RELATIVE CHOW-KÜNNETH

PROJECTORS FOR MODULAR VARIETIES

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

Introduction.

Let X be a smooth projective variety over an algebraically closed field. It is one of Grothendieck’s well-known “standard conjectures” [Gr] that the Künneth components of the diagonal Δ ⊂ X × X should be algebraic cohomology classes. Since then, as part of a much more far-reaching conjecture about the structure of Chow groups, the third-named author has proposed an even stronger conjecture [Mu2, Conj. A], that in the Q-algebra of algebraic correspondences on X × X modulo rational equivalence there should exist a collection of orthogonal idempotents Π^{i} whose sum is the identity, i.e., the diagonal class, and which represent the Künneth components of the diagonal in cohomology; such a collection of Π^{i} is called a Chow-Künneth decomposition for X, and the Π^{i} are Chow-Künneth projectors. This conjecture generalizes further to the case where λ : X → S is a smooth projective scheme over a smooth quasiprojective base, in which case one works with Rλ_{∗}Q instead of cohomology, and Π^{i} ∈ CH^{dim_{S} X}(X ×_{S} X) ⊗ Q, see [DM]; in this case we call the Π^{i} relative Chow-Künneth projectors. Motivated by this form of the conjecture and the decomposition theorem of [BBD], Corti and the second-named author [CM] have extended the conjecture still further to allow X and S to be quasiprojective varieties over C, where X is smooth, p : X → S is projective, and p is a stratified map with respect to compatible stratifications on S and X; Conjecture 1.4, the Motivic Decomposition Conjecture, gives a precise statement after the necessary preliminaries have been reviewed in Section 1.

In this paper, in Section 2, we prove the Motivic Decomposition Conjecture when (a) there is a finite set of points Σ ⊂ S such that p is smooth over S^{0} := S − Σ;

(b) Y := (p^{−1}Σ)_{red} is a divisor with normal crossings; (c) CH_{k}(Ỹ ×_{S} Ỹ) →^{∼} H_{2k}(Ỹ ×_{S} Ỹ); and (d) there exist relative Chow-Künneth projectors for the map p^{0} := p|_{X^{0}} : X^{0} → S^{0}, where X^{0} = p^{−1}(S^{0}). These hypotheses are (nontrivially) satisfied, for example, by certain families of abelian varieties, such as a smooth toroidal compactification of families of abelian varieties with real multiplication parameterized by elliptic and Hilbert modular varieties (over C). More generally, sometimes (d) can be deduced from knowledge of a Chow-Künneth decomposition of the generic fiber of p : X → S, whereas (c) is true if the components of Y are all toric varieties.

In Part II we will show that, under some additional assumptions on the cohomology of X, also satisfied by families of abelian varieties over elliptic and Hilbert modular varieties, it is possible to pass from having the relative Chow-Künneth projectors to a(n absolute) Chow-Künneth decomposition of X as a complex projective variety, cf. [GMu1] [Gmu2].

Typeset by AMS-TEX

Conventions and notations. We only consider varieties X over the complex numbers. 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^{i}(X, Q). Also, we will consider only complexes of sheaves of Q-vector spaces.

§1. Relative Chow motives, perverse sheaves, and the decomposition theorem.

Chow motives. Let S be a quasi-projective variety over a field k (k is arbitrary for this paragraph). Let CHM(S) be the pseudo-abelian tensor category of Chow motives over S. We recall the definition briefly; for generalities on this category, see [CH]. If S = Spec k, it coincides with the category of Chow motives of Grothendieck, cf. [Ma].

If X, Y are smooth varieties, Y equi-dimensional, with projective maps p : X → S, q : Y → S, consider the Chow group CH_{k}(X ×_{S} Y) as the group of relative correspondences from X to Y. If Z is another smooth equi-dimensional variety, one defines the composition

CH_{s}(X ×_{S} Y) ⊗ CH_{t}(Y ×_{S} Z) → CH_{s+t−dim Y} (X ×_{S} Z)

by

u ⊗ v ↦ v ∘ u := p_{XZ ∗}δ^{!}(u × v)

where δ : Y → Y is the diagonal map, p_{XZ} : X ×_{S} Y_{S} × Z → X ×_{S} Z is the projection and δ^{!} is the refined Gysin map [Fu, Chap. 6].

X ×_{S} Z ⟵^{p_{XZ}} X ×_{S} Y_{S} × Z ⟶ (X ×_{S} Y) × (Y ×_{S} Z)
↓  ↓p_{23}
Y  ⟶^{δ}  Y × Y .

The composition is associative.

An object of CHM(S) is of the form

(X, r, P) = (X/S, r, P)

where X is a smooth variety over k with a projective (not necessarily smooth) map p : X → S, r ∈ Z, and if X has connected components X_{i},

P ∈ ⊕_{i} CH_{dim X_{i}}(X ×_{S} X_{i})

such that P ∘ P = P. If (Y, s, Q) is another object, Y_{j} the components of Y, then

Hom((X, r, P), (Y, s, Q)) = Q ∘ (⊕_{j} CH_{dim Y_{j}−s+r}(X ×_{S} Y_{j})) ∘ P .

Composition of morphisms is induced from the composition of relative correspondences. Direct sum comes from disjoint union of varieties. Let Q(r) := (pt, r, id) be the Tate motif, and (X, r, P)(s) = (X, r + s, P). We also set

h(X/S) = (X, 0, id), h(X/S)(r) = (X, r, id) .

One has the functor

CH^{t} : CHM(S) → Q, CH^{t}((X, r, P)) = P_{∗} CH^{r+t}(X) .

If u ∈ Hom((X, r, P), (Y, s, Q)), the induced map is u_{∗} : P_{∗} CH^{r+t}(X) → Q_{∗} CH^{s+t}(Y) defined as u_{∗}α = p_{2 ∗}δ^{!}(u × α) where p_{2} : X ×_{S} X → X is the second projection.

Y ⟵^{p_{2}} X ×_{S} Y ⟶ X × (X ×_{S} Y)
↓  ↓ p_{12}
X  ⟶^{δ}  X × X .

One has (v∘u)_{∗}α = v_{∗}u_{∗}α. One easily sees:

Proposition 1.1. If j : S^{0} → S is an open immersion, there is a natural full functor

j^{∗} : CHM(S) → CHM(S^{0})

such that j^{∗}(X, r, P) = (X^{0}, r, P^{0}), X^{0} = X ×_{S} S^{0}, and P^{0} = P |_{X^{0}×_{S^{0}} X^{0}}. If i : T → S is a closed immersion, there is a faithful functor

i_{∗} : CHM(T) → CHM(S)

where i_{∗}(X/T, r, P) = (X/S, r, P). If S is projective there is a functor

π_{∗} : CHM(S) → CHM(k)

where π_{∗}(X/S, r, P) = (X, r, P), and the induced map on Hom groups is CH_{∗}(X ×_{S} Y) → CH_{∗}(X × Y).

Perverse sheaves and intersection complexes. For the following material, see [Bo], [BBD],[GMa] and [Sa] for details. From now on we consider quasi-projective varieties over C. For a quasi-projective variety S, denote by D_{c}^{b}(S) the triangulated category of bounded complexes of cohomologically constructible Q-sheaves on S. (We will consider only algebraic Whitney stratifications and constructible sheaves with respect to them.) Recall that a stratification is a decreasing sequence of closed subsets S = S_{0} ⊃ S_{1} ⊃ ⋯ ⊃ S_{dim S} such that S_{α} − S_{α+1} is smooth of pure dimension dim S − α, and that satisfies the Whintney condition.

For any map f : S → T, there are associated functors Rf_{∗}, Rf_{!}, f^{!} and f^{∗} between the categories D_{c}^{b}(S) and D_{c}^{b}(T); they satisfy adjoint properties. If X, Y are varieties with maps p : X → S, q : Y → S, and f : X → Y is an S-morphism, then there is natural map f^{∗} : Rq_{∗}Q_{Y} → Rp_{∗}Q_{X}; if X, Y are both smooth and equi-dimensional, and p, q proper, there is f_{∗} : Rp_{∗}Q_{X} → Rq_{∗}Q_{Y} [2(dim Y − dim X)].

If S is a stratification of S, and T a stratum of S, let i_{T} : T → S be the inclusion. Let D_{c}^{b}(S)^{≤0}, resp. D_{c}^{b}(S)^{≥0} be the full subcategory of D_{c}^{b}(S) consisting of objects K, constructible with respect to a stratification S, satisfying:

H^{i}i_{T}^{∗} K = 0 for i > − dim T,

resp.

H^{i}i_{T}^{!} K = 0 for i < − dim T.

The pair (D_{c}^{b}(S)^{≤0}, D_{c}^{b}(S)^{≥0}) gives a t-structure on D_{c}^{b}(S). In particular, K is in D_{c}^{b}(S)^{≤0}, L in D_{c}^{b}(S)^{≥0}[−1], then Hom(K, L) = 0. Let Perv(S) := D_{c}^{b}(S)^{≤0} ∩ D_{c}^{b}(S)^{≥0}; this is the category of perverse sheaves; it is an abelian category. One has the perverse cohomology functor

pH^{i} : D_{c}^{b}(S) → Perv(S) ,

with the following properties:

(i) it is a cohomological functor, namely ^{p}H^{i}(K[1]) = ^{p}H^{i+1}(K) and takes a distinguished triangle to a long exact sequence;

(ii) the restriction of ^{p}H^{0} to Perv(S) is the identity functor;

(iii) If S is smooth and A is a cohomologically locally constant, then

pH^{i}(K) = H^{i−dim S}(K)[dim S] .

If i : T → S is a closed immersion, i_{∗} : D_{c}^{b}(T) → D_{c}^{b}(S) preserves the subcategories of perverse sheaves, and i_{∗} commutes with^{p}H^{i}. If U → S is an open immersion, j^{∗} : D_{c}^{b}(S) → D_{c}^{b}(U) preserves the subcategories of perverse sheaves, and j^{∗} commutes with ^{p}H^{i}. For a map f : X → Y, one writes ^{p}R^{i}f_{∗}Q_{X} = ^{p}H^{i}Rf_{∗}Q_{X}.

If S^{0} is a smooth dense open subset of S and V a local system on S^{0}, its intersection complex IC_{S}(V) is defined in [GMa] and IC_{S}(V)[dim S] is an object of Perv(S). In fact K = IC_{S}(V)[dim S] satisfies and is characterized by the following: if it is S-constructible for a stratification S and,

(i) if T ⊂ S, T ≠ S is a stratum of S, H^{i}i_{T}^{∗} K = 0 for i ≥ − dim T, and H^{i}i_{T}^{!} K = 0 for i ≤ − dim T;

(ii) if S^{0} is the open stratum, K|_{S^{0}} = V[dim S]. The association V ↦ IC_{S}(V)[dim S] is an exact functor from the category of local systems on S^{0} to the category of Perv(S). If S^{0} = S − Σ, where Σ is a finite set of points, and j : S^{0} → S the open immersion, then IC_{S}(V) = τ_{≤dim S−1}Rj_{∗}V. (This is the case that we will encounter in this paper.)

If i : T → S is a closed immersion and T^{0} ⊂ T open dense and V a local system on T^{0}, IC_{T} (V)[dim T], or more precisely i_{∗}IC_{T} (V)[dim T] is an object of Perv(S). The following is known.

Proposition 1.2. If V, W are local systems on S^{0}, then

Hom(IC_{S}(V), IC_{S}(W)) →^{∼} Hom(V, W) .

If K is a perverse sheaf supported on S − S^{0}, namely K|_{S^{0}} = 0, then

Hom(K, IC_{S}(V)) = Hom(IC_{S}(V), K) = 0 .

The realization functor. There is a map (Y equi-dimensional)

CH_{dim Y +k}(X ×_{S} Y) → Hom(Rp_{∗}Q_{X}, Rq_{∗}Q_{Y} [−2k])

compatible with compositions. Hence there is a functor

real : CHM(S) → D_{c}^{b}(S)

such that on objects

(X, r, P) ↦ P_{∗}Rp_{∗}Q_{X}[2r] ,

(P_{∗} ∈ End_{D_{c}^{b}(S)}(Rp_{∗}Q_{X}) is a projector, and P_{∗}Rp_{∗}Q_{X} is its image, which exists since D_{c}^{b}(S) is pseudo-abelian.)

If X and Y are smooth varieties with projective maps to S, X_{i} the components of X, and f : X → Y a S-morphism, then the cycle [^{t}Γ_{f}] ∈ ⊕ CH_{dim X_{i}}(Y ×_{S} X_{i}) gives the morphism

f^{∗} : h(Y/S) → h(X/S) ,

and if X, Y are equidimensional, [Γ_{f}] ∈ CH_{dim X}(X ×_{S} Y) gives

f_{∗} : h(X/S) → h(Y/S)(dim Y − dim X) .

When they act on Chow groups, they become pull-back and push-forward by f

(f^{∗})_{∗} = f^{∗} : CH^{r}(Y) → CH^{r}(X) , (f_{∗})_{∗} = f_{∗} : CH^{r}(X) → CH^{r+dim Y −dim X}(Y) .

On realizations, they induce the maps f^{∗} and f_{∗} between Rp_{∗}Q_{X} and Rq_{∗}Q_{Y} mentioned before. For u ∈ Hom((X, r), (Y, s)) = CH_{dim Y −s+r}(X ×_{S} Y) and f : X → X′, g : Y → Y′,

(f × g)_{∗}u = g_{∗}uf^{∗} : h(X′/S)(r) → h(Y′/S)(s + dim Y′ − dim Y) .

The realization functor is compatible with the restriction functors j^{∗} : CHM(S) → CHM(S^{0}) and j^{∗} : D_{c}^{b}(S) → D_{c}^{b}(S^{0}), i.e. the following diagram commutes

CHM(S) ⟶^{j^{∗}} CHM(S^{0})
real↓  real↓
D_{c}^{b}(S) ⟶^{j^{∗}} D_{c}^{b}(S^{0})

If i : T → S is a closed immersion, i_{∗} : CHM(T) → CHM(S) and i_{∗} : D_{c}^{b}(T) → D_{c}^{b}(S) are compatible.

If S is projective, one has the following commutative diagram of functors

CHM(S) ⟶^{π_{∗}} CHM(k)
real↓  real↓
D_{c}^{b}(S) ⟶^{H^{∗}} V ect_{Q}

Theorem 1.3 (The decomposition theorem). Let X and S be quasi-projective varieties over C, X smooth, and p : X → S a projective map. Let S = S_{0} ⊃ S_{1} ⊃ ⋯ ⊃ S_{dim S} be a stratification of S such that there is a stratification of X so that p is a stratified map [GMa, §6]. Then:

(1) There are local systems V_{α}^{j} on S_{α}^{0} = S_{α} − S_{α+1} and an isomorphism

ψ : Rp_{∗}Q_{X} →^{∼} ⨁_{j, α} IC_{S_{α}}(V_{α}^{j})[−j + dim S_{α}] .

(2) Let V̄_{α}^{j} be another set of local systems, and ψ̄ : Rp_{∗}Q_{X} →^{∼} ⨁_{j, α} IC_{S_{α}}(V̄_{α}^{j})[−j + dim S_{α}] another isomorphism, and let ψ̄∘ψ^{−1} = (f_{ij}) where

f_{ij} : ⨁_{α} IC_{S_{α}}(V_{α}^{i})[−i + dim S_{α}] → ⨁_{α} IC_{S_{α}}(V̄_{α}^{j})[−j + dim S_{α}] .

Then f_{ij} = 0 for i < j and f_{ii} are isomorphisms. Furthermore, there are unique isomor-

phisms g_{iα} : V_{α}^{i} →^{∼} V̄_{α}^{i} such that f_{ii} = ⊕_{α}IC_{S_{α}}(g_{iα}).

The second part of the statement follows immediately from Proposition 1.2. Notice that, upon taking ^{p}H^{j}, ψ induces the isomorphism

pR^{j}p_{∗}Q_{X} →^{∼} ⨁_{α} IC_{S_{α}}(V_{α}^{j})[dim S_{α}] ,

and upon taking H^{− dim S_{α}}i_{S_{α}^{0}}^{∗}^{p}H^{j} (where i_{S_{α}^{0}} : S_{α}^{0} → S is the inclusion)

H^{− dim S_{α}}i_{S_{α}^{0}}^{∗}^{p}R^{j}p_{∗}Q_{X} →^{∼} V_{α}^{j} .

For general discussions on the following conjecture see [CH].

Conjecture 1.4 (Motivic Decomposition Conjecture). Let X and S be quasi-projective varieties over C, X smooth, and p : X → S a projective map. Let S = S_{0} ⊃ S_{1} ⊃ ⋯ ⊃ S_{dim S} be a stratification of S such that there is a stratification of X so that p is a stratified map. Then there are local systems V_{α}^{j} on S_{α}^{0}, a set of orthogonal projectors {Π_{α}^{j}} in the ring CH_{dim X}(X ×_{S} X) adding up to 1 and isomorphisms

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

In other words, there is an isomorphism

ψ : Rp_{∗}Q_{X} →^{∼} ⨁_{j, α} IC_{S_{α}}(V_{α}^{j})[−j + dim S_{α}]

under which {(Π_{α}^{j})_{∗}} on the left side coincides with the projection to IC_{S_{α}}(V_{α}^{j})[−j+dim S_{α}] on the right. We will refer to {Π_{α}^{j}} as relative Chow-Künneth projectors.

In the case S = Spec k this is part of Murre’s conjecture [Mu2].

§2. Existence of relative Chow-Künneth projectors.

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_{1}, ⋯, s_{n}} such that p is smooth over S − Σ, and (ii) Y = (p^{−1}Σ)_{red} is a divisor with normal crossings. (iii) all the components Y_{j} are toric varieties. (In fact, for Part I, only needs to assume

CH_{k}(Ỹ ×_{S} Ỹ) →^{∼} H_{2k}(Ỹ ×_{S} Ỹ).)

(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}). Let Ỹ → Y be the normalization, q : Ỹ → Σ the induced map, and Ỹ_{s} := q^{−1}(s) ⊂ Ỹ. Let i_{s} : s → S and i : Σ → S the closed immersions, and j : S^{0} → S the open immersion.

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

We will prove the motivic decomposition conjecture in this case:

Theorem I. There exist local systems V_{S}^{j} on S^{0}, V_{Σ}^{j} on Σ, respectively, a set of mutually orthogonal projectors

{Π_{S}^{i}, Π_{Σ}^{i}} ,

only finitely many of them are non-zero, adding up to 1 in the ring CH_{dim X}(X ×_{S} X), and isomorphisms

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

and

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

In other words, there exists an isomorphism

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

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

Remark. We have V_{S}^{i} ≅ R^{i−dim S}p_{∗}^{0}Q_{X^{0}}, so it is zero unless dim S ≤ i ≤ 2 dim X − dim S. The local system V_{Σ}^{i} is zero unless 0 ≤ i ≤ 2(dim X − 1), cf. Proposition 2.2.

One has maps of relative Chow motives over S:

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

We will 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}.

Since

CH_{dim X}(Ỹ ×_{S} Ỹ) = ⨁_{s∈Σ} CH_{dim X}(Ỹ_{s} × Ỹ_{s}),

or equivalently,

Hom(h(Ỹ /S), h(Ỹ /S)(−1)) = ⨁_{s} Hom(h(Ỹ_{s}/S), h(Ỹ_{s}/S)(−1),

one has Π̃_{Σ} = ⊕_{s}Π̃_{s}, with Π̃_{s} ∈ CH_{dim X}(Ỹ_{s} × Ỹ_{s}) . The above identities are equivalent to

(1_{s})  ι_{s}^{∗}ι_{s ∗}Π̃_{s}ι_{s}^{∗}ι_{s}_{∗} = ι_{s}^{∗}ι_{s}_{∗} ,
(2_{s})  Π̃_{s}ι_{s}^{∗}ι_{s}_{∗}Π̃_{s} = Π̃_{s}.

for all s. Furthermore

CH_{dim X}(Ỹ_{s} × Ỹ_{s}) = H_{2 dim X}(Ỹ_{s} × Ỹ_{s}) = ⊕_{j} Hom(H^{j+2}(Ỹ_{s}), H^{j}(Ỹ_{s})) .

So the two identities are equivalent to the respective identities of compositions of graded linear maps

ι_{s}^{∗}ι_{s ∗} : H^{∗}(Ỹ_{s}) → H^{∗+2}(Ỹ_{s})

and Π̃_{s} : H^{∗+2}(Ỹ_{s}) → H^{∗}(Ỹ_{s}).

Proposition I-1. There exists an element Π̃_{Σ} ∈ CH_{dim X}(Ỹ ×_{S} Ỹ) satisfying (1) and (2).

Given the preceding remarks, this is proved by the following simple lemma.

Lemma 2.1. Let H_{∗} and H^{∗} be vector spaces, φ : H_{∗} → H^{∗} a linear map. Then there exists a linear map ψ : H^{∗} → H_{∗} satisfying the following identities:

φψφ = φ ψφψ = ψ .

Proof. Let V be the image of φ and

H_{∗} ⟶^{b} V ⟶^{a} H^{∗}

be the factorization of φ. Let α : H^{∗} → V, β : V → H_{∗} be maps such that αa = id_{V}, bβ = id_{V}. Then the map ψ := βα satisfies the required identities. □

Let ι_{s} × ι_{s} : Ỹ_{s} × Ỹ_{s} → X ×_{S} X be the product map. Let

Π_{s} := (ι_{s} × ι_{s})_{∗}Π̃_{s} = ι_{s ∗}Π̃_{s}ι_{s}^{∗} ,

and Π_{Σ} = ⊕_{s∈Σ}Π_{s} be elements in CH_{dim X}(X ×_{S} X).

B.B. GORDON, M. HANAMURA, J.P. MURRE 9

Proposition I-2. The {Π_{s}}_{s∈Σ} is a set of orthogonal projectors. Proof. We have

Π_{s}Π_{s} = ι_{s ∗}Π̃_{s}ι_{s}^{∗}ι_{s ∗}Π̃_{s}ι_{s}^{∗}
= ι_{s ∗}Π̃_{s}ι_{s}^{∗} by (2_{s})
= Π_{s} .

It is evident that Π_{s}Π_{t} = 0 if s ≠ t. □ Proposition 2.2 [CH, Lemma 4.18]. Let ι^{∗} : Rp_{∗}Q_{X} → i_{∗}Rq_{∗}QỸ and ι_{∗} : i_{∗}Rq_{∗}QỸ [−2] → Rp_{∗}Q_{X} be the canonical maps. Taking H^{0}i^{∗}^{p}H^{j}, one has maps of local systems on Σ,

a : V_{Σ}^{j} ≅ H^{0}i^{∗}^{p}R^{j}p_{∗}Q_{X} → R^{j}q_{∗}QỸ

and

b : R^{j−2}q_{∗}QỸ → H^{0}i^{∗}^{p}R^{j}p_{∗}Q_{X} ≅ V_{Σ}^{j} .

The map a is injective, and b surjective.

We study the action of Π_{s}^{j} on Rp_{∗}Q_{X} and on its perverse cohomology ^{p}R^{i}p_{∗}Q_{X}. There exist local systems V^{i} on S^{0} and V_{Σ}^{i} on Σ, and an isomorphism

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

which will be fixed from now on. Proposition I-3. The action of the projector (Π_{Σ}^{j})_{∗} on ^{p}R^{i}p_{∗}Q_{X} coincides, under the above isomorphism ψ : ^{p}R^{i}p_{∗}Q_{X} →^{∼} IC(V^{i})[dim S]⊕i_{∗}V_{Σ}^{i}, with the projection to summand V_{Σ}^{j} if i = j and zero if i ≠ j. Proof. By Proposition 2.3 (below), the square

pR^{i}p_{∗}Q_{X}  ⟶^{(Π_{Σ}^{j})_{∗}}  ^{p}R^{i}p_{∗}Q_{X}
↓ι^{∗}  ↑ι_{∗}
i_{∗}^{p}R^{i}q_{∗}QỸ ⟶^{(Π̃_{Σ}^{j})_{∗}} i_{∗}^{p}R^{i}q_{∗}QỸ

commutes. The map from IC_{S}(V_{S}^{j})[dim S] to V_{Σ}^{j}, or a map the other way round is zero by Proposition 1.2. Since Π̃_{Σ}^{i} restricts to zero on X^{0} ×_{S} X^{0}, by Proposition 2.4, the induced map from IC_{S}(V_{S}^{j})[dim S] to itself is zero.

Applying the functor H^{0}i^{∗}, one obtains a commutative diagram

V_{Σ}^{j}  ⟶^{(Π_{s}^{i})_{∗}}  V_{Σ}^{j}
↓ a  ↑ b
R^{j−2}q_{∗}QỸ ⟶^{(Π̃_{s}^{i})_{∗}} R^{j}q_{∗}QỸ

One has by (2_{s}) (combined for all s) the equality

abΠ̃_{Σ}^{i}ab = δ_{ij}ab on R^{j}q_{∗}QỸ .

Since b is surjective and a injective,

bΠ̃_{Σ}^{i}a = δ_{ij}id on i_{∗}V_{Σ}^{j} .

This shows the claim. □

Proposition 2.3. Let Z and X be smooth equidimensional varieties with projective maps to S, r : Z → S, p : X → S. Let f : Z → X be a map over S and f ×f : Z ×_{S} Z → X ×_{S} X the product map. For v ∈ CH_{dim X}(Z ×_{S} Z), let u = (f × f)_{∗}v ∈ CH_{dim X}(X ×_{S} X). Then the following diagrams are commutative.

Rp_{∗}Q_{X} ⟶^{u_{∗}} Rp_{∗}Q_{X}
f^{∗}↓  f_{∗}↑
Rr_{∗}Q_{Z} ⟶^{v_{∗}} Rr_{∗}Q_{Z} ,

pR^{i}p_{∗}Q_{X} ⟶^{u_{∗}}  ^{p}R^{i}p_{∗}Q_{X}
f^{∗}↓  f_{∗}↑
pR^{i}r_{∗}Q_{Z} ⟶^{v_{∗}}  ^{p}R^{i}r_{∗}Q_{Z} ,

Proof. One has by definition f_{∗}vf^{∗} = u as morphisms in CHM(S). Since CHM(S) → D_{c}^{b}(S) is a functor, one has f_{∗}v_{∗}f^{∗} = u_{∗} as morphisms in D_{c}^{b}(S); this is the commutativity of the first diagram. The second diagram is obtained from the first by taking i-th perverse cohomology. □

Proposition 2.4. Let p : X → S be as above. Let j : S^{0} → S be an open immersion, X^{0} = p^{−1}S^{0} and p^{0} : X^{0} → S^{0} the restriction. For u ∈ CH_{dim X}(X ×_{S} X), let u^{0} ∈ CH_{dim X}(X^{0} ×_{S^{0}} X^{0}) be its restriction. Then

j^{∗}(u_{∗}) = u_{∗}^{0} : Rp_{∗}^{0}Q_{X^{0}} → Rp_{∗}^{0}Q_{X^{0}} .

Proof. Following through the definitions (see [CH]) this is reduced to the compatibility of the cycle class map with open immersions. □

For the following see [Ja].

Lemma 2.5. Let A be a ring with unit and I ⊂ A a nilpotent two-sided ideal. Then an orthogonal set of projectors {e_{1}, ⋯, e_{m}} of A/I can be lifted to an orthogonal set of projectors {π_{1}, ⋯, π_{m}} of A. If ∑ e_{i} = 1, then necessarily ∑ π_{i} = 1. Moreover if {π′_{1}, ⋯, π′_{m}} is another set of orthogonal projectors lifting the same {e_{1}, ⋯, e_{m}}, there is an element η ∈ I such that π′_{j} = (1 + η)π_{j}(1 + η)^{−1}.

Proposition 2.6. Let A be an object in a pseudo-abelian category. Let A_{j} (j = 1, ⋯, n) be subobjects A such that A = ⨁_{j=1}^{n} A_{j} and Hom(A_{i}, A_{j}) = 0 for i < j. Let P ∈ End(A) be a projector, P_{j} : A_{j} → A_{j} be components of P, and

P′ := ( \begin{matrix}P_{1} & & & & & 0 \\ & P_{2} \\ & & . & . & . \\ 0 & & & & & P_{n} \end{matrix} ) ∈ End(A) .

Then there exists an isomorphism Im P →^{∼} Im P′

Proof. By induction on n. Assume first n = 2. Let in_{P} : Im P → A the canonical injection, pr_{P} : A → Im P the canonical projection, and similarly for in_{P′} : Im P′ → A. We show the maps

pr_{P′} in_{P} : Im P → Im P′ = ⊕ Im P_{j} and pr_{P} in_{P′} : Im P′ → Im P

are isomorphisms, mutually inverse to each other. Write

P = ( \begin{matrix}P_{11} & P_{21} \\ 0 & P_{22} \end{matrix} )

where P_{ii} = P_{i}. Since P is a projector one has

P_{11}P_{12} + P_{12}P_{22} = P_{12} .

So one has P_{11}P_{12}P_{22} = 0. Hence one immediately shows

P P′P = P and P′P P′ = P′ .

Therefore,

pr_{P′} in_{P} pr_{P} in_{P′} = id and pr_{P} in_{P′} pr_{P′} in_{P} = id .

In the case n ≥ 3, decompose the projection P according to the direct sum into two factors A = A_{1} ⊕ (A_{2} ⊕ ⋯ ⊕ A_{n}) and then apply the above argument; then apply the induction hypothesis to the projection on the second factor. □

Applying this to Rp_{∗}Q_{X} and its decompositions to the sum of ^{p}Rp_{∗}Q_{X}[−i] and further to IC_{S}(V_{S}^{i})[−i + dim S] and i_{∗}V_{Σ}^{j}[−i], where the vanishing property is satisfied by Proposition 1.2, one concludes: there is an isomorphism

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

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}.

Proposition I-4. (1) One has an exact sequence

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

(2) The ideal I ∩ R is nilpotent.

Proof.

(1) follows from the more general fact that the functor CHM(S) → CHM(S^{0}) is full. To show (2), note first any element α ∈ I is the push-forward of an element α̃ ∈

CH_{dim X}(Ỹ × Ỹ), namely,

α = ι_{∗}α̃ ι^{∗}

If α ∈ I ∩ R, then

0 = αΠ_{Σ} = ι_{∗}α̃ ι^{∗}ι_{∗}Π̃_{Σ}ι^{∗}

hence in particular

ι^{∗}ι_{∗}α̃ ι^{∗}ι_{∗}Π̃_{Σ}ι^{∗}ι_{∗} = 0 .

Using the first identity Π_{Σ} satisfies, this amounts to

ι^{∗}ι_{∗}α̃ ι^{∗}ι_{∗} = 0 .

Suppose one has elements α_{ν} ∈ I ∩ R, ν = 1, 2, 3, which we may write α_{ν} = ι_{∗}α̃_{ν} ι^{∗}; then

α_{1} α_{2} α_{3} = ι_{∗}α̃_{1}ι^{∗}ι_{∗}α̃_{2}ι^{∗}ι_{∗}α̃_{3}ι_{∗} = 0 .

□

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], .

Using Lemma 2.5, take an orthogonal set of projectors Π_{S}^{j} ∈ CH_{dim X}(X ×_{S} X) lifting these and adding up to 1 − Π_{Σ}. One shows then the following, which concludes the proof of Theorem I.

Proposition I-5. The projectors thus obtained give rise to a decomposition as stated.

Proof. We have only to show Im(Π_{S}^{i})_{∗} →^{∼} IC_{S}(V_{S}^{i})[−i+dim S]. First we study the action of (Π_{S}^{i})_{∗} on the perverse cohomology ^{p}R^{j}p_{∗}Q_{X}; we write (Π_{S}^{i})_{∗} in the matrix form according to the isomorphism ψ : ^{p}R^{j}p_{∗}Q_{X} →^{∼} IC_{S}(V_{S}^{j})[dim X] ⊕ i_{∗}V_{Σ}^{j}. Since (Π_{S}^{i}|_{S^{0}})_{∗} = δ_{ij}id on V_{S}^{j}, one has (Π_{S}^{i})_{∗} = δ_{ij}id on IC_{S}(V_{S}^{j}). So the matrix for (Π_{S}^{i})_{∗} is

( \begin{matrix}id & 0 \\ 0 & p_{j}^{i} \end{matrix} )

where p_{j}^{i} ∈ End(V^{j}) is a projector. On the other hand we already know

(Π_{Σ}^{j})_{∗} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} )

so it follows p_{j}^{i} = 0 for all i, j. Hence (Π_{S}^{i})_{∗}^{p}R^{j}p_{∗}Q_{X} = IC_{S}(V_{S}^{i})[dim S] if i = j and = 0 if i ≠ j. Applying Proposition 2.6, we obtain the claim. □

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