V_{•} ×_{S} S̄; O^{∗}) ⟶^{∼} P ic_{V_{•}}(S̄)

(1) H^{1}(V_{•}; O) ⟶^{∼} Lie P ic_{V_{•}}^{0} = Lie P ic_{V_{•}}

(2) (k = ℂ) H^{1}(V_{•}; O^{∗}) ⟶^{∼} H^{1}(V_{•}^{an}; O^{∗}), H^{1}(V_{•}; O) ⟶^{∼} H^{1}(V_{•}^{an}; O)

(3) (k = ℂ) H^{1}(V_{•}; ℤ(1)) ⟶^{∼} H_{1}(P ic_{V_{•}}^{0}; ℤ).

Proof. The isomorphisms in (2) is a natural consequence of G.A.G.A. applied to E_{2,S}^{p,q} of (14). Assertion (0) is clear. The proof of (1) follows the proof of Theorem 1, §8.4 on p. 231 in [3]. In the exact sequence (induced by the exponential sequence on V_{•}):

H^{1}(V_{•}; ℤ(1)) ⟶^{∂} H^{1}(V_{•}^{an}; O) ⟶^{exp} H^{1}(V_{•}^{an}; O^{∗}) → H^{2}(V_{•}; ℤ(1)),

the map ∂ is injective since exp : ℂ → ℂ^{∗} is surjective. By parts (1) and (2), this is enough to prove (3). □

4. The Albanese and Picard 1-motives

Let us fix a simplicial scheme X_{•}, the complement of a divisor E_{•} with normal crossings in a smooth projective simplicial scheme X′_{•}. In this section, we define the Albanese and Picard 1-motives of X_{•}. We prove a conjecture of Deligne that the Hodge realization of the Picard 1-motive of X_{•} (over ℂ) is H^{1}(X_{•}; ℤ(1)).

4.1. The Picard 1-motive. Let R_{n} be the large group scheme ℤ^{E_{n}}. Put R”_{n} := Ker(b_{n} : R_{n} → D_{P ic_{Xn}}). We define

R′ := Ker(R_{0} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} R_{1}).

Any element r of R′(S̄) determines an invertible sheaf O(r) on X′_{0} ×_{S} S̄. There is a canonical isomorphism α_{r} : δ_{0}^{∗}O(r) ⟶^{∼} δ_{1}^{∗}O(r) on X′_{1} ×_{S} S̄; the isomorphism α_{r} satisfies the cocycle condition

p_{2}^{∗}(α_{r})p_{0}^{∗}(α_{r}) = p_{1}^{∗}(α_{r}).

We have a homomorphism

R′(S̄) ⟶^{h} P ic_{X′_{•}}(S̄) = P ic(X_{•} ×_{S} S̄) h′(m) = ([O(r)], α_{r}).

Let R′ ⟶^{h′} P ic_{X′_{•}} be the corresponding homomorphism of group schemes. Composing with the natural morphism P ic_{X′_{•}} → NS(X′_{•}) := π_{0}(P ic_{X′_{•}}), one gets a homomorphism

R′ ⟶^{h”} NS(X′_{•})

whose kernel is denoted R. Another description of R is

R = Ker(R”_{0} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} R”_{1}).

Recall from Corollary 3.5 that P := P ic_{X′_{•}}^{0,red} is a semiabelian scheme.

Definition 4.1. The motivic H^{1} (or the Picard 1-motive) of X_{•} is:

M^{1}(X_{•}) := [R ⟶^{h} P].

4.2. The Albanese 1-motive. We put A_{i} = A_{X_{i}} and D_{i} = D_{X_{i}}; note that D_{i} = D_{A_{i}}. By virtue of the simplicial structure of X_{•}, the groups A_{i}, A_{i}^{0}, D_{i} assemble to form simplicial groups A_{•}, A_{•}^{0} and D_{•}; they fit into an exact sequence of simplicial group schemes

0 → A_{•}^{0} → A_{•} → D_{•} → 0.

There is a natural functor (usually termed normalization) [23] which transforms a simplicial commutative group into a homological complex; the simplicial maps δ_{i} are replaced by the differentials d := Σ_{i}(−)^{i}δ_{i}. Thus, the previous sequence can be interpreted as an exact sequence of complexes of locally algebraic group schemes (concentrated in nonnegative degrees); as such, one gets a long exact sequence of homology. We set

G := H_{0}(A_{•}^{0}) = \frac{A_{0}^{0}}{d(A_{1}^{0})}, N := H_{1}(D_{•}) = \frac{Ker(d : D_{1} → D_{0})}{Im(d : D_{2} → D_{1})}.

One has the boundary map ∂ : N → G.

Definition 4.2. The Albanese 1-motive (or the motivic H_{1}) of X_{•} is

M_{1}(X_{•}) := [N ⟶^{∂} G].

Strictly speaking, N has to be torsion-free for M_{1}(X_{•}) to be a 1-motive; but we can still make sense of the dual 1-motive of M_{1}(X_{•}). So the torsion in N does not affect matters much.

Remark 4.3. (Constant schemes) If X_{•} happens to be a constant simplicial scheme, i.e., if X_{i} = X_{0} for all i > 0 and each of the simplicial maps is the identity, then M_{1}(X_{•}) = M_{1}(X_{0}) where M_{1}(X_{0}) of the special scheme X_{0} is defined as in (8).

Theorem 4.4. (Deligne’s conjecture)^{6} Let k = ℂ. One has a natural isomorphism

H^{1}(X_{•}; ℤ(1)) → 𝔗(M^{1}(X_{•})).

Proof. As before, we set P := P ic_{X′_{•}}^{0}; it is a reduced scheme.

The main step is the construction of a homomorphism β from H^{1}(X_{•}; ℤ(1)) to Lie P, the Lie algebra of P; note that Lie P is H^{1}(X′_{•}; O_{X′_{•}}). For this, we closely follow Deligne ([6], 10.3.9-10.3.15, p. 71-74); the modifications are minor.

Lemma 4.5. (i) Let j_{∗}^{m}O_{X}^{∗} be the subsheaf of meromorphic functions of j_{∗}O_{X}^{∗} on X′_{•}. One has

H^{1}(X_{•}; ℤ(1)) = ℍ^{1}(X′_{•}; [O_{X′} ⟶^{exp} j_{∗}^{m}O_{X}^{∗}]).

6This has been proved (independently) by Barbieri-Viale and Srinivas [2]; it was announced in [1].

(ii)

H^{1}(X_{•}; ℂ) = ℍ^{1}(X′_{•}; [O_{X′} ⟶^{d} Ω_{X′_{•}}^{1}(log E)]).

(iii) The inclusion of H^{1}(X_{•}; ℤ(1)) in H^{1}(X_{•}; ℂ) is defined by the morphism of complexes

O_{X′} ⟶^{exp} j_{∗}^{m}O_{X}^{∗}
∥  ^{df/f}↓
O_{X′} ⟶^{d} Ω_{X′_{•}}^{1}(log E).

Proof. Consider the commutative diagrams

H^{1}(X_{•}; ℤ(1)) ⟶^{∼} ℍ^{1}(X_{•}; [O_{X} ⟶^{exp} O_{X}^{∗}]) ⟶^{∼} ℍ^{1}(X′_{•}; [j_{∗}O_{X} ⟶^{exp} j_{∗}O_{X}^{∗}])
↓  ↓  ↓
H^{1}(X_{•}; ℂ) ⟶ ℍ^{1}(X_{•}; [O_{X} ⟶^{d} Ω_{X_{•}}^{1}]) ⟵ ℍ^{1}(X′_{•}; [j_{∗}O_{X} ⟶ j_{∗}Ω_{X_{•}}^{1}])
ℍ^{1}(X′_{•}; [j_{∗}O_{X} ⟶^{exp} j_{∗}O_{X}^{∗}]) ⟵^{∼} ℍ^{1}(X′_{•}; [O_{X′} ⟶^{exp} j_{∗}^{m}O_{X}^{∗}])
↓  ↓
ℍ^{1}(X′_{•}; [j_{∗}O_{X} ⟶ j_{∗}Ω_{X_{•}}^{1}]) ⟵^{∼} ℍ^{1}(X′_{•}; [O_{X′} ⟶ Ω_{X′_{•}}^{1}(log E)]).

The horizontal maps in the first diagram are isomorphisms due to

(1) the exact sequence of complexes on X′_{•}:

0 → ℤ(1) → O_{X_{•}} ⟶^{exp} O_{X_{•}}^{∗} → 0.

(2) the fact that the inclusion of complexes

0 → ℂ → [O_{X} ⟶^{d} Ω_{X_{•}}^{1}]

is a quasi-isomorphism in degree less than 2. More generally, the inclusion of the truncated De Rham complex Ω_{V_{•}}^{<n} in the De Rham complex of a smooth simplicial scheme V_{•} induces a quasi-isomorphism in degree less than n.

(3) the fact that R^{1}j_{∗}O_{X_{•}} = 0 ([6] 3.1.7).

The horizontal maps in the second diagram are isomorphisms because the morphism of complexes which define them are quasi-isomorphisms. □

The group ℍ^{1}(X′_{•}; [O_{X′_{•}}^{∗} → Ω_{X′_{•}}^{1}(log E)]) can be interpreted as the group of isomorphism classes of triples (L, α_{L}, ω_{L}), with L an invertible sheaf on X′_{•} (thereby determining L_{0} on X′_{0}), α_{L} is an isomorphism δ_{0}^{∗}(L_{0}) ⟶^{∼} δ_{1}^{∗}(L_{0}), and ω_{L} is an integrable connection on L, regular on X_{•} with at most simple poles along the divisor E_{•} (cf. [25] 2.5, p. 364 or [31], 1.5, p. 47), whereby, using the connections on invertible sheaves = one-forms dictionary, one can identify ω_{L} with an element of H^{0}(X′_{•}; Ω_{X′_{•}}^{1}(log E)).

If a denotes the projection X′_{•} to S, then the S_{F L} sheaf R^{1}a_{∗}[O_{X′_{•}}^{∗} →

Ω_{X′_{•}}^{1}(log E)] is representable by a group scheme, denoted M_{X_{•}}^{♮}, whose group of ℂ-points is the group ℍ^{1}(X′_{•}; [O_{X′_{•}}^{∗} → Ω_{X′_{•}}^{1}(log E)]). The group scheme

M_{X_{•}}^{♮} is an extension of a subgroup (containing P) of P ic_{X′_{•}} by the additive group scheme representing the S_{fppf} sheaf R^{0}a_{∗}Ω_{X′_{•}}^{1}(log E). The representability of R^{1}a_{∗}[O_{X′_{•}}^{∗} → Ω_{X′_{•}}^{1}(log E)] follows by applying Proposition 3.3 to the exact sequence

0 → R^{0}a_{∗}Ω_{X′_{•}}^{1}(log E) → R^{1}a_{∗}[O_{X′_{•}}^{∗} → Ω_{X′_{•}}^{1}(log E)] → R^{1}a_{∗}O_{X′_{•}}^{∗};

details may be found in [30].

Consider an element r of R(ℂ), i.e. a divisor r_{0} supported on E_{0}, algebraically equivalent to zero, such that δ_{0}^{∗}(r_{0}) = δ_{1}^{∗}(r_{0}) (equality of divisors on X′_{1}). The invertible sheaf O(r_{0}) on X′_{0} extends to an invertible sheaf O(r) on X′_{•}, equipped with a canonical isomorphism α_{r} : δ_{0}^{∗}(O(r_{0})) ≅ δ_{1}^{∗}(O(r_{0})), and an integrable connection ω_{r} on O(r), regular on X_{•} with atmost simple poles along the divisor E_{•}. The connection ω_{r} or, equivalently the corresponding one-form, may be described as follows: If f_{i} are defining equations for the divisor r_{0} on any open covering U_{i} of X′_{0}, then ω_{r} = dlog(f_{i}) = d(f_{i})/f_{i} on U_{i}. One defines a map

(15)  R → M_{X_{•}}^{♮}  r ↦ (O(r), α_{r}, ω_{r})

The group H^{1}(X_{•}; ℂ) = ℍ^{1}(X′_{•}; [O_{X′_{•}} → Ω_{X′_{•}}(log E)]) is the Lie algebra of M_{X_{•}}^{♮}. It can be identified with the isomorphism classes of triples (L, α_{L}, ω_{L}), for L an O_{X′_{•}}-torsor (thereby determining a O_{X′_{0}}-torsor L_{0}), an isomorphism α_{L} : δ_{0}^{∗}(L_{0}) ≅ δ_{1}^{∗}(L_{0}) and ω_{L} an integrable connection, as above, on L.

The group H^{1}(X_{•}; ℤ(1)) = ℍ^{1}(X′_{•}; [O_{X′_{•}} ⟶^{exp} j_{∗}^{m}O_{X_{•}}^{∗}]) is the group of isomorphism classes of triples (L, α_{L}, β_{L}), with L an O_{X′_{•}}-torsor (determining an O_{X′_{0}}-torsor L_{0}), an isomorphism α_{L} : δ_{0}^{∗}(L_{0}) ⟶^{∼} δ_{1}^{∗}(L_{0}), and β_{L} an isomorphism of the invertible sheaf exp(L) with an invertible sheaf O(r) (here r ∈ R(ℂ)). The map Aut(L) = ℂ → Aut(exp(L)) = ℂ^{∗} is surjective. This allows one to identify H^{1}(X_{•}; ℤ(1)) with the group of triples (t, α_{t}, r) for t an isomorphism class of O_{X′_{•}}-torsor (determining a O_{X′_{0}}-torsor t_{0}) and an isomorphism α_{t} : δ_{0}^{∗}(t_{0}) ≅ δ_{1}^{∗}(t_{0}), i.e. an element of Lie P ic_{X′_{•}}, and r ∈ R(ℂ), defining a divisor concentrated on E_{•} with exp(t) as image in P = P ic_{X′_{•}}^{0}(ℂ). In other words, we have defined an isomorphism (of abelian groups)^{7}

(16)  H^{1}(X_{•}; ℤ(1)) ⟶^{∼} 𝔗(M^{1}(X_{•})).

The following lemma tells us that this is compatible with the weight filtration and the Hodge filtration.

7See the definition of 𝔗 [6] §10.1.3.1.

Lemma 4.6. (i) One has

W_{1}(H^{1}(X_{•}; ℤ)) = Im((H^{1}(X′_{•}; ℤ)) ⟶^{j^{∗}} (H^{1}(X_{•}; ℤ))),

W_{0}((H^{1}(X_{•}; ℤ))) = Im(Ker(H^{1}(X′_{•}; ℤ) → H^{1}(X_{0}; ℤ)) ⟶^{j^{∗}} H^{1}(X_{•}; ℤ)).

(ii) The spectral sequence defined by the “stupid” filtration of [O_{X′_{•}} → Ω_{X′_{•}}^{1}(log E)] degenerates and converges to the Hodge filtration on H^{1}(X_{•}; ℂ).

Proof. For the first, let us check that H^{1}(X_{•} mod X′_{•}; ℤ) is pure of weight two. This follows from the exact sequence

0 → H^{1}(X′_{•}; ℤ(1)) ⟶^{j^{∗}} H^{1}(X_{•}; ℤ(1)) ⟶^{Div} H^{0}(E_{•}; ℤ) ⟶^{d} H^{2}(X′_{•}; ℤ(1))

(with a step of the Tate twist ℤ(−1), since H^{0}(E_{•}; ℤ) is pure of weight zero), part of the cohomology sequence of the exact sequence of complexes on X′_{•}

0 → [O_{X′} ⟶^{exp} O_{X′}^{∗}] → [O_{X′} ⟶^{exp} j_{∗}^{m}O_{X}^{∗}] → [0 → ℤ(E_{•})] → 0;

the last map is the divisor map j_{∗}^{m}O_{X}^{∗} → ℤ(E_{•}).

The second assertion in (i) follows from examining 8.1.19.1 and 8.1.20 of [6], p.35. As for (ii), it is clear from the definition 3.1.11, 8.1.8, 8.1.12 of [6] of the Hodge filtration. □

Back to the proof of Theorem 4.4: Combining the commutativity of the diagram

H^{1}(X_{•}; ℤ(1)) ⟶^{∼} 𝔗_{ℤ}(M^{1}(X_{•})) ⟶ Lie P = H^{1}(X′_{•}; O_{X′_{•}})
↓  ∥
H^{1}(X′_{•}; ℂ) ⟶^{∼} Lie M^{♮}(X_{•}) ⟶ Lie P = H^{1}(X′_{•}; O_{X′_{•}})

with part (ii) of Lemma 4.6, we see that the map in (16) is compatible with the Hodge filtration. □

5. A Duality theorem

We retain the notations of §4. In this section, we prove the duality of the Albanese and the Picard 1-motives (conjectured by Lichtenbaum). Recall the exact sequence (here P := P ic_{X′_{•}}^{0,red})

0 → C′ → P ⟶^{π} Q → 0.

We set a : X′_{•} → S to be the natural projection; we use a_{i} : X′_{i} → S for each of the components.

5.1. The duality theorem.

Theorem 5.1. (Lichtenbaum’s conjecture) The 1-motives

M_{1}(X_{•}) := [N ⟶^{∂} G], M^{1}(X_{•}) := [ R ⟶^{h} P]

are dual.

Remark 5.2. As mentioned earlier, M_{1}(X_{•}) is not always a 1-motive: N may contain torsion. But it is still possible to define the dual of M_{1}(X_{•}) (Deligne [6] §10.2).

Proof. It will be based on a number of lemmas. Let B denote the maximal abelian quotient of G; we define the torus T by means of the exact sequence

0 → T → G → B → 0.

Lemma 5.3. The abelian schemes B and Q are dual.

Proof. The semiabelian scheme G is defined by the top horizontal exact sequence

A_{X_{1}}^{0} ⟶^{d} A_{X_{0}}^{0} ⟶ G ⟶ 0
j_{1}↓  ^{j_{0}}↓  ↓
A_{X′_{1}}^{0} ⟶^{d} A_{X′_{0}}^{0} ⟶ B ⟶ 0;

the vertical maps are surjective, taking each semiabelian scheme to its maximal abelian quotient. The abelian scheme Q is the neutral component of the (reduced) kernel of P ic_{X′_{0}}^{0,red} ⟶^{d_{0}−d_{1}} P ic_{X′_{1}}^{0,red}. The classical duality of the Albanese and Picard varieties — for any smooth projective scheme Z, the abelian schemes A_{Z}^{0} and P ic_{Z}^{0,red} are naturally dual (compatible with morphisms). Applying this to X′_{0} and X′_{1}, and using functoriality, we see that Q and B are dual. □

Lemma 5.4. The torus Hom(R, 𝔾_{m}) is isomorphic to T.

Proof. In the previous commutative diagram, the torus corresponding to the kernel of j_{1} (resp. j_{0}) was identified to be Hom(R”_{1}, 𝔾_{m}) (resp. Hom(R”_{0}, 𝔾_{m})) by Corollary 2.2 (a) applied to X_{0} and X_{1}. Consequently Hom(R, 𝔾_{m}) can be identified with T. □

Lemma 5.5. There is a canonical isomorphism of tori:

Hom(N, 𝔾_{m}) ≅ C′.

Proof. Fix a smooth proper scheme W with projection g : W → S. The torus Hom(D_{W} , 𝔾_{m}) also represents the fppf sheaf R^{0}g_{∗}O_{W}^{∗}; a character of D_{W} provides a nonzero function, constant (since W is proper) on each connected component of W, i.e., on each irreducible component since W is smooth. The affine group scheme C′ represents the homology sheaf of the complex :

(17)  R^{0}a_{0∗}O_{X′_{0}}^{∗} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} R^{0}a_{1∗}O_{X′_{1}}^{∗} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}+δ_{2}^{∗}} R^{0}a_{2∗}O_{X′_{2}}^{∗};

this can be identified with the following complex:

(18) Hom(D_{X′_{0}}, 𝔾_{m}) ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} Hom(D_{X′_{1}}, 𝔾_{m}) ⟶^{δ_{0}^{∗}−δ_{1}^{∗}+δ_{2}^{∗}} Hom(D_{X′_{2}}, 𝔾_{m}).

Thus, C′ represents the homology sheaf of (18). We have defined N to be the étale group scheme representing the homology of the complex

D_{X_{2}} ⟶^{δ_{0}−δ_{1}+δ_{2}} D_{X_{1}} ⟶^{δ_{0}−δ_{1}} D_{X_{0}}.

But note D_{X_{n}} ≅ D_{X′_{n}} since the complement E_{•} of X_{•} in X′_{•} is a simplicial divisor. As the functor Hom(−, 𝔾_{m}) establishes an anti-equivalence of the category of group schemes of multiplicative type and étale group schemes, we get that C′ ≅ Hom(N, 𝔾_{m}). □

A consequence is the isomorphism Hom(C′, 𝔾_{m}) ≅ N. In the next lemma, π denotes the projection P → Q (as above).

Lemma 5.6. The 1-motive [R ⟶^{π h} Q] is dual to [0 → G].

Proof. As observed earlier, the 1-motive [R ⟶^{π h} Q] is the

Ker([R′′_{0} → P ic_{X′_{0}}^{0}] ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} [R′′_{1} → P ic_{X′_{1}}^{0}]),

rendering its dual to be the 1-motive [0 → G], by Theorem 2.1 applied to X_{0} and X_{1}. □

For any 1-motive M, Deligne ([6] 10.2.11, p. 67) defines the dual 1-motive M^{∗} by M^{∗} := RHom(M, 𝔾_{m}[−1]) where RHom is the derived functor of Hom on the abelian category of complexes of group schemes over S. We follow his prescription in determining the dual of M := M_{1}(X_{•}):

(a) The dual of M/W_{−2}(M)

Recall that P (V) = P ic_{X′_{•}}^{0,red}(V) classifies isomorphism classes of pairs (L, α) with L an invertible sheaf on X′_{0} ×_{S} V and α an isomorphism δ_{0}^{∗}(L) ≅ δ_{1}^{∗}(L) over X′_{1} ×_{S} V satisfying a cocycle relation (12) on X′_{2} ×_{S} V. We may also interpret α as a trivialization of the invertible sheaf δ_{0}^{∗}(L) ⊗ (δ_{1}^{∗}(L))^{−1}, i.e., an isomorphism δ_{0}^{∗}(L) ⊗ (δ_{1}^{∗}(L))^{−1} ≅ O_{X_{1}×_{S}V} (also required to satisfy a cocycle condition).

There is a natural morphism of functors Ext^{1}([N ⟶^{∂} B], 𝔾_{m}) ⟶^{ψ} P: elements of the former are isomorphism classes of pairs (L, α) where L is an invertible sheaf on X′_{0}, defined by an element of Q = Ext^{1}(B, 𝔾_{m}) (the dual abelian scheme), and α is a trivialization of ∂^{∗}(L). The element L naturally extends to an invertible sheaf on X′_{•} because L ∈ Q. Since ∂ is defined via the map δ_{0} − δ_{1} : A_{X_{1}} → A_{X_{0}}, we see that a trivialization of ∂^{∗}(L) amounts to a trivialization of the invertible sheaf δ_{0}^{∗}(L) ⊗ (δ_{1}^{∗}(L))^{−1} on X′_{1}. Comparing with the definition of P (recalled above), it is clear that the morphism ψ is actually an isomorphism of functors. The 1-motives [N ⟶^{∂} B] and [0 → P] are dual.

(b) For each r ∈ R ≅ Hom(T, 𝔾_{m}), the extension M of M/W_{−2}(M) by T defines an extension of M/W_{−2}(M) by 𝔾_{m}, and hence an element s(r) ∈ P. This combines into a homomorphism s : R → P. Therefore, the dual of M_{1}(X_{•}) is the 1-motive [R ⟶^{s} P].

Thus, to finish the proof of Theorem 5.1, it suffices to show that h = s.

Let T_{0} be the torus Hom(R′′_{0}, 𝔾_{m}). It is the kernel of the projection j_{∗} : A_{X_{0}} → A_{X′_{0}} (see 2.1). One may view A_{X_{0}}^{0} as the total space of a T_{0}-bundle over A_{X′_{0}}^{0}. Similarly, L (the universal torsor for X′_{0}) is the total space of a T_{0}-bundle over L′. Fix the universal morphisms u : X_{0} → L, u′ : X′_{0} → L′ and the isomorphisms

(19)  P ic_{A_{X′_{0}}^{0}}^{0,red} ≅ P ic_{L′}^{0,red} ≅ P ic_{X′_{0}}^{0,red}.

It is a complete tautology that both the T_{0}-bundles have the following property: For any r ∈ R′′_{0} ≅ Hom(T_{0}, 𝔾_{m}), the line bundle obtained via r over A_{X′_{0}}^{0} and L′ corresponds to the element [O(r)] ∈ P ic_{X′_{0}}^{0,red} under the isomorphisms in (19). Lemma 5.6 and the fact that R ↪ R′′_{0} together show that π(s(r)) = π(h(r)).

Consider the restriction of the T_{0}-bundle L̃ := (u′)^{∗}L over X′_{0} to the open subscheme X_{0}. The universal morphism u : X_{0} → L is defined by lifting the map u′ : X_{0} → L′ by means of a consistent choice of sections of the T_{0}- bundle L̃. For any r ∈ R, the map u determines a rational section τ_{r} of O(r) on X′_{0} whose divisor is r. On X_{0}, the section τ_{r} of O(r) is nowhere vanishing and regular. Similarly, we have a rational section τ_{−r} of the invertible sheaf O(−r). One has a a priori rational section of the invertible sheaf L_{1} := δ_{0}^{∗}(O(r)) ⊗ δ_{1}^{∗}(O(r))^{−1} on X′_{1} given by ζ_{r} := δ_{0}^{∗}(τ_{r}) ⊗ δ_{1}^{∗}(τ_{−r}); this is because O(r)^{−1} = O(−r). But L_{1} is the trivial invertible sheaf O_{X_{1}} on X_{1} and the section ζ_{r} on X_{1} is the identity section.

The element s(r) corresponds to a pair (π(s(r)), β_{r}) where β_{r} is a trivialization of the invertible sheaf corresponding to π(s(r)) over N. As remarked earlier, the element π(s(r)) corresponds to the invertible sheaf O(r) on X′_{0}. Utilizing the definition of the map ∂, we see that the pull back ∂^{∗}(O(r)) is the invertible sheaf L_{1} on X′_{1}. Since β_{r} is the pullback (via ∂) of the trivialization τ_{r} of O(r) on X_{0}, we obtain that β_{r} = ζ_{r} on X_{1}. Therefore, β_{r} also corresponds to the identity trivialization of O_{X′_{1}}. Comparing with the definition of h (recalled below), we obtain that h(r) = s(r).

One could alternatively proceed using the fact that D_{X_{i}} = D_{X′_{i}} for any i, and the sequences (17) and (18).

Let us recall the definition of the map h: Any element r ∈ R satisfies δ_{0}^{∗}(r) = δ_{1}^{∗}(r) = r_{1} (equality of divisors on X_{1}). Therefore, there is a canonical isomorphism α_{r} : δ_{0}^{∗}(O(r)) ≅ δ_{1}^{∗}(O(r)), corresponding to the identity in Hom_{X′_{1}}(O(r_{1}), O(r_{1})). The invertible sheaf δ_{0}^{∗}(O(r)) ⊗ (δ_{1}^{∗}(O(r)))^{−1} is canonically isomorphic to O_{X′_{1}}; the element α_{r}, interpreted as a trivialization of the former, corresponds to the identity section of O_{X′_{1}}. The map h sends r to the pair (O(r), α_{r}).

Consequently, the dual 1-motive of M_{1}(X_{•}) is M^{1}(X_{•}). □

Corollary 5.7. The Picard 1-motive M^{1}(U_{•}) is a contravariant functor.

Proof. Combine Theorem 5.1 with the (evident) covariant functoriality of the Albanese 1-motives.

Remark 5.8. Theorem 5.1 depends on the following two facts:

(1) For any r ∈ R, the pullback δ_{0}^{∗}(O(r)) ⊗ δ_{1}^{∗}(O(r)^{−1}) is the trivial invertible sheaf O_{X′_{1}} on X′_{1}; this is by definition of the group R.

(2) the pullback of any rational section τ_{r} of O(r) (such that the divisor of τ_{r} is precisely r) under the map ∂ is the identity section of O_{X′_{1}}; this is by definition of the map ∂.

6. Curves

A curve denotes a scheme of pure dimension one. For any curve C over an algebraically closed field, there is the motivic H^{1} of C denoted H_{m}^{1}(C)(1)(1) defined by Deligne [6] §10.3, and the homological 1-motive h_{1}(C) defined by Lichtenbaum [21]; it is clear by inspection that their definitions are valid over a perfect field k.

In this section, we provide a new proof (based on Theorem 5.1) of an unpublished theorem of Lichtenbaum that h_{1}(C) is dual to H_{m}^{1}(C)(1).

We refer to ([3] pp. 247-248; [39]; [22]) for the details of seminormalization. We recall that the seminormalization Ĉ of a curve C is the largest curve between the normalization C̃ and C which is universally homeomorphic to C. The association of Ĉ and C̃ with C is compatible with base change; this requires k to be perfect.

The 1-motives H_{m}^{1} and h_{1} of curves do not change upon seminormalization and depend only on the underlying reduced scheme structure. Therefore, we may assume that we deal with reduced curves which are seminormal (i.e., that they do not have cuspidal singularities).

6.1. Jacobians of proper curves: a review. Let L be the (reduced) closed subscheme of a seminormal proper curve C corresponding to the singular locus of C. Take g : C̃ → C to be the canonical projection. Put K := L ×_{C} C̃; it is a reduced closed subscheme of C̃. There is a natural homomorphism g : D_{K} → D_{L} with kernel D′. The inclusion of K in C̃ provides a homomorphism D_{K} → DC̃. This yields a map D′ ⟶^{μ} DC̃.

Since C̃ is normal, the connected components of C̃ are the irreducible components of C̃. So one has an isomorphism π_{0}(P icC̃) ⟶^{∼} π_{0}(AC̃) = DC̃. Using this, one can state the Abel-Jacobi theorem as follows:

Theorem 6.1. (Abel-Jacobi) For any smooth projective curve X, one has a natural isomorphism

A_{X} ⟶^{∼} P ic_{X};

this isomorphism is induced by the map X ↪ P ic_{X} (each point of X is viewed as a divisor).

Consider the exact sequence of sheaves on C:

(20)  1 → O_{C}^{∗} → g_{∗}O∗ ; C̃ → g_{∗}O∗ ; C̃/O_{C}^{∗} → 1 .

The quotient sheaf V := g_{∗}O∗ ; C̃/O_{C}^{∗} is supported on L. Let f : C → S be the structure morphism. The sequence (20) furnishes an exact sequence of sheaves on S_{F L}:

(21) 1 → R^{0}f_{∗}O_{C}^{∗} → R^{0}f_{∗}(g_{∗}O∗ ; C̃) ⟶^{ν} R^{0}f_{∗}V →

→ R^{1}f_{∗}O∗ ; Ĉ → R^{1}f_{∗}(g_{∗}O∗ ; C̃) → 1

Since g is finite, its higher derived functors are zero. Thus R^{1}f_{∗}(g_{∗}O∗ ; C̃) = R^{1}(fg)_{∗}O∗ ; C̃. In other words, the last sheaf in (21) corresponds to the Picard scheme of C̃.

The sheaf R^{0}f_{∗}V is the torus dual to the group D′. The sheaf R^{0}f_{∗}(g_{∗}O∗ ; C̃) is the torus dual to DC̃. The map ν between these two sheaves in corresponds to the pullback of functions by the natural inclusion of K ↪ C̃. Therefore ν is dual to the map D′ ⟶^{μ} DC̃ mentioned above. The cokernel of ν is the torus T dual to the group scheme D := Ker(D′ ⟶^{μ} DC̃).

An immediate corollary of (21) is that the Picard scheme of C also represents the functor which assigns, to a S-scheme Z, the group of isomorphism classes of pairs (L, α) where L is an invertible sheaf on C̃ ×_{S} Z and α is a trivialization of the restriction of L to K ×_{S} Z.

Remark 6.2. Note that the Picard scheme of a proper curve is always reduced [16] Proposition 2.10.

6.2. 1-motives of curves. Let C be a curve (assumed to be seminormal). Denote by C′ the unique proper curve containing C as a dense open subscheme such that the singular locus of C′ is the same as that of C. As usual, C̃ and C̃′ denote their normalizations ^{8}. We may also describe the latter as the unique smooth proper curve containing C̃ as an open dense subscheme. Denoting by E (resp. E_{0}) the closed complement of C (resp C̃) in C′ (resp. C̃′), there is a natural projection c : E_{0} → E, an isomorphism of finite étale schemes. One has an isomorphism

(22)  c^{∗} : \underline{ℤ}^{w(E)} ⟶^{∼} \underline{ℤ}^{w(E_{0})}.

8Note that there is no ambiguity here: C̃′ = (C̃)′

Consider the natural morphism α : \underline{ℤ}^{w(E)} → P ic_{C′} → π_{0}(P ic_{C′}); let v denote the induced map Ker(α) → P ic_{C′}^{0}.

Definition 6.3. (Deligne) The motivic H^{1} of C is the 1-motive

H_{m}^{1}(C)(1) := [Ker(α) ⟶^{v} P ic_{C′}^{0}].

We recall that the special simplicial scheme ((C̃/C)^{Δ_{n}})_{n≥0} is smooth (the (n + 1)-fold fibre product (C̃/C)^{Δ_{n}} is the disjoint sum of C̃, the diagonal, and a finite étale scheme) and that it admits ((C̃′/C′)^{Δ_{n}})_{n≥0} as a smooth compactification with a (smooth!) simplicial divisor E_{•} as complement. Let us put

C_{n} := (C̃/C)^{Δ_{n}}, C′_{n} := (C̃′/C′)^{Δ_{n}} (n ≥ 0).

Both of the schemes C_{•} and C′_{•} are smooth simplicial schemes. We have C_{0} = C̃ and C′_{0} = C̃′.

The natural augmentations a : C_{•} → C and b : C′_{•} → C′ are proper hypercoverings of C and C′ respectively.

Lemma 6.4. The map b^{∗} : P ic_{C′}^{0} → P ic_{C′_{•}}^{0} is an isomorphism.

Proof. The map from b_{0} : C̃′ = C′_{0} → C′ is finite and surjective. Interpreting the conditions on the elements of P ic_{C′_{•}}^{0} as descent data, we see that the map b^{∗} is injective ([3], §6) (see also 6.1).

For the surjectivity, one uses the well known fact that descent data for invertible sheaves is effective for finite surjective morphisms ([3], §6.5, Theorem 1, p. 157). □

Let F denote the singular locus of C; we take F_{0} := F ×_{C} C̃. Denote by V the proper seminormal curve corresponding to the modulus E_{0} on C̃′ ([36] IV, no. 4, p. 61 and 76). There is a distinguished S-point U which is the unique singular point of V. The pullback of U under the map C̃′ → V is E_{0}. The curve C̃′ is the normalization of V. We set H := P ic_{V}^{0} .

We recall the following

Theorem 6.5. (Rosenlicht-Serre) There is a natural isomorphism

H ⟶^{∼} A0 ; C̃

of semiabelian schemes over S.

Proof. This follows by combining

• Theorem 1 in ([36] Chapter V, p.88): H is isomorphic to the generalized Jacobian of C̃′ corresponding to the modulus E_{0} and

• Example 1 in [35]: the generalized Jacobian of C̃′ corresponding to the modulus E_{0} is none other than A0 ; C̃. □

Notice that F and E (similarly E_{0} and F_{0}) are disjoint.

Put D′ := Ker(D_{F_{0}} → D_{F}); there is a natural map D′ ⟶^{μ} AC̃ induced by the inclusion of F_{0} in C̃. We set D := Ker(D′ ⟶^{μ} AC̃ → DC̃), the last being the group of connected (= irreducible) components of C̃. The group D(S̄) is generated by differences (x − y) ∈ D_{F_{0}}(S̄) such that b_{0}(x) = b_{0}(y) ∈ F (S̄) and x, y are in the same geometric component of C̃.

Using Theorem 6.5, we may define h_{1}(C) [21].

Definition 6.6. (Lichtenbaum) The homological 1-motive h_{1}(C) is the 1motive

[D ⟶^{μ} A0 ; C̃].

The key identification is in the

Theorem 6.7. One has natural isomorphisms

(i) H_{m}^{1}(C)(1) ⟶^{∼} M^{1}(C_{•}).

(ii) h_{1}(C) ⟶^{∼} M_{1}(C_{•}).

Proof. We begin with (i). We have defined R to be the subgroup scheme of \underline{ℤ}^{w(E_{0})} corresponding to the intersection of the kernels of \underline{ℤ}^{w(E_{0})} →

π_{0}(P ic_{C′_{0}}^{0}) and \underline{ℤ}^{w(E_{0})} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} \underline{ℤ}^{w(E_{1})}.

It is a simple consequence of the definition of C′_{•} that we have a complex:

\underline{ℤ}^{w(E)} ⟶^{c^{∗}} \underline{ℤ}^{w(E_{0})} ⟶^{δ_{0}^{∗}−δ_{1}^{∗}} \underline{ℤ}^{w(E_{1})}.

But c^{∗} is an isomorphism (22). Therefore, R = Ker(\underline{ℤ}^{w(E_{0})} → π_{0}(P ic_{C′_{0}}^{0})). Since the group of connected components of the Picard schemes of C′ and C′_{0} = C̃′ are isomorphic ([3], Proposition 10, §9.2, p.248-9), we find an isomorphism c^{∗} : Ker(α) ⟶^{∼} R.

The diagram below is commutative:

Ker(α) ⟶^{v} P ic_{C′}^{0}
c^{∗}↓  _{b^{∗}}↓
R ⟶^{h} P ic_{C′_{•}}^{0}

with isomorphisms (see Lemma 6.4) for vertical arrows. This proves (i).

We remark that C_{0} := C̃ and C_{1} := C̃ ×_{C} C̃. As usual, we write the Albanese 1-motive of C_{•} as [N ⟶^{∂} G] (see §4).

The Albanese 1-motive of a special scheme W is the neutral component of the Albanese scheme A_{W} of W (2.1). Any normal curve is a special scheme. It is clear from the definitions of the Albanese 1-motive and the homological 1-motive that, for a normal curve Y, one has h_{1}(Y) = M_{1}(Y) = A_{Y}^{0} .

Returning back to our curve C and C_{•}, let us set A_{i} := A_{C_{i}}. Since each C_{i} is the disjoint union of the diagonal (isomorphic to C̃) and a finite étale scheme X_{i} (i ≥ 1)^{9}, we have that A_{i} = AC̃ ×_{S} D_{X_{i}}. We have A_{0} = AC̃. Therefore, the simplicial large group A_{•} contains a constant simplicial large group B_{i} := AC̃ (for all non-negative integers i). Viewing these simplicial groups (and D_{X_{i}}) as complexes (by normalization), we see (1) that AC̃ = H_{0}(B_{•}) = H_{0}(A_{•}). (since C_{0} = C̃) that π_{0}(A_{0}) = DC̃. This in turn implies that

9This is because C_{0} = C̃.

N = \frac{Ker (D_{X_{1}} ⟶^{δ_{0}−δ_{1}} DC̃)}{Im (D_{X_{2}} ⟶^{δ_{0}−δ_{1}+δ_{2}} D_{X_{1}})}.

In particular, we deduce G = A0 ; C̃. So we have already proved that W_{−1}h^{1}(C) = W_{−1}M^{1}(C_{•}). We will use the previous description of N to show that D is actually N (compatible with the map μ) and this will finish the proof of (ii).

Let us put F_{n} to be the (n + 1)-fold fibre product of F_{0} over F. We have F_{1} := F_{0} ×_{F} F_{0} and F_{2} := F_{0} ×_{F} F_{0} ×_{F} F_{0}.

The group D_{X_{1}}(S̄) is generated by points (x, y) ∈ F_{1}(S̄) with x ≠ y (i.e. x, y are distinct points of F_{0}). By definition, the points x, y ∈ C̃(S̄) satisfy b_{0}(x) = b_{0}(y) ∈ C(S̄). Set K := Ker(D_{X_{1}} ⟶^{δ_{0}−δ_{1}} DC̃). The group K(S̄) is generated by points (x, y) ∈ F_{1}(S̄) which map to the same (geometrically) connected component of C̃. There is a natural morphism γ : K → D, which we describe on the level of the geometric points: we send (x, y) to the element x − y ∈ D(S̄). The morphism γ is surjective (by the seminormality of C).

Let us turn to the group D_{X_{2}}(S̄). It is generated by elements (x, y, z) ∈ F_{2}(S̄) with x ≠ y, y ≠ z, x ≠ z. By definition, we have b_{0}(x) = b_{0}(y) = b_{0}(z) ∈ C(S̄). One has a map

δ : D_{X_{2}} ⟶^{δ_{0}−δ_{1}+δ_{2}} K (x, y, z) ↦ (y, z) − (x, z) + (x, y).

The composition γ.δ can be checked to be zero; in fact, Ker(γ) = Im(δ). Since the group N is Coker(δ), we see that N = D. It is clear that the maps from N and D to A0 ; C̃ are compatible with this isomorphism. This proves (ii). □

Theorem 6.8. (Lichtenbaum, unpublished) The 1-motives H_{m}^{1}(C)(1) and h_{1}(C) are dual.

Proof. This follows from Theorems 6.7 and 5.1. □

Corollary 6.9. (Hodge Realization) For a curve C over ℂ, one has an isomorphism

H_{1}(C) ≅ 𝔗(h_{1}(C)).

Proof. Combine Theorems 10.3.8 of [6] and 5.1. □

The motivic H^{1} of C can also be interpreted as the motivic relative cohomology group H_{m}^{1}(C̃′ − E_{0} rel F_{0})(1). Likewise, one can interpret h_{1}(C) as the motivic relative homology group H_{1}^{m}(C̃′ − F_{0} rel E_{0}). With these identifications, the duality theorem translates to a Lefschetz duality theorem H_{m}^{1}(C̃′ −E_{0} rel F_{0})(1) ⟶^{∼} (H_{1}^{m}(C̃′ −F_{0} rel E_{0}))^{∗} (see 1.5 of [8]). The resulting duality of the Hodge realizations is a classical result [4], [42] (especially p. 390-92).

7. 1-motives associated with varieties

Let Y be a scheme over S. By de Jong [10], there exists a diagram

X_{•} ⟶^{j} X′_{•} ⟵ E_{•}
α↓
Y

where the smooth simplicial scheme X_{•} is the open complement of a reduced simplicial divisor with normal crossings E_{•} in a smooth projective simplicial scheme X′_{•} such that, via α, X_{•} is a proper hypercovering of Y. Using this, it is possible (as in [6] §6.3) to translate the theory developed for simplicial schemes such as X_{•} to obtain invariants for the scheme V. This has been treated in [30].

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Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

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