DECOMPOSITION OF THE DIAGONAL AND EIGENVALUES OF FROBENIUS FOR FANO HYPERSURFACES.

SPENCER BLOCH, HÉLÈNE ESNAULT, AND MARC LEVINE

Abstract. Let X ⊂ ℙ^{n} be a possibly singular hypersurface of degree d ≤ n, defined over a finite field 𝔽_{q}. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its ℓ-adic cohomology H^{i}(X̄, ℚ_{ℓ}), for ℓ ≠ char(𝔽_{q}), are divisible by q, without using the result on the existence of rational points by Ax and Katz [18].

1. Introduction

If X is a variety defined over a finite field k = 𝔽_{q}, one encodes the number of its rational points over all finite extensions 𝔽_{q^{s}} ⊃ 𝔽_{q} in the zeta function, defined by its logarithmic derivative

\frac{ζ′(X, t)}{ζ(X, t)} = ∑_{s≥1} |X(𝔽_{q^{s}})|t^{s−1}. (1.1)

By the theorem of Dwork [11], we know that ζ(t) is a rational function

ζ(X, t) ∈ ℚ(t). (1.2)

We assume that X is projective and we denote by U = ℙ^{n} \ X the complement of a projective embedding. The Grothendieck-Lefschetz trace formula [17] gives a cohomological formula for the numerator and the denominator of the rational function

ζ(U, t) = ∏_{i=0}^{2 dim(U)} det(1 − F_{i}t)^{(−1)^{i+1}}, (1.3)

where F_{i} is the arithmetic Frobenius acting on the compactly supported ℓ-adic cohomology H_{c}^{i}(\overline{U}, ℚ_{ℓ}). Letting H_{prim}^{i}(\overline{X}, ℚ_{ℓ}) denote the primitive cohomology H^{i}(\overline{X}, ℚ_{ℓ})/H^{i}(\overline{ℙ}^{n}, ℚ_{ℓ}) of X, we have

Date: Feb. 10, 2003.

(1.4) H_{c}^{i}(\overline{U}, ℚ_{ℓ}) ≅ \begin{cases}H_{prim}^{i−1}(\overline{X}, ℚ_{ℓ}) & for (i − 1) ≤ 2 dim(X), \\ H^{i}(\overline{ℙ^{n}}, ℚ_{ℓ}) & for i ≥ 2 dim(X). \end{cases}

For X smooth and complete, the Weil conjectures [9] assert that the the eigenvalues of F_{i} in any complex embedding ℚ_{ℓ} ⊂ ℂ have absolute values q^{\frac{(i−1)}{2}} . In particular, there is no possible cancellation of eigenvalues between the numerator and the denominator of the zeta function. Consequently, the property

|X(𝔽_{q^{s}})| ≡ |ℙ^{n}(𝔽_{q^{s}})| mod q^{κ} (1.5)

for all s ≥ 1, and some κ ∈ ℕ \ {0}, is equivalent to the property that

(1.6) the eigenvalues of F_{i} are divisible by q^{κ}, as algebraic integers.

However, if X is singular, one does not have in general the purity of weights of Frobenius on H_{c}^{i}. Thus, a cancellation between the numerator of the zeta function and its denominator is at least in principle possible, and the property (1.5) is no longer a priori equivalent to the property (1.6). The purpose of this article is to study the relation between (1.5) and (1.6) in the case of hypersurfaces of degree d ≤ n.

Let X be a complete intersection in ℙ^{n} defined by r equations of degrees d_{1} ≥ d_{2} ≥ … ≥ d_{r} with the property

1 ≤ κ = [\frac{n − d_{2} − … − d_{r}}{d_{1}} ]. (1.7)

The theorem of Ax and Katz says precisely that (1.5) holds true. On the other hand, we also know by [7], [10], [12], that if the finite field is replaced by a field of characteristic 0, the Hodge type of X is κ for all cohomology groups of X. (See [4] for a more precise discussion of those theorems). This gives a strong indication that (1.6) should be true as well. Indeed, as explained to us by Daqing Wan, (1.6) is true for κ = 1. One knows by [8], Theorem 5.5.3, that q divides the eigenvalues of Frobenius acting on H^{a}(\overline{X}, ℚ_{ℓ}) for a > dim(X), and since this cohomology vanishes for complete intersections for a < dim(X), the theorem of Ax and Katz implies divisibility by q for a = dim(X) as well. Similarly, using vanishing and Ax-Katz’s result, and replacing [8], Theorem 5.5.3 by the corresponding statement for the slopes of the Frobenius action on rigid cohomology ([22], p. 820), one obtains that the slopes of the Frobenius action on rigid cohomology are ≥ 1.

The purpose of this note is to give a motivic interpretation for Fano hypersurfaces and κ = 1 of the divisibility result, which does not use the theorem by Ax and Katz.

We now describe our method. Let us first assume that X is smooth. By Roitman’s theorem [21], we know that CH_{0}(X ×_{k} K) = ℤ for any field extension K ⊃ k which is algebraically closed. By [2, appendix to lecture 1], this implies that the class of the diagonal in CH_{n−1}(X × X) goes to zero in CH_{n−1}((X × X \ (ξ × X ∪ X × A))_{ℚ} for some divisor A on X and some 0-cycle ξ. Letting this class act as a correspondence, it follows that the restriction map H^{i}(\overline{X}, ℚ_{ℓ}) → H^{i}(\overline{X \ A}, ℚ_{ℓ}) is zero for i ≥ 1. This shows divisibility, as in [14], lemma 2.1.

For singular varieties, the proof of the Hodge-type statement in the complete intersection singular case ([12]) shows that the cohomology with compact support H_{c}^{i}(U) =: H^{i}(ℙ^{n}, X) carries the necessary information, and is easier to deal with than its dual H^{j}(U). To carry out the argument used in the smooth case, one needs a version of the Chow groups which is related to compactly supported cohomology. If X is a strict normal crossing divisor, one can use the relative motivic cohomology H_{M}^{2n}(ℙ^{n} × U, X × U, ℤ(n)), as defined in [20], chapter 4, 2.2 and p. 209; this relative motivic cohomology acts as correspondences on H_{c}^{∗}(\overline{U}, ℚ_{ℓ}). Due to the lack of resolution of singularities in positive characteristic, we will in general need an alteration π : (ℙ, Y) → (ℙ^{n}, X) of (ℙ^{n}, X), that is, a projective, generically finite morphism π : ℙ → ℙ^{n}, with ℙ smooth, such that Y := π^{−1}(X) is a strict normal crossing divisor. We then use the relative motivic cohomology H_{M}^{2n}(ℙ × U, Y × U, ℤ(n)).

Recall that H_{M}^{m}(ℙ × U, Y × U, ℤ(n)) is the homology H_{2n−m}(Z^{n}(ℙ × U, Y ×U, ∗)), where Z^{n}(ℙ×U, Y ×U, ∗) is the single complex associated to the double higher Chow cycle complex

(1.8)

⋯  ⋯  ⋯
∂↓  ^{∂}↓  ↓
Z^{n}(ℙ × U, 1) ⟶^{rest} Z^{n}(Y^{(1)} × U, 1) ⟶^{rest} Z^{n}(Y^{(2)} × U, 1)
∂↓  ^{∂}↓  ↓
Z^{n}(ℙ × U, 0) ⟶^{rest} Z^{N}(Y^{(1)} × U, 0) ⟶^{rest} Z^{n}(Y^{(2)} × U, 0).

Here Y^{(a)} is the normalization of all the strata of codimension a, Z^{n}(Y^{(a)} × U, b) is a group of cycles on Y^{(a)} × U × S^{b} where S^{•} is the cosimplicial scheme S^{n} = Spec (k[t_{0}, …, t_{n}]/(∑ t_{i} − 1)) with face maps S^{n} ↪ S^{n+1} defined by t_{i} = 0. More precisely, Z^{n}(Y^{(a)} × U, b) is generated by the codimension n subvarieties Z ⊂ Y^{(a)} × U × S^{b} such that, for each face F of S^{b}, and each irreducible component F′ ⊂ Y^{(a)} of the strata of Y we have codim_{F′×U×F} (Z ∩ (F′ × U × F)) ≥ n. The horizontal restriction maps are the intersection with the smaller strata, the vertical ∂’s are the boundary maps.

For technical reasons, we find it convenient to use a subcomplex Z^{n}(ℙ × U, I(Y × U), ∗) of Z^{n}(ℙ × U, Y × U, ∗). For T a smooth kscheme of finite type, and A a closed subset, let Z^{n}(T, I(A), m) be the subgroup of Z^{n}(T, m) consisting of the cycles W ∈ Z^{n}(T, m) with Supp(W) ∩ (A × S^{m}) = ∅. The Z^{n}(T, I(A), m) evidently form a subcomplex Z^{n}(T, I(A), ∗) of Z^{n}(T, ∗), functorial for flat pull-back and proper push-forward. Set

H_{M}^{m}(T, I(A), ℤ(n)) := H_{2n−m}(Z^{n}(T, I(A), ∗)). (1.9)

The inclusion Z^{n}(ℙ×U, I(Y ×U), ∗) → Z^{n}(ℙ×U, ∗) extends to a map of complexes Z^{n}(ℙ × U, I(Y × U), ∗) → Z^{n}(ℙ × U, Y × U, ∗). We call H_{M}^{m}(T, I(A), ℤ(n)) the motivic cohomology with modulus.

Let Δ ⊂ ℙ×U be the inverse image of the diagonal ⊂ ℙ^{n} ×U, i.e., Δ is the graph of the alteration π restricted to ℙ×U. Since rest(Δ) = 0,Δ yields a class

[Δ] ∈ H_{M}^{2n}(ℙ × U, I(Y × U), ℤ(n)). (1.10)

We show

Theorem 1.1. Let X ⊂ ℙ^{n} be a hypersurface of degree d ≤ n over a field k. Then there is an alteration π : (ℙ, Y) → (ℙ^{n}, X) and a divisor A ⊂ ℙ which cuts all the strata of Y in codimension ≥ 1 and such that the image of [Δ] in H_{M}^{2n}((ℙ \ A) × U, I((Y \ A) × U), ℚ(n)) is zero.

The main idea behind this geometric statement relies on the following. By a counting argument, Roitman [21] shows that, for a hypersurface X ⊂ ℙ^{n} of degree d ≤ n, the correspondence

{(x, ℓ) ∈ X × Grass(1, n) | ℓ ⊂ X or ℓ ∩ X = {x} for some x ∈ X} (1.11)

dominates X. It follows that the map ℤ ≅ CH_{1}(ℙ^{n}) → CH_{0}(X) has cokernel killed by multiplication by d = deg X, where CH_{0}(X) is Fulton’s homological Chow group ([16]). This implies CH_{0}(X) ⊗ ℚ = ℚ.

We replace Roitman’s correspondence by

(1.12) P := {(y, ℓ) ∈ ℙ × Grass(1, n) | π(y) ∈ ℓ, and either ℓ ⊂ X or ℓ ∩ X = {x} for some x ∈ X}.

We show that P dominates ℙ, and then use the technique of blowing up strata of Y introduced in [3] to find the rational equivalence relation which holds on the complement of some good divisor A. Finally, we show

Theorem 1.2. Let X ⊂ ℙ^{n} be a projective variety over a field k, and let U = ℙ^{n} \ X. Suppose there is an alteration π : (ℙ, Y) → (ℙ^{n}, X) and a divisor A ⊂ ℙ which cuts all strata of Y in codimension ≥ 1, such that the image of [Δ] in H_{M}^{2n}((ℙ \ A) × U, I((Y \ A) × U), ℚ(n)) is zero.

(1) If the characteristic of the ground field k is 0, then gr_{0}^{F} H^{i}(X) = 0 for all i ≥ 1.

(2) If k = 𝔽_{q} is a finite field, then the eigenvalues of the arithmetic Frobenius F_{i} acting on the compactly supported ℓ-adic cohomology H_{c}^{i}(\overline{U}, ℚ_{ℓ}) are all divisible by q as algebraic integers for all i ≥ 1.

(3) If k is a perfect field of characteristic p, then the slopes of the Frobenius operator acting on the rigid cohomology H_{c}^{i}(U/K) are all ≥ 1 for all i ≥ 1.

To conclude, we remark that this article solves the natural question posed in the introduction of [15], but only in the case κ = 1. Thinking of the discussion developed in [4], (5.2) for κ ≥ 2 in the smooth case, it is not entirely clear what the substitute would be for 1.1. One may also try to generalize these results for κ = 1, replacing the hypersurface X with a more general singular Fano variety. A singular Fano variety X over a field is a gometrically connected, projective, Cohen-Macaulay variety such that the reflexive hull of ω_{X_{reg}}^{N} is invertible and ample for some N ∈ ℤ \ ℕ. Examples are hypersurfaces of degree d ≤ n. The question is then whether a Fano variety fulfills Theorem 1.1. If yes, as in [14], this would show that a singular Fano variety over a finite field has a rational point. Acknowledgements. We thank Pierre Berthelot, Pierre Deligne, V. Srinivas and in particular Daqing Wan for interesting discussions on topics related to this work.

2. The proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. We fix a basefield k and write ℙ^{n} for ℙ_{k}^{n}. We want to show that a certain class [Δ] in motivic cohomology with modulus is trivial. Suppose for a moment we know this vanishing for k an infinite field. If k is a finite field, there exist Galois extensions k_{ℓ}/k with Galois group ℤ_{ℓ} for any prime ℓ. In particular, k_{ℓ} is infinite, so [Δ_{k_{ℓ}}] = 0 by hypothesis. Since the motivic cohomology with modulus over k_{ℓ} is a direct limit over motivic cohomology with modulus over finite subfields k′ ⊂ k_{ℓ}, and since motivic cohomology with modulus admits a norm, we conclude [Δ] is killed by some power of ℓ. Since this is true for two different ℓ, and the union of Galois translates of A in good position with respect to Y is still in good position, the theorem follows. Thus, we may assume k is infinite. In particular, we will use without comment various general position arguments.

Fix X ⊂ ℙ^{n} a hypersurface of degree d ≤ n. We want to define a closed subvariety Z inside the Grassmann of lines Grass(1, n) consisting of lines “maximally tangent” to X. We have the incidence correspondence U := {(z, ℓ) | z ∈ ℓ ∈ Grass(1, n)}. Define

(2.1)  V := U ×_{Grass(1,n)} U = {(y, ℓ, z) | y, z ∈ ℓ}.

The ℙ^{1}-bundle pr_{2} : V → U, (y, ℓ, z) ↦ (z, ℓ) has a section (x, ℓ) ↦ (x, ℓ, x). Locally on U we may identify V ≅ ℙ^{1} × U with homogeneous coordinates s, t in such a way that the section is given by t = 0. The section O_{ℙ^{n}} ⟶^{X} O_{ℙ^{n}}(d) pulls back to a section of p^{∗}O(d)_{ℙ^{n}} under the projection p : V → ℙ^{n}, (y, ℓ, z) ↦ y, and the section X restricts to an equation F (s, t) = F_{0}s^{d} + F_{1}s^{d−t}t + … + F_{d}t^{d}, where the F_{i} are (local) functions on U. Note F_{0} is a local defining equation of U ×_{ℙ^{n}} X ⊂ U. We are interested in the closed sets defined locally by F_{0} = … = F_{d−1} = 0 ( resp. F_{0} = … = F_{d} = 0). Denote by

(2.2)  Z′_{X} ⊂ Z′ ⊂ Grass(1, n)

the projection on the Grassmannian of these sets. Intuitively, an open set Z^{0} := Z′\Z′_{X} consists of lines d-fold tangent to X at a point. Define Z ⊂ Z′ to be the closure of Z^{0}, and let Z_{X} := Z′_{X} ∩ Z.

Proposition 2.1. The projection

p_{2} : Z ×_{Grass(1,n)} U → ℙ^{n}

is surjective. Intuitively, a general point on ℙ^{n} has a line through it maximally tangent to X.

Proof. It suffices to consider geometric points. Let y ∈ ℙ^{n} \ X be a geometric point. There is a linear transformation of ℙ^{n} such that the equation of X is x_{0}^{d} + x_{0}^{d−1}f_{1}(x_{1}, …, x_{n}) + … + f_{d}(x_{1}, …, x_{n}), with f_{i} ∈ k[x_{1}, …, x_{n}] homogeneous of degree i, and such that y has homogeneous coordinates (1 : 0 : …: 0). Thus a line passing through y has parametrization (s : tu_{1} : …: tu_{n}) ∈ ℙ^{n}, for (s : t) ∈ ℙ^{1} and (u_{1} : …: u_{n}) ∈ ℙ^{n−1}. The intersection of this line with X has equation s^{d} +s^{d−1}tf_{1}(u_{1}, …, u_{n})+…+t^{d}f_{d}(u_{1}, …, u_{n}) and its intersection with X will be d-tangent if and only if this equation has the shape (s + tu)^{d} with (u : u_{1} : …: u_{n}) ∈ ℙ^{n}. This is equivalent to the d homogeneous equations f_{i} = \binom{d}{i}u^{i}, i = 1, …, d in (u : u_{1} : … u_{n}) ∈ ℙ^{n}. Since d ≤ n there exists a homogeneous solution. □

Example 2.2. Let (T_{0} : T_{1} : T_{2}) be homogeneous coordinates on ℙ^{2}, and let X : T_{0}T_{1} = 0, so d = n = 2. Clearly in this case Z is simply the variety of lines through (0 : 0 : 1), and Z_{X} ⊂ Z is the two points corresponding to the components of X.

It follows from proposition 2.1 that dim Z ≥ n − 1. Let Z ⊂ Z be a general linear section of dimension n−1, and write Q := Z ×_{Grass(1,n)}U. Recall that an alteration π : (ℙ, Y) → (ℙ^{n}, X) of (ℙ^{n}, X) is a projective, generically finite morphism π : ℙ → ℙ^{n}, with ℙ smooth, such that Y := π^{−1}(X) is a strict normal crossing divisor.

Lemma 2.3. There exists a commutative diagram of schemes

\xymatrix{ℙ \ar[d]_{π} & P \ar[l]_{f} \ar[d]_{g} \ar[dr]^{q} \\ ℙ^{n} & Q \ar[r]_{p̄} \ar[l]^{r̄} & Z}

satisfying the following conditions:

(1) Q and Z are as above, and p̄ and r̄ are the natural maps. In particular, p̄ is a ℙ^{1}-bundle and r̄ is surjective.

(2) P is irreducible and normal.

(3) f : P → ℙ is projective, generically finite and surjective.

(4) q : P → Z is surjective.

(5) There exists a normal crossings divisor Y ⊂ ℙ such that π : (ℙ, Y) → (ℙ^{n}, X) is an alteration.

(6) There exists a divisor A ⊂ ℙ such that A meets all the strata of

Y properly, and such that P \ f^{−1}(A) → ℙ \ A is finite. Given a surjection of projective k-schemes Q′ ↠ Q, the map g can be taken to factor P → Q′ → Q.

Proof. Since k is infinite and Z ⊂ Z is a general linear space section, the map r̄ in 1) is surjective. Let π be an alteration. Then there will be an irreducible component of ℙ ×_{ℙ^{n}} Q dominating both Q and ℙ. Taking P to be the normalization of this component gives 2), 3), and 4). (To see the final assertion, one can replace Q′ by a plane section and assume Q′ → Q has finite degree. Then substitute ℙ ×_{ℙ^{n}} Q′ in the above.) Condition 5) comes from the work of de Jong [5].

To prove 6), we use the following result ([3], Theorem 2.1.2):

Theorem 2.4. Let Y ⊂ ℙ be a normal crossings divisor in a smooth variety. Let f : W → ℙ be a finite type morphism, and assume that W \ f^{−1}(Y) ⊂ W is dense. Given p : ℙ′ → ℙ the blowup of a face (stratum) of Y, let Y′ = p^{∗}(Y)_{red} be the reduced pullback, and let f′ : W′ → ℙ′ be the strict transform of f. Then (ℙ′, Y′, f′) satisfy the same hypotheses as (ℙ, Y, f). There exists a composition of such blowups, (ℙ_{N}, Y_{N}) → … → (ℙ, Y) such that the strict transform morphism f_{N} : W_{N} → ℙ_{N} meets the faces of Y_{N} properly, i.e. for Z ⊂ ℙ_{N} a face of codimension r, f_{N}^{−1}(Z) ⊂ W_{N} has codimension ≥ r.

Replacing the alteration ℙ → ℙ^{n} with a composition ℙ_{N} → … → ℙ → ℙ^{n} and changing notation, we may assume f : P → ℙ meets faces properly. Since f has finite degree, this amounts to saying that the fibre of f over the generic point of any face is finite. The existence of a divisor A as in 6) is now clear. □

Lemma 2.5. Let p : Q → B be a smooth projective morphism of k-schemes, with geometrically connected fibers of dimension one. Let s_{0}, s_{∞} : B → Q be sections, take t̃ ∈ H^{0}(Q, O_{Q}(s_{∞}(B) − s_{0}(B))) and suppose that the rational function t on Q determined by t̃ satisfies

div(t) = s_{0}(B) − s_{∞}(B).

Let B̄ ⊂ B be the closed subscheme of B defined by the equation s_{0} = s_{∞}. Then the restriction t̄ of t to p^{−1}(B̄) is a unit, and there is a unit ū on B̄ with t̄= p^{∗}(ū).

Proof. The hypotheses imply p_{∗}O_{Q}^{×} = O_{B}^{×} and this continues to hold after pullback. Smoothness of p implies that s_{i}(B) ⊂ Q are Cartier divisors. We view t̃ as an isomorphism t̃: O(s_{∞}(B)) ≅ O(s_{0}(B)). By definition, the Cartier divisors agree over B̄, so there is a tautological identification τ : O(s_{0}(B))_{p^{−1}(B̄)} ≅ O(s_{∞}(B))_{p^{−1}(B̄)}. The composition

t̃∘ τ ∈ Γ(p^{−1}(B̄), Aut(O(s_{0}(B)))) = Γ(p^{−1}(B̄), O_{p^{−1}(B̄)}^{×}) ≅ Γ(B̄, O× ; B̄)

yields the desired unit ū. □

Recall Z ⊂ Z is a general linear section, where Z is a space of lines in ℙ^{n} which are maximally tangent to our given hypersurface X. We have removed from Z any possible irreducible components consisting entirely of lines on X, so the subset Z_{X} ⊂ Z of lines on X is nowhere dense. We define Z_{X} = Z ∩ Z_{X} and Z^{0} = Z \ Z_{X}. By generality, Z^{0} ⊂ Z is dense. Define Q^{0} = p̄^{−1}Z^{0} (resp. P^{0} = π^{−1}(Z^{0})). The ℙ^{1}-bundle Q^{0} → Z^{0} has a set-theoretic section s̄_{∞}^{0} associating to a line ℓ the unique point in ℓ ∩ X.

Consider the diagram

\xymatrix{P \ar[d]_{π} \ar[dr]_{g} \ar[r]_{p} & P ×_{Z} Q \ar[l]_{s_{0}} \ar[d]^{p_{1}} \\ Z & Q \ar[l]^{p̄}} (2.3)

Here the section s_{0} corresponds to the map g. Similarly, the settheoretic section s̄_{∞}^{0} gives rise to a set-theoretic section s_{∞}^{0} : P^{0} → P^{0} ×_{Z} Q. By making a further blow-up of faces of Y, enlarging A and changing notation (cf. the last part of lemma 2.3), we may assume that the closure P̃ of s_{∞}^{0}(P^{0}) in P ×_{Z} Q is finite over ℙ \ A, hence finite over P \ f^{−1}(A). Replacing P with P̃ and changing notation, we may assume s_{∞}^{0} gives rise to another section s_{∞} : P → P ×_{Z} Q. The picture is now

(2.4)  \xymatrix{Y \ar@{^{(}->}[d] \\ ℙ & P \ar[l]^{f} \ar[r]^{s_{0}} \ar[r]_{s_{∞}} & P ×_{Z} Q \ar[l]|{p} \\ A \ar@{^{(}->}[u]}

Let \overline{P} ⊂ P be the closed subscheme where s_{0} = s_{∞}. Then

(2.5)  (P^{0} ∩ f^{−1}(Y))_{red} ⊂ \overline{P}.

Indeed, we can check this down on ℙ^{n}, i.e. we can ignore the alteration π. Points in P^{0} map to pairs consisting of a line ℓ maximally tangent to X but not lying on X, together with a point y ∈ ℓ. The fibre p^{−1}(ℓ, y) = {(ℓ, y, z) | z ∈ ℓ}. The sections s_{0} and s_{∞} are given respectively by s_{0}(ℓ, y) = (ℓ, y, y) and s_{∞}(ℓ, y) = (ℓ, y, ℓ ∩ X). Since Y = π^{−1}(X), we get the desire inclusion (2.5) after alteration.

Lemma 2.6. Possibly enlarging the divisor A (preserving the hypothesis that A meets faces of Y properly), there exists a rational function t on P ×_{Z} Q such that

(2.6) (div t) ∩ (f ∘ p)^{−1}(ℙ \ A) = (s_{0} − s_{∞}) ∩ (f ∘ p)^{−1}(ℙ \ A),

and such that further, t|((f ∘ p)^{−1}(Y))_{red} ∩ (P^{0} ×_{Z} Q) ≡ 1.

Proof. By assumption, the map P \ f^{−1}(A) → ℙ \ A is finite. There are thus a finite set of points of P lying over generic points of faces of Y.

Let L = p_{∗}(O(s_{∞} − s_{0})). As R^{i}p_{∗}(O(s_{∞} − s_{0})) = 0 for i > 0, L is an invertible sheaf on P; adding divisors to A meeting faces properly, we can assume that L is trivial on P \ f^{−1}A. A generating section of L

10 SPENCER BLOCH, HÉLÈNE ESNAULT, AND MARC LEVINE

thus gives a generating section t̃of O(s_{∞}−s_{0}) over P ×_{Z} Q\(fp)^{−1}(A). We let t be the corresponding rational function on P ×_{Z} Q.

Clearly t satisfies (2.6). The fact that t can be taken to be ≡ 1 on the indicated divisor follows from (2.5) and lemma 2.5. Indeed, the lemma shows that the restriction of t comes from a unit on f^{−1}Y ∩ P^{0}. Enlarging A, this unit lifts to a unit on P \ f^{−1}A. Normalizing t, we can assume this unit is 1. □ Proof of Theorem 1.1. We use lemma 2.6 to construct an effective cycle D ∈ Z^{n}((ℙ \ A) × U, 1) with Supp(D) ∩ (Y \ A) × U × S^{1} = ∅, (2.7)

∂(D) = N · Δ ∈ Z^{n}((ℙ \ A) × U, 0)

for some integer N ≠ 0. This suffices to prove the theorem.

To construct D, we have the closed embedding i : P ×_{Z} Q → P ×ℙ^{n}. Let P ×_{Z} Q^{#} = i^{−1}((P \ f^{−1}(A)) × U), (2.8) and let Γ^{#} ⊂ P ×_{Z} Q^{#} × ℙ^{1} (2.9) be the closure of the graph of t. Let \overline{Γ} := i_{∗}Γ^{#} ⊂ (P \ f^{−1}(A)) × U × ℙ^{1} (2.10) be the image of Γ^{#}, and let Γ ⊂ (P \ f^{−1}(A)) × U × S^{1}; S^{1} = Spec k[t_{0}, t_{1}]/(t_{0} + t_{1} − 1) (2.11) be the pull-back of \overline{Γ} via (t_{0}, t_{1}) ↦ (−t_{0} : t_{1}). We let Γ_{∗} ⊂ P × ℙ^{n} × ℙ^{1} be the closure of Γ.

By lemma 2.6, we have (2.12) Γ_{∗} ∩ (f^{−1}(Y \ A) × ℙ^{n} × ℙ^{1} ⊂ (f^{−1}(Y \ A) × X × ℙ^{1})

∪ (f^{−1}(Y \ A) × ℙ^{n} × {1}).

Thus Γ_{∗} ∩ (f^{−1}(Y \ A) × U × S^{1}) = ∅. (2.13) Also we have ∂(Γ) = (f × id)^{∗}(Δ) ∈ Z^{n}((P \ f^{−1}(A)) × U, 0). (2.14) Thus, setting D := (f × id)_{∗}(Γ) ∈ Z^{n}((ℙ \ A) × U, 1), (2.15) we have

(2.16) rest(D) = 0, ∂(D) = (f × id)_{∗} ∘ (f × id)^{∗}(Δ) = N · Δ,

where N = deg(f) ≠ 0. This completes the proof. □

3. The proof of Theorem 1.2

This section is devoted to the proof of Theorem 1.2.

In what follows, we write simply H^{a}(X, b) to denote either geometric étale cohomology (viz. X/𝔽_{q}, H^{a}(X, b) := H_{ét}^{a}(X ×_{𝔽_{q}} \overline{𝔽}_{q}, ℚ_{ℓ}) for (ℓ, char(𝔽_{q})) = 1), or de Rham cohomology H_{DR}^{a}(X) for X over a field of characteristic 0 (thus b plays no rôle here), or rigid cohomology with the Frobenius action multiplied by p^{−b} for X over a perfect field of characteristic p. On de Rham cohomology we denote by F the Hodge filtration ([6]).

Lemma 3.1. Let ℙ be smooth and le A ⊂ ℙ be a divisor meeting faces of Y properly, where Y ⊂ ℙ is a normal crossing divisor. Let s : H_{A}^{a}(ℙ, Y; 0) → H^{a}(ℙ, Y; 0). Then

(1) In the de Rham case, Image(s) ⊂ F^{1}H^{a}(ℙ, Y; 0).

(2) In the étale case, the eigenvalues of Frobenius F_{a} on Image(s) are all divisible by q.

(3) In the rigid case, the slopes of F_{a} on Image(s) are all divisible by q.

Proof. We have a diagram

(3.1)  \xymatrix{H^{a−1}(Y, 0) \ar[r] & H^{a}(ℙ, Y; 0) \ar[r] & H^{a}(ℙ, 0) \\ H_{A}^{a−1}(Y, 0) \ar[r] \ar[u]^{s} & H_{A}^{a}(ℙ, Y; 0) \ar[r] \ar[u]^{s} & H_{A}^{a}(ℙ, 0). \ar[u]^{s}}

The assertion for the middle vertical arrow reduces to the comparable assertions for the left and right hand vertical arrows. (In the de Rham case, one must use the fact that gr_{F} is an exact functor.) Then the spectral sequences

E_{1}^{s,t} = H_{A·Y^{(s)}}^{t}(Y^{(s)}, 0) ⇒ H_{A}^{s+t}(Y, 0) (3.2)

E_{1}^{s,t} = H^{t}(Y^{(s)}, 0) ⇒ H^{s+t}(Y, 0)

reduce the problem to the case where the relative divisor Y is smooth. Thus it suffices to consider the right hand vertical arrow.

Suppose for a while we work with étale cohomology. We mimic Berthelot’s method as in [14], Lemma 2.1. Let … ⊂ A_{j} ⊂ A_{j−1} … ⊂ A_{0} = A be a finite stratification by closed subsets such that A_{i−1} \ A_{i} is smooth. The localization sequence

(3.3) … → H_{A_{j}}^{b}(\overline{ℙ}, ℚ_{ℓ}) →

H_{A_{j−1}}^{b}(\overline{ℙ}, ℚ_{ℓ}) → H_{(A_{j−1}\A_{j})}^{a}(\overline{ℙ \ A_{j}}, ℚ_{ℓ}) → …

commutes with the Frobenius action. Therefore we may assume that both A and ℙ are smooth, but no longer projective. We consider an affine covering ℙ = ∪_{i=0}^{N}U_{i}. The spectral sequence

(3.4) E_{2}^{ab} = H^{b}( … → H_{A}^{b}(\overline{U^{a−1}}, ℚ_{ℓ}) → H_{A}^{b}(\overline{U^{a}}, ℚ_{ℓ}) →

H_{A}^{b}(\overline{U^{a+1}}, ℚ_{ℓ}) → …) ⟹ H_{A}^{a+b}(\overline{ℙ}, ℚ_{ℓ})

allows us to reduce to the case where ℙ is smooth affine, A ⊂ ℙ is smooth, where r = codim(A) ≥ 1. By purity, we have a functorial Gysin isomorphism

H^{a−2r}(\overline{A}, ℚ_{ℓ}) ⟶^{Gysin} H_{A}^{a}(\overline{ℙ}, ℚ_{ℓ}(r)) (3.5)

By functoriality, this commutes with Frobenius, and we know that the eignevalues of Frobenius acting on the left term are algebraic integers (use duality with H_{c}^{∗} and e.g. corollary 5.5.3(iii) in [8]). But this is equivalent to saying that the Frobenius eigenvalues on H_{A}^{a}(\overline{ℙ}, ℚ_{ℓ}) are all divisible by q^{r}, finishing the proof when k = 𝔽_{q}.

The same sort of argument in the de Rham case reduces us to showing

Image(H_{DR}^{b}(A) → H_{DR}^{b+2r}(ℙ)) ⊂ F^{r}H_{DR}^{b+2r}(ℙ)) (3.6)

when A is a smooth codimension r subvariety of the smooth affine ℙ. Blowing up A in ℙ we may assume that A has codimension 1. We take a smooth compactification ℙ ⊂ ℙ′ such that A has a smooth compactification A ⊂ A′ ⊂ ℙ and the divisor A′ ∪ W, W = (ℙ′ \ ℙ) is a normal crossing divisor. Then the Gysin map is the connecting homomorphism of the residue sequence

0 → Ω_{ℙ′}^{•}(log(W)) → Ω_{ℙ′}^{•}(log(W + A′)) → Ω_{A′}^{•−1}(log(W ∩ A′)) → 0. (3.7)

Since one has the exact subsequence

(3.8) 0 → Ω_{ℙ′}^{≥(p+1)}(log(W)) → Ω_{ℙ′}^{≥(p+1)}(log(W + A′)) →

Ω_{A′}^{≥p}(log(W ∩ A′)) → 0,

one sees that F^{p}H_{DR}^{b}(A) cobounds to F^{p+1}H_{DR}^{b+2}(ℙ). This finishes the proof when k has charateristic 0. Finally, when k is perfect of characteristic p, we use a similar argument, replacing étale cohomology by rigid cohomology and then applying [12], Lemma 2.1. This finishes the proof of the lemma. □

proof of theorem 1.2. ℙ will be a smooth, projective variety of dimension n, Y ⊂ ℙ is a normal crossings divisor, and U := ℙ \ Y. Consider the diagonal (1.10). Using the cycle class map from motivic cohomology (cf. section 4), we view the diagonal as being a class in our theory

(3.9) [Δ] ∈ H^{2n}((ℙ, Y) × U, n) ≅ ⨁_{a+b=2n} H^{b}(U, b) ⊗ H^{a}(ℙ, Y; a) ≅

⨁_{a+b=2n} Hom(H^{a}(ℙ, Y; 0), H^{a}(ℙ, Y; 0)).

(The isomorphism on the right uses the existence of a good theory of compactly supported cohomology in our cohomology. Of course, the homomorphism on the right is the identity.) By the hypotheses of theorem 1.2, [Δ] dies in H^{2n}((ℙ \ A, Y \ Y ∩ A) × U, n), which means that the map s below is onto:

(3.10)  H_{A}^{a}(ℙ, Y; 0) ↠^{s} H^{a}(ℙ, Y; 0) → H^{a}(ℙ \ A, Y \ Y ∩ A; a).

The assertions of the theorem are now consequences of lemma 3.1 above. □

4. Cycle maps

Let H_{?}^{∗} be one of the cohomology theories used above: étale cohomology with relevant twists and Galois action ([8]), de Rham cohomology with Hodge filtration ([6]) or rigid cohomology with Frobenius action ([1]). We explain how to define cycle maps

cl^{n} : H_{M}^{2n}(T, I(A), ℤ(n)) → H_{?}^{2n}(T, A), (4.1)

natural with respect to smooth pull-back and projective push-forward, where H_{?}^{2n}(T, A) denotes the theory on T with compact supports relative to A; if T̄ is a compactification of T, we have the usual compactly supported cohomology H_{?,c}^{∗}(T) := H_{?}^{∗}(T̄, T̄ \ T), which is canonically defined independent of the choice of T̄.

For T smooth over k and W a closed subset, we have the cohomology with supports H_{?,W}^{∗} (T). We have as well the relatively compact version H_{?,W}^{∗} (T, A) and the natural commutative diagram

\xymatrix{H_{?,W}^{∗} (T, A) \ar[r] \ar[d] & H_{?}^{∗}(T, A) \ar[d] \\ H_{?,W}^{∗} (T) \ar[r] & H_{?}^{∗}(T).} (4.2)

If W ∩ A = ∅, the map

H_{?,W}^{∗} (T, A) → H_{?,W}^{∗} (T) (4.3)

is an isomorphism. H_{?}^{∗} satisfies the homotopy property: the map

p^{∗} : H_{?}^{∗}(T, A) → H_{?}^{∗}(T × 𝔸^{1}, A × 𝔸^{1}) (4.4)

is an isomorphism.

Let R be the coefficient ring H_{?}^{0}(k). We have the group of codimension n cycles Z^{n}(T) = Z^{n}(T, 0). For W ⊂ T a closed subset, we let Z_{W}^{n} (T) ⊂ Z^{n}(T) be the subgroup of cycles with support on W. If codim_{T} W ≥ n, we have the purity isomorphism

cl_{W}^{n} : Z_{W}^{n} (T) ⊗ R → H_{?,W}^{2n} (T), (4.5)

which is natural with respect to maps f : (T′, W′) → (T, W), f^{−1}(W) ⊂ W′, with codim_{T′}W′ ≥ n. Taking the limit over W and forgetting supports defines the map

cl^{n} : Z^{n}(T) → H_{?}^{2n}(T) (4.6)

with cl^{n}(f^{∗}Z) = f^{∗}cl^{n}(Z) for Z ∈ Z^{n}(T), and f : T′ → T with codim_{T′}f^{−1}(Supp(Z)) ≥ n. The maps cl^{n} are also natural with respect to projective push-forward and products.

Now let A be a closed subset of some smooth T, and let W be a closed subset of T with codim_{T} W ≥ n and W ∩ A = ∅. Via the isomorphisms (4.3), (4.5), we have the isomorphism

cl_{W}^{n} : Z_{W}^{n} (T) ⊗ R → H_{?,W}^{2n} (T, A) (4.7)

Taking the limit of such W and forgetting supports gives us the natural map

cl_{A}^{n}(0) : Z^{n}(T, I(A), 0) → H_{?}^{2n}(T, A). (4.8)

Similarly, we have the natural map

cl_{A}^{n}(1) : Z^{n}(T, I(A), 1) → H_{?}^{2n}(T × S^{1}, A × S^{1}). (4.9)

This yields the commutative diagram

\xymatrix{Z^{n}(T, I(A), 1) \ar[r]^{cl_{A}^{n}(1)} \ar[d]_{δ_{1}^{∗}−δ_{0}^{∗}} & H_{?}^{2n}(T × S^{1}, A × S^{1}) \ar[d]^{δ_{1}^{∗}−δ_{0}^{∗}} \\ Z^{n}(T, I(A), 1) \ar[r]_{cl_{A}^{n}(0)} & H_{?}^{2n}(T, A).} (4.10)

By the homotopy property (4.4), the right-hand vertical arrow is zero, so cl_{A}^{n}(0) descends to the desired map

cl_{A}^{n} : H_{M}^{2n}(T, I(A), ℤ(n)) → H_{?}^{2n}(T, A). (4.11)

The naturality of cl_{A}^{n} with respect to flat pull-back, projective pushforward and products follows from that of the cycle-classes with support.

References

[1] Berthelot, P. : Finitude et pureté en cohomologie rigide, Invent. math. 128 (1997), 329-377.

[2] Bloch, S.: Lecture on Algebraic cycles. Duke University Mathematics Series, IV, (1980).

[3] Bloch, S.: The moving lemma for higher Chow groups. J. Algebraic Geom. 3 (1994), no. 3, 537-568.

[4] Bloch, S; Esnault, H.: Congruences for the number of rational points, Hodge type and motivic conjectures for Fano varieties, 11 pages, appears in the Cubo journal.

[5] de Jong, A.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51-93.

[6] Deligne, P.: Théorie de Hodge II. Publ. Math. IHES 40 (1972), 5-57.

[7] Deligne, P.: Cohomologie des intersections complètes. in SGA 7 XI, Lect. Notes Math. vol. 340, 39-61, Berlin Heidelberg New York Springer 1973.

[8] Deligne, P.: Théorème d’intégralité, Appendix to Katz, N.: Le niveau de la cohomologie des intersections complètes, Exposé XXI in SGA 7, Lect. Notes Math. vol. 340, 363-400, Berlin Heidelberg New York Springer 1973.

[9] Deligne, P.: La conjecture de Weil, I. Publ. Math. IHES 43 (1974), 273-307.

[10] Deligne, P.; Dimca, A.: Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulières. Ann. Sci. Éc. Norm. Supér. (4) 23 (1990), 645-656.

[11] Dwork, B.: On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631-648.

[12] Esnault, H.: Hodge type of subvarieties of ℙ^{n} of small degrees. Math. Ann. 288 (1990), no. 3, 549-551.

[13] Esnault, H.; Nori, M.; Srinivas, V.: Hodge type of projective varieties of low degree. Math. Ann. 293 (1992), no. 1, 1-6.

[14] Esnault, H.: Varieties over a finite field with trivial Chow group of 0-cycles have a rational point. Invent. math. 151 (2003), 187-191.

[15] Esnault, H; Wan, D.: Hodge type of the exotic cohomology of complete intersections. preprint 2002, 7 pages, appears in the C. R. Aca. Sc. Paris.

[16] Fulton, W.: Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag (1984).

[17] Grothendieck, A.: Formule de Lefschetz et rationalité des fonctions L. Séminaire Bourbaki 279, 17-ième année (1964/1965), 1-15.

[18] Katz, N.: On a theorem of Ax. Amer. J. Math. 93 (1971), 485-499.

[19] Katz, N., Messing, W.: Some consequences of the Riemann hypothesis vor varieties defined over finite fields, Invent. math. 23 (1974), 73-77.

[20] Levine, M.: Mixed Motives, Mathematical Surveys and Monographs 57 (1998), American Mathematical Society.

[21] Roitman, A.: Rational equivalence of zero-dimensional cycles. Mat. Zametki 28 (1980), no. 1, 85-90.

[22] Wan, D.: Higher rank case of Dwork’s conjecture, J. Amer. Math. Soc. 13 (2000), no. 4, 807-852.

[23] Wan, D.:Poles of Zeta Functions of Complete Intersections, Chinese Ann. Math. Series B Vol. 21, no. 2(2000), 187-200.

University of Chicago, Mathematics, IL 60 636, Chicago, USA

  E-mail address: bloch@math.uchicago.edu

Mathematik, Universität Essen, FB6, Mathematik, 45117 Essen, Germany

  E-mail address: esnault@uni-essen.de

Northeastern University, Mathematics, MA 02115, Boston, USA E-mail address: marc@neu.edu