ABSOLUTE CHOW-KÜNNETH

PROJECTORS FOR MODULAR VARIETIES

B. Brent Gordon, Masaki Hanamura and Jacob P. Murre

Introduction.

In this paper we verify the conjecture on the existence of Chow-Künneth projectors for certain modular varieties derived from the universal families of abelian varieties over Hilbert modular varieties. The case of elliptic modular varieties were studied in [GMu 1, 2].

Let X be a smooth projective variety of dimension d over a field k. The Chow-Künneth projectors of X ([Mu 2]) are elements Π_{i}, i = 0, ⋯, 2d, in the correspondence ring CH_{d}(X × X), satisfying the following conditions:

— The Π_{i} are mutually orthogonal projectors and ∑ Π_{i} = Δ_{X} (the class of the diagonal).

— If H^{∗}(X) = ⊕_{i=0}^{2d}H^{i}(X) is a Weil cohomology, then Π_{i} acts on it as the projection to H^{i}(X). (In this paper we take k = C and H^{i}(X) to be Betti cohomology.)

It is conjectured that Chow-Künneth projectors exist for any X. Note the cohomology classes Π_{i} are the Künneth components of the diagonal class in H^{2d}(X × X). So the conjecture is stronger than a conjecture of Grothendieck – Conjecture (C) in his standard conjectures, [Gr],[Kl] – that the Künneth components of the diagonal be classes of algebraic cycles. Further, the projectors Π_{i} should satisfy additional properties, [Mu 2].

The existence of Chow-Künneth projectors are known for curves, surfaces and varieties of the form C × S with C a curve and S a surface [Mu 1]. It is of considerable interest to verify the conjecture for some special varieties. The problem for modular varieties as we consider here is important from the viewpoint of modular forms as well. In the case of elliptic modular varieties Scholl identified the space of elliptic modular forms with a motive [Sch].

In this paper we take Hilbert modular varieties. To be specific, let F be an imaginary quadratic number field of degree d and Γ ⊂ SL_{2}(F) a congruence subgroup. Fix an embedding F ↪ C. Then Γ acts on ℌ^{d} (d-fold product of the upper half-plane) properly discontinuously, so let

S^{0} := Γ\ℌ^{d} .

This is a complex manifold, and indeed a quasi-projective variety. There is a natural projective variety S of S^{0} obtained by adding a finite set of points (the cusps), called the Baily-Borel compatification. Over S^{0} there is a natural family of abelian varieties p^{0} : X^{0} → S^{0}. One can construct a compactification p : X → S of p^{0}, as described in

Typeset by AMS-TEX

detail in §2. For each m ≥ 1, the m-fold product over S

\overbrace{X ×_{S} × ⋯ ×_{S} X}^{m times}

has a desingularization which we will denote by X_{m} (so-called the Kuga-Sato variety). The main result of this paper is the X_{m} has Chow-Künneth projectors. In particular, Conjecture (C) of Grothendieck holds for X_{m}.

In order to prove the result, we start from the relative Chow-Künneth projectors over S as were proved to exist in [GHM1]. They are mutually orthogonal projectors

P^{i}/S, P_{∞}^{i}/S ∈ CH_{dim X}(X ×_{S} X)

adding up to the identity Δ(X/S) ∈ CH_{dim X}(X ×_{S} X), such that

(P^{i}/S)_{∗}Rp_{∗}Q_{X} ≅ IC_{S}(R^{i}p_{∗}^{0}Q_{X})[−i], , (P_{∞}^{i}/S)_{∗}Rp_{∗}Q_{X} ≅ V^{i}[−i] .

Here IC_{S}(−) is the intersection complex of a local system on S^{0} and V^{i} are sheaves supported on the set of cusps Σ. Relative Chow-Künneth projectors for X/S are analogues of Chow-Künneth projectors for X/k.

In general, even if one has relative Chow-Künneth projectors for a map p : X → S, there is no way to create out of those absolute Chow-Künneth projectors for X. The relative projectors give rise to a decomposition of Rp_{∗}Q_{X}, and each of the direct summands has hypercohomology possibly in more than one degree. Thus one needs additional decomposition of each relative projector so that each piece has hypercohomology in one degree.

The task of this paper is, for the particular example X_{m} → S, such a decomposition of each relative projector is indeed possible. The proof rests on the special nature of the cohomology of S and of the generic fiber of X_{m} → S. Assembling the projectors thus obtained, one finds the absolute Chow-Künneth projectors for X_{m}.

The paper is organized as follows. In §1, we take a map p : X → S satisfying two sets of geometric conditions, (1.1)(α) − (γ) and (1.3)(i)-(iii). The conditions (α) − (γ) imply the existence of relative Chow-Künneth projectors. Using the remaining (i)-(iii), we we decompose the relative projectors further and then construct absolute Chow-Künneth projectors for X_{m}. In §2, we verify the map X_{m} → S over the Hilbert modular variety satisfies the conditions in §1.

§1. On the existence of Chow-Künneth decomposition under certain conditions.

We consider varieties X over the field k = C. (In many parts of this paper we can take any field k.) Then CH_{k}(X) := A_{k}(X) ⊗ Q is its rational Chow group, where A_{k}(X) is as defined in [Fu], and H^{i}(X) := H_{B}^{i}(X(C), Q) (Betti cohomology with rational coefficients). Also, we will consider only complexes of sheaves of Q-vector spaces.

We will make full use of the theory of relative Chow motives. All we need are summarized in §1 of [GHM1] (we follow the notation there), and full details are in [CH]. For a quasiprojective variety S, CHM(S) denotes the category of Chow motives over S. For S = Spec kk, write CHM(k).

For an object M of CHM(k), its Chow-Künneth decomposition is a direct sum decomposition M = ⊕_{i}M_{i} such that M_{i} has cohomology only in degree i. In case M = h(X), the motive of a smooth projective variety X, the Chow-Künneth decomposition of h(X)

h(X) = ⊕h^{i}(X)

is given by an orthogonal set of projectors Π^{i} ∈ End(h(X)) = CH_{dim X}(X × X), the absolute Chow-Künneth projectors.

1.1 Assumptions. Let k = C, X and S projective varieties over k, X smooth, and p : X → S a surjective projective map. Assume that we have the following situation:

S^{0} ⊂ S is an open set, smooth over k, such that its complement Σ = S − S^{0} is a finite set of points, and p is smooth over S^{0}. Let X^{0} = p^{−1}(S^{0}), and p^{0} : X^{0} → S^{0} be the restriction of p. There is a resolution of singularities p′ : S̃ → S (an isomorphism over S^{0}) such that p factors through S̃. Let d = dim X − dim S.

X^{0} ⟶ X
↘
p^{0} ↓ p ↓ S̃
↙
S^{0} ⟶ S

Assume:

α) The irreducible components of S̃ − S^{0} are smooth projective toric varieties.

β) The irreducible components of X − X^{0} are smooth projective toric varieties.

γ) The variety X^{0}/S^{0} has a relative Chow-Künneth decomposition. Namely there is an orthogonal set of projectors (P^{i})^{0} ∈ CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) with ∑(P^{i})^{0} = 1, such that (P^{i})^{0} acts on R^{j}p_{∗}^{0}Q_{X^{0}} as identity if j = i, and as zero if j ≠ i.

In this situation we have relative Chow-Künneth decomposition for both X/S and S̃/S, as was shown in [GHM1]. Specifically there exist mutually orthogonal projectors Π_{S}^{i}(X/S), Π_{Σ}^{i}(X/S) in CH_{dim X}(X ×_{S} X) adding up to Δ(X/S) (the identity element) such that

Π_{S}^{i}(X/S)_{∗}Rp_{∗}Q_{X} = IC_{S}(R^{i−dim S}P^{0}_{∗}Q_{X^{0}})[−i + dim S] ,
Π_{S}^{i}(X/S)_{∗}Rp_{∗}Q_{X} = V^{i}[−i]

where V^{i} are sheaves supported on Σ. For simplicity let

P^{i}/S = Π_{S}^{i+dim S}(X/S) , P_{∞}^{i}/S = Π_{Σ}^{i}(X/S) ,

and P_{∞}/S = ∑ P_{∞}^{i}/S.

Similarly for S̃/S, we have a mutually orthogonal projectors Π_{∞}^{i}(S̃/S) ∈ CH_{dim S}(S̃ ×_{S} S̃) such that

Π_{∞}^{i}(S̃/S)_{∗}Rp′_{∗}QS̃ = W^{i}[−i]

with sheaves W^{i} supported on Σ, and

(1 − ∑ Π_{∞}^{i}(S̃/S))_{∗}Rp′_{∗}QS̃ = IC_{S} ,

the intersection complex of S. Set Π_{∞}(S̃/S) = ∑ Π_{∞}^{i}(S̃/S).

The question we want to address is: under what conditions can we construct – starting from the (relative Chow-Künneth) decomposition of X over S – an absolute Chow-Künneth decomposition of X ?

1.2 Concerning the above question we have the following general lemma.

Lemma 1. Let p : X → S be a projective map, with X, S projective over k, with X smooth. Then the natural map

Φ : CH_{dim X}(X ×_{S} X) → CH_{dim X}(X ×_{k} X)

is a ring homomorphism. Hence it transforms (orthogonal) projectors over S into (orthogonal) projectors over k. Hence we have a functor

Φ : CHM(S) → CHM(k),
M/S = (X/S, P/S, r) ↦ (X, P, r) = Φ(M/S) = M .

Proof. This follows from the properties of the refined Gysin maps used in the construction of composition of correspondences. See [CH], [GHM-1, §1].

Remark. As a matter of notation, since mostly we work with relative projectors, we shall usually write M instead of M/S; only if both relative and absolute motives are involved we write M/S and M respectively.

Problem. The problem of passing from relative to absolute Chow-Künneth decomposition is the following: if P/S is a relative Chow-Künneth projector, then in general the motive M = Φ(M) = (X, P, 0) still has cohomology in more than one degree so that we have to split it further.

1.3 Main Theorem. Let p : X → S satisfy the assumptions of 1.1. Assume moreover:

i) S̃ has a Chow-Künneth decomposition over k.

ii) If t is a point of S^{0}, the natural map

CH^{r}(X) → H_{B}^{2r}(X_{t}(C), Q)^{π_{1}^{top}(S^{0},t)}

(0 ≤ r ≤ d = dim X − dim S) is surjective. (The target is the invariant part under the action of the topological fundamental group of S^{0}.)

iii) For i odd, H_{B}^{i}(X_{t}(C), Q)^{π_{1}^{top}(S^{0},t)} = 0.

iv) If V is an irreducible, non-constant representation of π_{1}^{top}(S^{0}, t) and V the corresponding local system on S^{0}, then H^{q}(S^{0}, V) = 0 if q ≠ dim S. Equivalently, the intersection cohomology IH^{q}(S^{0}, V) = 0 if q ≠ dim S. (This follows from H^{q}(S^{0}, V) ≅ IH^{q}(S, V) for q < dim S, and the duality between IH^{q}(S, V) and IH^{2 dim S−q}(S, V^{∗}).)

Under these assumptions X has a Chow-Künneth decomposition over k.

Remark. For S̃/S we have constructed the projector “at infinity” Π_{∞}(S̃/S) := Π_{Σ}(S̃/S) in [GHM1]. The condition i) is equivalent to

i)’ The motive (S̃, 1 − Π_{∞}(S̃/S), 0) has a Chow-Künneth decomposition over k. In fact, (S̃, Π_{∞}(S̃/S)) has Chow-Künneth decomposition ⊕(S̃, Π_{∞}^{j}(S̃/S)). In the course of the proof of the theorem, we will directly invoke the assumption i)’.

1.4 Preliminaries for the proof. We work (except for Lemma 2) always over a field k. The next lemma is from [Ma, §5].

Lemma 2. Let C be a pseudo-abelian category, X, Y ∈ Ob(C), and let f : X → Y and g : Y → X be morphisms in C with f ∘g = id_{Y}. Then π := g∘f is a projector of X and

Im π →^{∼} Y.

Lemma 3. Let j : S^{0} → S be an open immersion of quasi-projective varieties over k. Then the functor

j^{∗} : CHM(S) → CHM(S^{0}), (X, P, r) ↦ (X^{0}, P^{0}, r)

where X^{0} = X ×_{S} S^{0}, P^{0} = P |X^{0} ×_{S^{0}} X^{0}, is full. Namely for objects (X, P, r), (Y, Q, s) in CHM(S), the map

Hom_{CHM(S)}((X, P, r), (Y, Q, s)) → Hom_{CHM(S^{0})}((X^{0}, P^{0}, r), (Y^{0}, Q^{0}, s))

is surjective.

Proof. The last map is the natural one

Q∘ CH_{dim X−s+r}(X ×_{S} Y)∘P → Q^{0}∘ CH_{dim X−s+r}(X^{0} ×_{S^{0}} Y^{0})∘P^{0}

induced from the open immersion X^{0} ×_{S^{0}} Y^{0} → X ×_{S} Y. It is obviously a surjection.

Lemma 4. Let X, S be quasi-projective varieties over k, with X smooth over k, and p : X → S a projective map.

a) Giving a projector Π ∈ CH_{dim X}(X ×_{S} X) and an isomorphism in CHM(S)

f : (X, Π, 0) → ⊕^{m}(S, Δ(S), −q)

is equivalent to giving elements f_{1}, ⋯, f_{m} ∈ CH_{dim S+q}(X) and g_{1}, ⋯, g_{m} ∈ CH_{dim X−q}(X) subject to the condition p_{∗}(f_{i} · g_{j}) = δ_{ij}[S]. Here f_{i} · g_{j} is the intersection product in X, and [S] ∈ CH_{dim S}(S) is the fundamental class. Under this correspondence, Π = δ_{S}^{!}(∑ f_{i} ×_{k} g_{i}).

b) If (X, P, 0) ∈ CHM(S) and Π is a constituent of P (i.e. P ∘Π = Π∘P = Π) then we can take the above f_{i} and g_{j} such that moreover f_{i}∘P = f_{i} and g_{j} = P ∘g_{j}. Conversely if we have such f_{i} and g_{j}, then the corresponding Π is a constituent of P.

Remark. If S is smooth and the map p is smooth, for elements f ∈ CH_{dim S+q}(X) and g ∈ CH_{dim X−q}(X), for any point t ∈ S,

p_{∗}(f · g) = deg((f|_{X_{t}}) · (g|_{X_{t}})) [S]

where f|_{X_{t}} ∈ CH_{q}(X_{t}) and g|_{X_{t}} ∈ CH_{d−q}(X_{t}), and (f|_{X_{t}}) · (g|_{X_{t}}) ∈ CH_{0}(X_{t}) their intersection product in X_{t}. See [Fu, Corollary 10.1 and Proposition 10.2].

Proof. Giving Π and f is equivalent to giving a pair of morphisms

(X, Δ(X), 0) f ; ⟶ ; ⟵ ; g ⊕^{m} (S, Δ(S), −q)

with f ∘g = id. Here f = (f_{1}, ⋯, f_{m}), f_{i} ∈ CH_{dim S+q}(X) and g = (g_{1}, ⋯, g_{m}), g_{j} ∈ CH_{dim X−q}(X), and f ∘g = (f_{i}∘g_{j}) where f_{i}∘g_{j} = p_{∗}(f_{i} · g_{j}). Then

Π = g∘f = ∑ g_{i}∘f_{i}

where g_{i}∘f_{i} = δ_{S}^{!}(g_{i} × f_{i}); δ_{S}^{!} is the refined Gysin map with respect to the square

X ×_{S} X ⟶ X × X
↓  ↓
S  ⟶^{δ_{S}} S × S

The proof of b) is the same except one replaces (X, Δ(X), 0) by (X, P, 0). Then the corresponding morphisms are

f_{i} : (X, P, 0) → (S, Δ(S), −q)  and  g_{j} : (S, Δ(S), −q) → (X, P, 0) ,

namely elements f_{i} ∈ CH_{dim S+q}(X) with f_{i} = f_{i}∘P and g_{j} ∈ CH_{dim X−q}(X) with g_{j} = P ∘g_{j}.

1.5 Lifting Lefschetz motives.

The proposition below is a special case of Proposition 2, which will be used in the sequel of the paper. In Proposition 1, we assume S is also smooth.

Proposition 1. Let X, and S be smooth quasi-projective varieties, p : X → S a projective map, and j : S^{0} → S an open immersion. Let

X^{0} ⟶ X
p^{0}↓  ↓^{p}
S^{0} ⟶^{j} S

be the induced fiber square.

a) Let

f^{0} : (X^{0}, Π^{0}, 0) →^{∼} ⊕^{m}(S^{0}, Δ(S^{0}), −q)

be an isomorphism in CHM(S^{0}). Then there are a projector Π ∈ CH_{dim X}(X ×_{S} X) lifting Π^{0} and an isomorphism

f : (X, Π, 0) →^{∼} ⊕^{m}(S, Δ(S), −q)

lifting f.

b) Moreover if Π^{0} is a constituent of P^{0} and P is a lifting of P^{0}, then one can take Π to be a constituent of P.

Proof. a) We have

(X^{0}, Δ(X^{0}), 0) f^{0} ; ⟶ ; ⟵ ; g^{0} ⊕^{m} (S^{0}, Δ(S^{0}), −q)

such that f^{0}∘g^{0} = id and Π^{0} = g^{0}∘f^{0}. Using Lemma 1, take arbitrary liftings f, g of f^{0}, g^{0} respectively, so that one has

(X, Δ(X), 0) f ; ⟶ ; ⟵ ; g ⊕^{m} (S, Δ(S), −q)

Since the map

End_{CHM(S)}((S, Δ, −q)) = CH_{dim S}(S) → End_{CHM(S^{0})}((S^{0}, Δ, −q)) = CH_{dim S}(S^{0})

is an isomorphism, we still have the equality f ∘g = id. By Lemma 2, Π := g∘f is a projector and one has f : (X, Π, 0) →^{∼} ⊕^{m}(S, Δ(S), −q) lifting f^{0}.

b) If Π^{0} is a constituent of P^{0}, then f^{0}∘P^{0} = f^{0} since f^{0} = f^{0}∘Π^{0} = f^{0}∘Π^{0}∘P^{0} = f^{0}∘P^{0}; similarly P^{0}∘g^{0} = g^{0}. Therefore if we replace the f and g from above by f ∘P, resp. P ∘g, we still get liftings of f^{0} and g^{0} respectively. Taking these “new” f and g, we therefore still get f ∘g = id and for the “new” Π = g∘f we clearly have P ∘Π = Π = Π∘P.

1.6 Lifting Lefschetz motives, continued.

Let now p : X → S be as in section 1.1, but we assume (for the moment) only the condition α) of 1.1. With the projector Π_{∞}(S̃/S) as in (1.1), it is proved in [GHM1] there is an exact sequence

(1) 0 → I → (1 − Π_{∞}(S̃/S))∘ CH_{dim S}(S̃ ×_{S} S̃)∘(1 − Π_{∞}(S̃/S)) → CH_{dim S}(S^{0}) → 0

with I a nilpotent (two-sided) ideal. Set A = (1 − Π_{∞}(S̃/S))∘ CH_{dim S}(S̃ ×_{S} S̃)∘(1 − Π_{∞}(S̃/S)) and A^{0} = CH_{dim S}(S^{0})(= Q). Clearly if m ≥ 1 is an integer and M_{m}(−) denotes the ring of m by m matrices one again has an exact sequence

0 → J → M_{m}(A) → M_{m}(A^{0}) → 0

with J also nilpotent. Thus from (1) we get an exact sequence

(2) 0 → J → End_{CHM(S)}(⊕^{m}(S̃, 1 − Π_{∞}(S̃/S), 0)) → End_{CHM(S^{0})}(⊕^{m}(S^{0}, Δ, 0)) → 0

with J a nilpotent two-sided ideal.

Proposition 2. Let p : X → S be as above (in particular satisfying condition α) ). Let P^{0}, Π^{0} ∈ CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) be projectors with Π^{0} a constituent of P^{0}, and let P ∈ CH_{dim X}(X ×_{S} X) be a projector lifting P^{0}. Let moreover given an isomorphism

f^{0} : (X^{0}, Π^{0}, 0) →^{∼} ⊕^{m}(S^{0}, Δ(S^{0}), −q)

in CHM(S^{0}). Then there are a projector Π ∈ CH_{dim X}(X ×_{S} X), that lifts Π^{0} and is a constituent of P, and an isomorphism

f : (X, Π, 0) →^{∼} ⊕^{m}(S̃, 1 − Π_{∞}(S̃/S), −q)

in CHM(S) lifting f^{0}.

Proof. The argument starts as in the proof of Proposition 1 (with S replaced by S̃); i.e. if g^{0} is the inverse of f^{0}, lift first f^{0}, g^{0} to f, g such that f = f ∘P and g = P ∘g (see part of proof Proposition 1). However now have now only

f ∘g = 1 + η

with η ∈ J where J is in the sequence (2) above. Since η is nilpotent, we can replace g by g∘(1 + η)^{−1}, i.e., without loss of generality we can assume f ∘g = id, and we still have g = P ∘g. Now the proof goes as in Proposition 1.

1.7 Proof of the theorem.

Let p : X → S now satisfy the conditions of the theorem.

Step I. Construction of relative relative Chow-Künneth decomposition for X/S.

As recalled in (1.1), we have that X/S has a relative Chow-Künneth decomposition over S with projectors

P^{j}/S (j = 0, ⋯ 2d)  and  P_{∞}^{j}/S

The restrictions (P^{j})^{0}/S^{0} ∈ CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) of P^{j}/S are relative Chow-Künneth projectors for X^{0}/S^{0}.

Step II. Construction of projectors (P^{2r}/S)_{alg} which are constituents of (P^{2r}/S) for 0 ≤ r ≤ d.

Let t be a point of S^{0}. There is a non-degenerate pairing via cup product

H_{B}^{2r}(X_{t}(C), Q) × H_{B}^{2d−2r}(X_{t}(C), Q) → Q .

Since the action of π_{1} = π_{1}^{top}(S^{0}, t) on H_{B}^{i}(X_{t}, Q) is semi-simple by [De], the pairing obtained by restriction

H_{B}^{2r}(X_{t}(C), Q)^{π_{1}} × H_{B}^{2d−2r}(X_{t}(C), Q)^{π_{1}} → Q

is still non-degenerate.

By condition ii) of the theorem, we can take g_{j} ∈ CH^{r}(X), j = 1, ⋯, m = dim H_{B}^{2r}, resp. f_{i} ∈ CH^{d−r}(X), i = 1, ⋯, m such that the classes of their restrictions to the fiber X_{t} form dual bases for H^{2r}(X_{t})^{π_{1}}, resp. H^{2d−2r}(X_{t})^{π_{1}}. Moreover without loss of generality we can assume that

(P^{2r}/S)∘g_{j} = g_{j}, resp. f_{i} ∘(P^{2r}/S) = f_{i} .

Namely replacing g_{j} by (P^{2r}/S)∘g_{j} has in the fiber X_{t} only the effect that we replace the corresponding cohomology class of g_{j}|X_{t} by its image under the action of P^{2r}|X_{t}; however P^{2r}|X_{t} acts as the identity on H^{2r}(X_{t}) (being the 2r-th Künneth component). Similarly replacing f_{i} by f_{i}∘(P^{2r}/S) gives in H^{2d−2r}(X_{t}) instead of the cohomology class of f_{i}|X_{t} the cohomology class of (f_{i}∘P^{2r})|X_{t} which is (by Lieberman’s Lemma, see [Kl]) the cohomology class of ′(P^{2r}|X_{t})_{∗}(f_{i}|X_{t}), with ′(P^{2r}|X_{t}) the transpose of (P^{2r}|X_{t}), and we get the same as before since the cohomology class of ′(P^{2r}|X_{t}) is the (2d − 2r)-th Künneth component.

Now looking first over the open part X^{0}/S^{0} (i.e., restricting to f_{i}^{0}, g_{j}^{0}), we can apply Lemma 4 to get a relative projector Π^{0}/S^{0}, which is a constituent of (P^{2r})^{0}/S^{0}, and next applying Proposition 2, we get a relative projector Π/S which is a constituent of P^{2r}/S and such that

(X/S, Π/S, 0) ≅ ⊕^{m}(S̃, 1 − Π_{∞}(S̃/S), −r) .

Let (P^{2r}/S)_{alg} := Π/S (the subscript “alg” indicates it splits the algebraic cohomology in the fiber X_{t}) and (P^{2r}/S)_{trans} := P^{2r}/S − (P^{2r}/S)_{alg}.

Step III. Local systems on S^{0}.

Consider on S^{0} the following local systems

V^{i} = \{ \begin{matrix}R^{i}p^{0}_{∗}Q & if i = 2r − 1 (0 ≤ r ≤ d), \\ R^{2r}p^{0}_{∗}Q /(R^{2r}p^{0}_{∗}Q)^{π_{1}} & if i = 2r (0 ≤ r ≤ d). \end{matrix}

In the second case (R^{2r}p^{0}_{∗}Q)^{π_{1}} is the constant system with fiber H^{2r}(X_{t}, Q)^{π_{1}}. By our assumption iii) we have H^{∗}(S^{0}, V^{i}) = 0 if ∗ ≠ dim S.

Step IV. From relative to absolute motives.

Using (see section 1.2) the functor

Φ : CHM(S) → CHM(k),
M/S = (X/S, P/S, r) ↦ (X, P, r) = Φ(M/S) = M ,

we get now the following four types of motives (all in CHM(k)):

M^{2r−1} = (X, P^{2r−1}, 0) (1 ≤ r ≤ d), M_{trans}^{2r} = (X, P_{trans}^{2r}, 0) (0 ≤ r ≤ d),
M_{alg}^{2r} = (X, P_{alg}^{2r}, 0) (0 ≤ r ≤ d), and M_{∞} = (X, P_{∞}, 0) .

The first two types have “pure” cohomology, i.e. cohomology in only one degree. This follows from Step III and the claim below. Indeed for M^{2r−1}, by the claim

(P^{2r−1})_{∗}Rp_{∗}Q_{X} = IC_{S}(R^{2r−1}p_{∗}^{0}Q_{X^{0}})[−2r + 1] ,

thus H^{j}(M^{2r−1}) = IH^{j−2r+1}(S, R^{2r−1}p_{∗}^{0}Q_{X^{0}}). It is zero if j − 2r + 1 ≠ dim S. Similarly M_{trans}^{2r} has j-th cohomology equal to IH^{j−2r}(S, V^{2r}), which is zero if j − 2r ≠ dim S.

The M_{∞} has a decomposition according to the decomposition P_{∞} = ∑ P_{∞}^{j}. It remains to examine M_{alg}^{2r}.

Claim. Let (P^{i})^{0} = (P^{i})_{1}^{0} + (P^{i})_{2}^{0} be the decomposition of the projector (P^{i})^{0} ∈ CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) to the sum of mutually orthogonal projectors. Accordingly, one has a decomposition of the local system

R^{i}p_{∗}^{0}Q_{X^{0}} = W_{1}^{i} ⊕ W_{2}^{i} .

Let P^{i} = P^{i}/S ∈ CH_{dim X}(X ×_{S} X) be a lifting as above, and P^{i} = (P^{i})_{1} + (P^{i})_{2} be a decomposition to mutually orthogonal projectors. Then

Im[(P^{i})_{a}]_{∗}Rp_{∗}Q_{X} ≅ IC_{S}(W_{a}^{i})[−i], .

This is proved as in [GHM-1,Proposition I-5], where Im(P^{i})_{∗}Rp_{∗}Q_{X} ≅ IC_{S}(R^{i}p_{∗}^{0}Q_{X^{0}})[−i] is proven.

Step V. Splitting the M_{alg}^{2r}.

There is the isomorphism (X, P_{alg}^{2r}, 0) ≅ ⊕^{m}(S̃, 1 − Π_{∞}(S̃/S), −r). By assumption i)’ we can split the motive on the right into motives with “pure” cohomology; using the isomorphism we can split the left.

1.8 We show that the condition ii) in section 1.3 follows from another condition. For the formulation, we consider a generic fiber of p : X → S, to be made precise below.

We take a subfield k_{1} ⊂ C, the algebraic closure of a field of finite type over Q, so that there are k_{1}-varieties X_{1}, S_{1}, and a map p_{1} : X_{1} → S_{1} defined over k_{1} with

S = S_{1} ⊗_{k_{1}} C, X = X_{1} ⊗_{k_{1}} C, and p = p_{1} ⊗_{k_{1}} C .

Let ξ be the generic point of S_{1}, ξ̄ the corresponding geometric generic point, so

ξ = Spec k_{1}(S_{1}) → S_{1} , ξ̄ = Spec \overline{k_{1}(S_{1})} → S_{1} .

Let t : \overline{k_{1}(S_{1})} ↪ C be an embedding over k_{1}, which gives a point t : Spec C → S_{1}, equivalently t : Spec C → S. Taking the fibers over these points, we have

X ⟶ X_{1} ⟵ (X_{1})_{ξ} ⟵ (X_{1})ξ̄ ⟵ X_{t}
p↓  ↓^{p_{1}}  ↓  ↓  ↓
S ⟶ S_{1} ⟵ ξ ⟵ ξ̄ ⟵ t

Note (X_{1})_{t} = X_{t}. The alternative condition we consider is:

ii)’ With t as chosen above, the image of the cycle map

γ : CH^{r}(X_{t}) → H_{B}^{2r}(X_{t}, Q)

contains the invariant part H_{B}^{2r}(X_{t}, Q)^{π_{1}^{top}(S^{0},t)}.

Proposition 4. The condition ii)’ implies the condition ii).

Proof. For any α ∈ H_{B}^{2r}(X_{t}, Q)^{π_{1}^{top}(S^{0},t)} take Z^{t} ∈ CH^{r}(X_{t}) such that γ(Z^{t}) = α. Step 1. One may assume Z^{t} is defined over L = k_{1}(ξ̄). The Z^{t} is defined over a field L(v) of finite type over L. Let V be a model of L(v), i.e.

a smooth variety over L with function field L(v). Replacing V by an open set if necessary, we have a cycle Z ∈ CH^{r}((X_{1})ξ̄ ×_{L} V) which restricts to Z^{t} on the generic fiber over V. Take an L-valued point v_{0} ∈ V and let Z(v_{0}) ∈ CH^{r}((X_{1})ξ̄) be the restriction of Z to the fiber over v_{0}. Then γ(Z(v_{0})) = α. Write Zξ̄ = Z(v_{0}) in the following.

Step 2. One may assume Zξ̄ is defined over k_{1}(ξ), i.e. that Zξ̄ ∈ CH^{r}((X_{1})_{ξ}).

Consider the following commutative diagram

CH^{r}(X_{t}) ⟶^{γ} H_{B}^{2r}(X_{t}, Q) ⊂ H_{B}^{2r}(X_{t}, Q_{ℓ}) ≅ H_{et}^{2r}(X_{t}, Q_{ℓ})
↑  ↑≅
CH^{r}((X_{1})ξ̄) ⟶^{γ_{et}}  H_{et}^{2r}((X_{1})ξ̄, Q_{ℓ})

The isomorphism H_{B}^{2r}(X_{t}, Q_{ℓ}) ≅ H_{et}^{2r}(X_{t}, Q_{ℓ}) is the comparison between Betti and etale cohomoloy for complex varieties, and the vertical maps are the ones induced from X_{t} → (X_{1})ξ̄.

Let G = Gal(k_{1}(ξ̄)/k_{1}(ξ)). Then we have natural maps

G → π_{1}^{alg}(S^{0}, ξ̄) ←^{∼} π_{1}^{alg}(S^{0}, t)

(the former is onto, the latter an isomorphism) and the cycle maps

γ : CH^{r}((X_{1})ξ̄) → H_{et}^{2r}((X_{1})ξ̄, Q_{ℓ}) ≅ H_{B}^{2r}(X_{t}, Q_{ℓ})

are compatible with the actions of G, resp. π_{1}^{alg}(−). Since the map π_{1}^{top}(S^{0}, t) → π_{1}^{alg}(S^{0}, t) has dense image, α ∈ H_{B}^{2r}(X_{t}, Q_{ℓ}) is invariant with respect to the action of π_{1}^{alg}(S^{0}, t). Therefore if γ ∈ G, γ(σ(Zξ̄)) = α. Hence averaging over the (finite number of) conjugates of Zξ̄, we get the claim.

Step 3. Since the restriction map CH^{r}(X_{1}) → CH^{r}((X_{1})_{ξ}) is surjective, there is Z_{1} ∈ CH^{r}(X_{1}) which maps to α. Let Z ∈ CH^{r}(X) be the image of Z_{1}. Then γ(Z) = α.

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Department of Mathematics, University of Oklahoma, 601 Elm, Room 423, Norman, Oklahoma 73019 U.S.A.

E-mail address: bgordon@ou.edu

Graduate School of Mathematics, Kyushu University, Fukuoka, 812 JAPAN E-mail address: hanamura@math.kyushu-u.ac.jp

Department of Mathematics and Computer Science, Leiden University, P.O. Box 9512, 2300 RA Leiden, THE NETHERLANDS

E-mail address: murre@rulwinw.leidenuniv.nl