CHOW-KÜNNETH PROJECTORS

FOR MODULAR VARIETIES

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

Abstract. We show the existence of the Chow-Künneth projectors for certain varieties, including Kuga-Shimura varieties of Hilbert modular varieties. The Chow-Künneth projectors of a smooth projective variety are, by definition, mutually orthogonal idempotents of the Chow ring of self-correspondences which give decomposition of the total cohomology of the variety into degree pieces.

Projecteurs de Chow-Künneth pour des variétés modulaires

Résumé. Nous démontrons l’existence des projecteurs de Chow-Künneth pour certaines variétés, inclurent les variétés de Kuga-Shimura des variétés modulaires de Hilbert. Les projecteurs de Chow-Künneth d’une variété lisse projective sont par définition des idempotents orthogonales de l’anneau de Chow des auto-correspondances qui donnent la décomposition par les degrés de la cohomologie totale de la variété.

Let X be a smooth projective variety of dimension d over a field k. The Chow-Künneth projectors of X ([13]) 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. (The existence is known for curves, surfaces [12], and abelian varieties.) 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, [10],[11] – that the Künneth components of the diagonal be the classes of algebraic cycles. Further, the projectors Π_{i} should satisfy additional properties, [13].

For smooth projective maps between smooth quasi-projective varieties, one can analogously define the notion of relative Chow-Künneth projectors. They exist for abelian schemes [5].

One may further consider relative Chow-Künneth projectors for any (not necessarily smooth) projective map p : X → S, with X smooth over C, as in [4]. They are projectors in the ring CH_{dim X}(X ×_{S} X) that give the decomposition of the derived image Rp_{∗}Q_{X} into the sum of shifts of intersection complexes of local systems on strata of S.

Under certain, rather strong, assumptions on the fibers of p, we show such relative projectors exist, Theorem I. With additional assumptions, it can be proven X has absolute Chow-Künneth projectors, Theorem II. The assumptions for Theorem II are satisfied for Kuga-Shimura families over a Hilbert modular variety. We give only an outline of the proofs of Theorems I and II; details will be published elsewhere.

We note the case of elliptic modular varieties had been studied in [8],[9].

1. Relative Chow-Künneth projectors.

Throughout this paper, for a variety X over C, CH_{k}(X) := A_{k}(X) ⊗ Q is its rational Chow group, where A_{k}(X) is as defined as in [6], 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.

Let X and S be a quasi-projective varieties over C and assume X smooth. Let p : X → S be a projective surjective map. Assume

(i) there is a finite set of points Σ ⊂ S such that p is smooth over S − Σ.

(ii) Y = (p^{−1}Σ)_{red} is a divisor with normal crossings.

(iii) all the irreducible components Y_{j} of Y are toric varieties.

(iv) there exist relative Chow-Künneth projectors for the map p^{0} := p|_{X^{0}} : X^{0} → S^{0} where S^{0} = S − Σ and X^{0} = p^{−1}(S^{0}). 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.

Let Ỹ → Y be the normalization. The assumption (iii) implies CH_{k}(Ỹ ×_{S} Ỹ) →^{∼} H_{2k}(Ỹ ×_{S} Ỹ). Let q : Ỹ → Σ and ι : Ỹ → X be the induced maps. Let i : Σ → S be the closed immersion, and j : S^{0} → S the open immersion.

Ỹ ⟶^{ι} X ⟵ X^{0}
↓q  ↓p  ↓p^{0}
Σ ⟶^{i} S ⟵^{j} S^{0}

The group CH_{dim X}(X ×_{S} X) has a natural ring structure, and acts on the derived image Rp_{∗}Q_{X}. For a projector Π ∈ CH_{dim X}(X ×_{S} X), the corresponding endomorphism Π_{∗} ∈ End(Rp_{∗}Q_{X}) is a projector; we denote by Π_{∗}Rp_{∗}Q_{X}its image. For a local system V on S^{0}, IC_{S}(V) denotes its intersection complex [7].

Theorem I. There exist local systems (indexed by a finite set of integers j) V_{S}^{j} on S^{0}, V_{Σ}^{j} on Σ, respectively, a finite set of mutually orthogonal projectors adding up to 1 in the ring CH_{dim X}(X ×_{S} X),

{Π_{S}^{j}, Π_{Σ}^{j}} ,

and isomorphisms

(Π_{S}^{j})_{∗}Rp_{∗}Q_{X} →^{∼} IC_{S}(V_{S}^{j})[−j + dim S]

and

(Π_{Σ}^{j})_{∗}Rp_{∗}Q_{X} →^{∼} i_{∗}V_{Σ}^{j}[−j] .

In other words, there exists an isomorphism

Rp_{∗}Q_{X} →^{∼} ⨁ IC_{S}(V_{S}^{j})[−j + dim S] ⊕ ⨁ i_{∗}V_{Σ}^{j}[−j]

under which (Π_{S}^{j})_{∗} and (Π_{Σ}^{j})_{∗} are projections to the direct summands on the right hand side.

The projectors as stated are called relative Chow-Künneth projectors. Note the decomposition of Rp_{∗}Q_{X} as above is known to exist in the derived category of Q-sheaves D^{b}(S) by [2], [15].

2. Absolute Chow-Künneth projectors.

Let X and S be projective varieties over C, 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 C, 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. The complement X − X^{0} is assumed to be a divisor with normal crossings. There is a resolution of singularities p′ : S̃ → S (an isomorphism over S^{0}) such that p factors through S̃. We assume S̃ − S^{0} is a divisor with normal crossings. Let d = dim X − dim S.

X^{0} ⟶ X
↘
p^{0}↓ p↓ S̃
↙
S^{0} ⟶ S

Theorem II. Let p : X → S satisfy the following assumptions.

(i) The irreducible components of S̃ − S^{0} are smooth projective toric varieties. (ii) The irreducible components of X − X^{0} are smooth projective toric varieties. (iii) The variety X^{0}/S^{0} has a relative Chow-Künneth decomposition. (iv) S̃ has a Chow-Künneth decomposition (namely, it has Chow-Künneth pro-

jectors).

(v) 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}.)

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

(vii) 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.

Under these assumptions X has a Chow-Künneth decomposition.

3. Outline of proofs of Theorems I and II.

Let CHM(S) be the pseudo-abelian category of Chow motives over S. See [4] for details. If X is a smooth variety with a projective (not necessarily smooth) map p : X → S, and r ∈ Z, there is an associated object h(X/S)(r) in CHM(S). If q : Y → S is another such, then

Hom(h(X/S)(r), h(Y/S)(s)) = CH_{dim Y −s+r}(X ×_{S} Y) .

Given a map f : X → Y over S, one has morphisms

f^{∗} : h(Y/S) → h(X/S),  and  f_{∗} : h(X/S)(dim X) → h(Y/S)(dim Y) .

Under the assumption of Theorem I, we have morphisms

h(Ỹ /S)(−1) ⟶^{ι_{∗}} h(X/S) ⟶^{ι^{∗}} h(Ỹ /S) .

We can show that there exists an element

Π̃_{Σ} ∈ CH_{dim X}(Ỹ ×_{S} Ỹ) = Hom(h(Ỹ /S), h(Ỹ /S)(−1))

satisfying the following identities:

(1)  ι^{∗}ι_{∗}Π̃_{Σ}ι^{∗}ι_{∗} = ι^{∗}ι_{∗} ,
(2)  Π̃_{Σ}ι^{∗}ι_{∗}Π̃_{Σ} = Π̃_{Σ}.

According to the decomposition

CH_{dim X}(Ỹ ×_{S} Ỹ) = ⊕_{j} Hom(R^{j+2}q_{∗}RỸ , R^{j}q_{∗}RỸ)

we decompose Π̃_{Σ} = ⊕_{j}Π̃_{Σ}^{j}.

Let

Π_{Σ}^{j} := (ι × ι)_{∗}Π̃_{Σ}^{j} = ι_{∗}Π̃_{Σ}^{j}ι^{∗} ∈ CH_{dim X}(X ×_{S} X) = End(h(X/S) ) ,

and Π_{Σ} = ι_{∗}Π̃_{Σ}ι^{∗} = ∑ Π_{Σ}^{j}. The condition (2) implies that Π_{Σ}^{j} are mutually orthogonal projectors, one can prove, using (1),

Im(Π_{Σ}^{j})_{∗} Rp_{∗}Q_{X} →^{∼} i_{∗}V_{Σ}^{j}[−j] .

Here the V_{Σ}^{j} are local systems on Σ such that

Rp_{∗}Q_{X} →^{∼} ⨁ IC_{S}(V_{S}^{i})[−i + dim S] ⊕ ⨁ i_{∗}V_{Σ}^{i}[−i]

(the decomposition theorem, [2], [15]).

Next let I be defined by the exact sequence

0 → I → CH_{dim X}(X ×_{S} X) ⟶^{φ} CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) → 0

where φ is the restriction. I is an ideal of CH_{dim X}(X ×_{S} X). Let R be the two-sided orthogonal complement of Π_{Σ}, namely

R := (1 − Π_{Σ})∘ CH_{dim X}(X ×_{S} X)∘(1 − Π_{Σ}) .

It is a subring of CH_{dim X}(X ×_{S} X) with identity 1 − Π_{Σ}. One thus has an exact sequence

0 → I ∩ R → R ⟶^{φ} CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) → 0 .

It can be shown, using (1) above, that the ideal I is nilpotent.

Take an orthogonal set of projectors Π_{S^{0}}^{j} ∈ CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) adding up to 1 which gives rise to the decomposition on S^{0}:

Rp_{∗}Q_{X^{0}} = ⊕R^{j−dim S}p_{∗}^{0}Q_{X^{0}}[−j + dim S], .

The Π_{S^{0}}^{j} can be lifted to orthogonal projectors Π_{S}^{j} in R adding up to the identity 1 − Π_{Σ}. The projectors thus obtained give rise to a decomposition as stated in Theorem I.

For Theorem II, we start with the relative Chow-Künneth projectors Π_{S}^{i}(X/S) and Π_{Σ}^{i}(X/S) in CH_{dim X}(X ×_{S} X). Set

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

For each r, using the assumption (v), we construct the projector P_{alg}^{2r}, which is a constituent of P^{2r} (namely P^{2r} − P_{alg}^{2r} is also a projector), so that its restriction (P_{alg}^{2r})^{0} := (P_{alg}^{2r})|S^{0} ∈ CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) satisfies

(P_{alg}^{2r})_{∗}^{0}R^{2r}p_{∗}^{0}Q_{X^{0}} = (R^{2r}p_{∗}^{0}Q_{X^{0}})^{π_{1}}

the right hand side being the constant local system of the π_{1}-invariant part. Let P_{trans}^{2r} = P^{2r} − P_{alg}^{2r}. The mutually orthogonal projectors adding to 1 in CH_{dim X}(X ×_{S} X), P^{2r−1}, P_{alg}^{2r}, P_{trans}^{2r}, and P_{∞}^{i} remain so in CH_{dim X}(X × X) under the obvious ring homomorphism. We keep the same notation for their images.

We obtain the decomposition of the motive M = h(X) ∈ CHM(C) into the sum of the following four types of motives.

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_{∞}^{i} = (X, P_{∞}^{i}, 0) .

The first two and the last types have “pure” cohomology, i.e. cohomology in only one degree. To verify we use the assumptions (vi) and (vii).

For the third type, we can show there is an isomorphism (for some m) (X, P_{alg}^{2r}, 0) ≅ ⊕^{m}(S̃, 1 − Π_{∞}(S̃/S), −r). Here Π_{∞}(S̃/S) is the projector at infinity for the map S̃ → S (which also exists by applying Theorem I to it). By assumption (iv) we can split the motive on the right into motives with “pure” cohomology.

4. Applications: families over Hilbert modular varieties.

Let F be totally real number field of degree d of degree d = [F : Q], let O = O_{F} denote the ring of integers of F, and let

Φ = (ϕ^{(1)}, …, ϕ^{(d)}) : F ↪ R^{d}

be the embedding into R^{d} induced by the d embeddings of F into R. Note Φ induces an embedding SL_{2}(F) ↪ SL_{2}(R)^{d}, thus SL_{2}(F) acts on ℌ^{d}, the product of d complex upper half-planes, by componentwise linear fractional transformation.

Let Γ ⊂ SL_{2}(F) be a neat congruence subgroup. It acts on ℌ^{d} freely and properly discontinuously on ℌ^{d}. The quotient as a complex manifold

S^{0} := Γ\ℌ^{d}

has the structure of a smooth quasi-projective variety over C. There is a projective variety S, the Satake-Baily-Borel compatification, obtained by adding a finite set of points (the cusps) to S^{0}; we refer to it as a Hilbert modular variety. As is wellknown, there is a smooth compactification S̃ of S^{0} lying over S such that S̃ − S^{0} is a divisor whose components are smooth projective toric varieties.

Let L ⊂ F ×F be a projective O-module of rank 2 such that for g ∈ Γ, g ·L = L. The semi-direct product Γ ⋉ L acts on ℌ^{d} × C^{d} by

(( \begin{matrix}a & b \\ c & d \end{matrix} ) , ^{t}(μ, ν)) (τ, z) = (\frac{aτ + b}{cτ + d} , \frac{z + μτ + ν}{cτ + d} ).

(Here the embedding Φ and the multi-indexing of everything are omitted from the notation.) The quotient

X^{0} := (Γ ⋉ L)\(ℌ^{d} × C^{d})

is a quasi-projective variety that maps to S^{0} by projection on the first factor; one thus has a family of d-dimensional abelian varieties with real multiplication by O.

For each m ≥ 1, let

X_{m}^{0} := \overbrace{X^{0} ×_{S^{0}} × ⋯ ×_{S^{0}} X^{0}}^{m times} ,

the m-fold self-product of X^{0} over S^{0}, and p^{0} = p_{m}^{0} : X_{m}^{0} → S^{0} be the projection. By the method of toroidal compactifications, [1], [14], we can take a smooth projective variety X_{m} containing X_{m}^{0} as an open set, such that the complement is a divisor with components smooth toric varieties, and that p^{0} extends to a map X_{m} → S̃. In other words, for p : X_{m} → S, the assumptions (i), (ii) of Theorem II are satisfied.

Over S^{0}, p is an abelian scheme, so the assumption (iii) is also satisfied by [5]. The hypotheses (iv) - (vii) can be verified by the knowledge of cohomology of the Hilbert modular variety and the vanishing theorem (by Matsushima-Shimura) of cohomology of S^{0} with coefficients in a local system.

Applying Theorem II one concludes the X_{m} has Chow-Künneth projectors. In particular, Conjecture (C) of Grothendieck holds for X_{m}.

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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, Leiden University, P.O. Box 9512, 2300 RA Leiden, THE NETHERLANDS

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