WEIGHT-MONODROMY CONJECTURE FOR p-ADICALLY UNIFORMIZED VARIETIES

TETSUSHI ITO

Abstract. The aim of this paper is to prove the weight-monodromy conjecture (Deligne’s conjecture on the purity of monodromy filtration) for varieties with p-adic uniformization by the Drinfeld upper half spaces of any dimension. The ingredients of the proof is to prove a special case of the Hodge standard conjecture, and apply an argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theoretic invariants. We also consider a p-adic analogue by using the weight spectral sequence of Mokrane.

1. Introduction

Let K be a complete discrete valuation field with finite residue field 𝔽_{q} of characteristic p > 0, O_{K} the ring of integers of K, and l a prime number different from p. Let X be a proper smooth variety over K, and V := H_{ét}^{w}(X\overline{K}, ℚ_{l}) the ladic cohomology of X\overline{K} := X⊗_{K} \overline{K} on which the absolute Galois group Gal(\overline{K}/K) acts. We define the inertia group I_{K} of K by the exact sequence :

1 ⟶ I_{K} ⟶ Gal(\overline{K}/K) ⟶ Gal(\overline{𝔽}_{q}/𝔽_{q}) ⟶ 1.

The monodromy filtration M_{•} on V is determined by the action of I_{K} on V, and the weight filtration W_{•} on V is determined by the action of a lift of the geometric Frobenius element Fr_{q} ∈ Gal(\overline{𝔽}_{q}/𝔽_{q}), which is the inverse of the q-th power map on \overline{𝔽}_{q} (for details, see Definition 5.1, Definition 5.3).

The weight-monodromy conjecture claims the coincidence of these two filtrations up to some shift ([De1], [De3], [Il1], [Il2], [RZ1], [Rap]). Sometimes, the weight-monodromy conjecture is also called Deligne’s conjecture on the purity of monodromy filtration in the literature. Conjecture 1.1 (Weight-monodromy conjecture).

M_{i}V = W_{i+w}V for all i.

Date: January 7, 2003.

1991 Mathematics Subject Classification. Primary: 11G25; Secondary: 11F80, 11F85, 14G20, 14F20.

Let us briefly recall the theory of p-adic uniformization by Tate, Raynaud, Mumford, Mustafin, Kurihara, and others ([Mum], [Mus], [Ku]). Let Ω̂_{K}^{d} be the Drinfeld upper half space of dimension d, which is a rigid analytic space obtained by removing all K-rational hyperplanes from ℙ_{K}^{d}. We have a natural action of PGL_{d+1}(K) on Ω̂_{K}^{d}. For a cocompact torsion free discrete subgroup Γ ⊂ PGL_{d+1}(K), the rigid analytic quotient Γ\Ω̂_{K}^{d} has a structure of a projective smooth variety X_{Γ} over K (for details, see §6). The main theorem of this paper is as follows. Theorem 1.2 (Weight-monodromy conjecture for X_{Γ}). Let Γ ⊂ PGL_{d+1}(K) be a cocompact torsion free discrete subgroup. Then, the weight-monodromy conjecture (Conjecture 1.1) holds for X_{Γ}.

Note that Conjecture 1.1 was known to hold if X/K satisfies at least one of the following conditions.

(1) X has a proper smooth model over O_{K} ([SGA4-III], [De2], [De3]).

(2) X is a curve or an abelian variety over K ([SGA7-I], IX).

(3) X is a surface ([RZ1] for the semistable case, [dJ] for the general case).

(4) X is a certain threefold with strictly semistable reduction ([It2]).

(5) K is of characteristic p > 0 ([De3], [Te], [It1]).

Therefore, Theorem 1.2 is new only in mixed characteristic although our proof of Theorem 1.2 is valid even if K is of characteristic p > 0 (However, see also Remark 6.2). In mixed characteristic and in dimension ≥ 3, Conjecture 1.1 is still open up to now. Therefore, Theorem 1.2 gives us an interesting class of varieties of any dimension in mixed characteristic for which Conjecture 1.1 holds.

Our proof of Theorem 1.2 is based on a careful analysis of the weight spectral sequence of Rapoport-Zink ([RZ1]). We prove a special case of the Hodge standard conjecture and use it to follow an argument of Steenbrink, M. Saito in [St], [SaM], where they proved a Hodge analogue of Conjecture 1.1 by using polarized Hodge structures. Namely, in our proof of Theorem 1.2, the Hodge standard conjecture plays a role of polarized Hodge structures. Note that in [It2], by basically the same idea as in this paper, the author proved Conjecture 1.1 for certain threefolds by using the Hodge index theorem which is nothing but the Hodge standard conjecture for surfaces.

The outline of this paper is as follows. In §2, we fix notation and recall some basic facts about the Hodge standard conjecture and related theorems. Note that, in this paper, we mainly consider varieties whose l-adic cohomology groups are generated by algebraic cycles. We introduce appropriate notation in §2.1.

In §3, we recall some properties of blowing-up and prove the Hodge standard conjecture of blow-ups under certain assumptions by looking at the asymptotic behavior of the Lefschetz operator and the cup product pairings (Proposition 3.4). Such argument seems well-known at least in the simplest case of a blow-up at a point on a surface. However, since the multiplicative structure of a blow-up is not so simple in higher dimensions, we need tedious computations to prove it.

In §4, we consider a projective smooth variety B^{n} of dimension n over \overline{𝔽}_{q} which is obtained by successive blowing-ups of ℙ_{\overline{𝔽}_{q}}^{n} along linear subvarieties. The main result in §4 is a proof of the Hodge standard conjecture for such varieties for certain choice of an ample line bundle (Proposition 4.7). Although we can, in principle, compute everything combinatorially about the variety B^{n}, the proof of Proposition 4.7 is far from trivial. We look at the combinatorial structure of B^{n} very carefully, and the proof proceeds by induction on n. This proof is inspired by McMullen’s proof of an analogue of the hard Lefschetz theorem for non-rational polytopes ([Mc]).

In §5, we recall some basic facts on the weight spectral sequence of Rapoport-Zink. By an argument of Steenbrink, M. Saito, we give a proof of a special case of Conjecture 1.1 assuming the existence of ℚ-structures on l-adic cohomology and the Hodge standard conjecture (Proposition 5.7). Note that, although the class of varieties considered in Proposition 5.7 is extremely small, this class is large enough to include all p-adically uniformized varieties as we will see in §6.

In §6, we firstly give a proof of Theorem 1.2. The key point is that there is an explicitly constructed formal scheme Ω̂_{O_{K}}^{d} locally of finite type over Spf O_{K}, whose associated rigid analytic space is Ω̂_{K}^{d}. From the construction, we see that X_{Γ} has a proper semistable model 𝔛_{Γ} over O_{K}. Moreover, if 𝔛_{Γ} is strictly semistable, we see that all irreducible components of the special fiber of 𝔛_{Γ} are isomorphic to the variety B^{d} in §4. Hence we can prove the Hodge standard conjecture for them by Proposition 4.7. The same is true for intersections of them. By Proposition 5.7, the proof of Theorem 1.2 is completed.

In the rest of §6, we also give several immediate applications. By combining Theorem 1.2 with the results of Schneider-Stuhler on the l-adic cohomology of X_{Γ} ([SS]), we give a proof of Schneider-Stuhler’s conjecture on the filtration F^{•} on H_{ét}^{d}(X\overline{K}, ℚ_{l}) (Theorem 6.3). As a consequence, we prove the following formula of the local zeta function ζ(s, X_{Γ}) of X_{Γ} :

ζ(s, X_{Γ}) := ∏_{k=0}^{2d} det (1 − q^{−s} · Fr_{q}; H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l})^{I_{K}})^{(−1)^{k+1}}

= (1 − q^{−s})^{μ(Γ)·(−1)^{d+1}} · ∏_{k=0}^{d} \frac{1}{1 − q^{k−s}} ,

where μ(Γ) is the multiplicity of the Steinberg representation in the representation of PGL_{d+1}(K) induced from the trivial character on Γ (Theorem 6.4). In particular, ζ(s, X_{Γ}) is independent of l. In §6.5, we give an application to mod l Galois representations (Theorem 6.7). Finally, in §6.6, we consider a padic analogue by using the weight spectral sequence of Mokrane ([Mo]). After proving a p-adic analogue of Conjecture 1.1, we also prove that the local zeta function ζ_{p-adic}(s, X_{Γ}) defined by p-adic Hodge theory coincides with ζ(s, X_{Γ}) above (Theorem 6.11, Theorem 6.12). However, we note that the computation of ζ_{p-adic}(s, X_{Γ}) is accomplished by comparing the l-adic and the p-adic case (This argument is inspired by [SaT]). The reason why we have to make such a detour is that Schneider-Stuhler’s computation doesn’t work directly on the p-adic étale cohomology. In [SS], Schneider-Stuhler also conjectured that their filtration F^{•} on de Rham cohomology coincides with Hyodo’s monodromy filtration ([Hy]), but we can’t prove this here (see Remark 6.14, Remark 6.15).

Needless to say, there is a very important theory so called the theory of padic uniformization of Shimura varieties established by Čerednik, Drinfeld for Shimura curves, and Varshavsky, Rapoport-Zink in higher dimensions ([Č], [Dr], [Va], [RZ2]). We strongly expect that Theorem 1.2 has applications to the zeta functions of Shimura varieties with p-adic uniformization ([Rap], [RZ1]). We also expect that this will establish a special case of the compatibility of the global and local Langlands correspondences ([Harr], Problem 1). We plan to return to these problems in the future.

Acknowledgments. The author is grateful to his advisor Professor Takeshi Saito for advice and support during the graduate course at University of Tokyo. The author also would like to thank Gerd Faltings, Michael Harris, Luc Illusie, Fumiharu Kato, Kazuya Kato, Minhyong Kim, Barry Mazur, Yoichi Mieda, Michael Rapoport, Richard Taylor, and Teruyoshi Yoshida for invaluable comments and encouragement. A part of this work was done during the author’s stay at Universität zu Köln in June 2002, Université de Paris-Sud in June 2002, and Max-Planck-Institut für Mathematik in Bonn in November and December 2002. The author would like to thank them for their cordial hospitality. The author was supported by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists.

2. Review of the Hodge standard conjecture

2.1. Notation and assumptions. Let F be an algebraically closed field of any characteristic, l a prime number different from the characteristic of F. Let X be a projective smooth variety over F of dimension n. Let Z^{k}(X) be the group of algebraic cycles on X of codimension k, and cl^{k}: Z^{k}(X) → H_{ét}^{2k}(X, ℚ_{l}(k)) the cycle map for l-adic cohomology. Let Z_{num}^{k}(X) ⊂ Z^{k}(X) be the subgroup consisting of algebraic cycles which are numerically equivalent to zero. It is known that

N^{k}(X) := Z^{k}(X)/Z_{num}^{k}(X)

is a finitely generated free ℤ-module ([Kl], Lemma 5-2, see also Lemma 2.3 below). Here we put the following assumption on X.

Assumption 2.1.

dim_{ℚ_{l}} H_{ét}^{k}(X, ℚ_{l}) = \begin{cases}rank_{ℤ} N^{k/2}(X) & if k is even \\ 0 & if k is odd. \end{cases}

Remark 2.2. Assumption 2.1 is independent of l because the l-adic Betti numbers are independent of l (This is a consequence of the Weil conjecture, see, for example, [KM], Corollary 1). Lemma 2.3. Under Assumption 2.1, cl^{k} induces an isomorphism :

N^{k}(X) ⊗_{ℤ} ℚ_{l} ≅ H_{ét}^{2k}(X, ℚ_{l}(k)).

Proof. Fix an integer k. Let V ⊂ H_{ét}^{2k}(X, ℚ_{l}(k)) be a ℚ_{l}-vector subspace generated by the image of the cycle map cl^{k}. Take elements x_{1}, …, x_{r} ∈ Z^{k}(X) such that {cl^{k}(x_{1}), …, cl^{k}(x_{r})} is a basis of V over ℚ_{l}. Then, it is easy to see that the image of the map

Z^{n−k}(X) ⟶ ℤ^{⊕r}, y ↦ (y · x_{1}, …, y · x_{r})

is isomorphic to N^{n−k}(X), where · denotes the intersection product. Hence we have the following inequalities :

rank_{ℤ} N^{n−k}(X) ≤ r ≤ dim_{ℚ_{l}} H_{ét}^{2k}(X, ℚ_{l}) = dim_{ℚ_{l}} H_{ét}^{2n−2k}(X, ℚ_{l}).

Therefore, Assumption 2.1 implies that the above inequalities are equalities. Hence V = H_{ét}^{2k}(X, ℚ_{l}(k)) for all k, and all cohomology classes are generated by algebraic cycles. In particular, the numerical equivalence coincides with the l-adic homological equivalence. We have a surjective map N^{k}(X) ⊗_{ℤ} ℚ_{l} → H_{ét}^{2k}(X, ℚ_{l}(k)), between ℚ_{l}-vector spaces of the same dimension, hence it is an isomorphism. □

Namely, N^{k}(X) defines l-independent ℚ-structures on the l-adic cohomology of X. Similarly, by using the cycle map for crystalline cohomology ([GM], [Gr]), it is easy to see that N^{k}(X) also defines ℚ-structures on the crystalline cohomology of X.

We define

H^{k}(X) := \begin{cases}N^{k/2}(X) ⊗_{ℤ} ℝ & if k is even \\ 0 & if k is odd, \end{cases}

and H^{∗}(X) := ⨁_{k} H^{k}(X). We consider H^{∗}(X) virtually as cohomology with coefficients in ℝ. The cup product ∪ on H^{∗}(X) is induced by the intersection product. Most properties of l-adic cohomology such as pullback, Poincaré duality, Künneth formula, etc. are valid for H^{∗}(X). To avoid confusion, we always put the subscript “ét” for étale cohomology.

A ℚ-divisor (resp. ℝ-divisor) is a formal sum of divisors with coefficients in ℚ (resp. ℝ). A ℚ-divisor (resp. ℝ-divisor) L is called ample if L is of the form L = a_{1}L_{1} + ⋯ + a_{r}L_{r} with r ≥ 1, a_{1}, …, a_{r} ∈ ℚ_{>0} (resp. ℝ_{>0}), and L_{1}, …, L_{r} ample. If L is an ample ℚ-divisor, some positive integral multiple of L is an ample divisor. But this doesn’t hold for ample ℝ-divisors.

There is a natural map from the group of ℝ-divisors on X to H^{2}(X). In this paper, we don’t usually distinguish an ℝ-divisor, a formal sum of line bundles with coefficients in ℝ (i.e. ℝ-line bundle), and its class in H^{2}(X). We can naturally define pullbacks and restrictions for ℝ-divisors. To avoid confusion, for an ℝ-divisor H, we sometimes use the notation O(H) instead of H. For example, f^{∗}O(H) (resp. O(H)|_{X}) denotes the pullback (resp. restriction) as a formal sum of line bundles.

2.2. Hard Lefschetz conjecture. As in §2.1, let X be a projective smooth variety over F of dimension n satisfying Assumption 2.1, and L an ample ℝdivisor on X. By taking cup product with L, we have an ℝ-linear map

L: H^{k}(X) ⟶ H^{k+2}(X)

called the Lefschetz operator. Conjecture 2.4 (Hard Lefschetz conjecture). For all k, L^{k} induces an isomorphism :

L^{k}: H^{n−k}(X) ⟶^{≅} H^{n+k}(X).

If L is a ℚ-divisor, Conjecture 2.4 is proved by Deligne in all characteristics ([De3]). If F = ℂ, Conjecture 2.4 holds by transcendental methods ([We]). The general characteristic 0 case follows from this by Lefschetz principle. However, in characteristic p > 0, Conjecture 2.4 seems open in general.

Assume that Conjecture 2.4 holds for (X, L). For an integer k with 0 ≤ k ≤ n, we define the primitive part P^{k}(X) by

P^{k}(X) = Ker (L^{n−k+1} : H^{k}(X) ⟶ H^{2n−k+2}(X)).

We define P^{k}(X) = 0 for k < 0 or k > n. Then, by Conjecture 2.4, we have the primitive decomposition :

(2.1)  H^{k}(X) = ⨁_{i≥0} L^{i}P^{k−2i}(X) ≅ ⨁_{i≥0} P^{k−2i}(X) for all k.

For even k with 0 ≤ k ≤ n, we define the pairing ⟨, ⟩_{H^{k}(X)} on H^{k}(X) by

⟨, ⟩_{H^{k}(X)} : H^{k}(X) × H^{k}(X) → ℝ, ⟨x, y⟩_{H^{k}(X)} = (−1)^{k/2}L^{n−k}x ∪ y.

Otherwise, we define ⟨, ⟩_{H^{k}(X)} to be the zero pairing. The pairings ⟨, ⟩_{H^{k}(X)} are nondegenerate by Conjecture 2.4 and Poincaré duality.

We denote the restriction of ⟨, ⟩_{H^{k}(X)} to P^{k}(X) by ⟨, ⟩_{P^{k}(X)}. The decomposition (2.1) is an orthogonal decomposition with respect to ⟨, ⟩_{H^{k}(X)}. Therefore, each ⟨, ⟩_{P^{k}(X)} is nondegenerate. Moreover, for all k with 0 ≤ k ≤ n, ⟨, ⟩_{H^{k}(X)} is isomorphic to the alternating sum of ⟨, ⟩_{P^{k−2i}(X)} for i ≥ 0 :

(2.2)  ⟨, ⟩_{H^{k}(X)} = ∑_{i≥0}(−1)^{i}⟨, ⟩_{P^{k−2i}(X)} on H^{k}(X) ≅ ⨁_{i≥0} P^{k−2i}(X).

Remark 2.5. The primitive decomposition may depend on the choice of L. However, the dimensions of P^{k}(X) are independent of L because

dim_{ℝ} P^{k}(X) = dim_{ℝ} H^{k}(X) − dim_{ℝ} H^{k−2}(X).

2.3. Hodge standard conjecture. As in §2.2, let X be a projective smooth variety over F of dimension n satisfying Assumption 2.1, and L an ample ℝdivisor on X. Assume further that the hard Lefschetz conjecture (Conjecture 2.4) holds for (X, L). Conjecture 2.6 (Hodge standard conjecture). For all k, the pairing ⟨, ⟩_{P^{k}(X)} is positive definite. Remark 2.7. Usually, Conjecture 2.6 is stated for all projective smooth varieties X without Assumption 2.1 (Hdg(X) in [Kl]). It is a part of Grothendieck’s standard conjectures, and known to hold in the following cases.

(1) If F = ℂ, it is a consequence of the existence of polarized Hodge structures on singular cohomology ([We]). The general characteristic 0 case follows by Lefschetz principle.

(2) If X is of dimension ≤ 2, it is known as the Hodge index theorem ([Kl], 5, [Fu], Example 15.2.4). As a corollary, we can prove the case k ≤ 1 by taking hyperplane sections.

Remark 2.8. Note that Conjecture 2.6 is stated for a pair (X, L). It seems that Conjecture 2.6 for (X, L) doesn’t automatically imply Conjecture 2.6 for (X, L′) with another choice of L′ (see also Corollary 2.12 below). Example 2.9. For an ample ℝ-divisor L on ℙ^{n}, it is clear that Conjecture 2.4 holds for (ℙ^{n}, L). The only nontrivial primitive cohomology is P^{0}(ℙ^{n}) ≅ ℝ and the pairing ⟨, ⟩_{P^{0}(ℙ^{n})} is

P^{0}(ℙ^{n}) × P^{0}(ℙ^{n}) ≅ ℝ × ℝ ∋ (x, y) ↦ L^{n} · xy ∈ ℝ ≅ H^{2n}(ℙ^{n}).

Since L^{n} is positive, this pairing is positive definite. Hence Conjecture 2.6 holds for (ℙ^{n}, L) for all ample ℝ-divisors L on ℙ^{n}.

2.4. Signatures. Let notation be as in §2.3. Firstly, we recall the definition of signatures. Let V be a finite dimensional ℝ-vector space with a bilinear symmetric perfect pairing ⟨, ⟩. Take a basis {e_{1}, …, e_{r}} of V and consider the matrix M = (m_{ij}) defined by m_{ij} = ⟨e_{i}, e_{j}⟩. By Sylvester’s law of inertia, all eigenvalues of M are in ℝ, and the number of positive eigenvalues minus the number of negative eigenvalues is independent of the choice of {e_{1}, …, e_{r}}. We denote this number by sign(V, ⟨, ⟩) and call it the signature of the pairing ⟨, ⟩ on V.

Proposition 2.10. The Hodge standard conjecture (Conjecture 2.6) holds for (X, L) if and only if

sign(H^{k}(X), ⟨, ⟩_{H^{k}(X)}) = ∑_{i≥0}(−1)^{i} dim_{ℝ} P^{k−2i}(X) for all k with 0 ≤ k ≤ n.

Remark 2.11. By Remark 2.5, the right hand side of Proposition 2.10 doesn’t depend on a choice of L.

Proof. By definition, Conjecture 2.6 is equivalent to the following equality :

(2.3)  dim_{ℝ} P^{k}(X) = sign(P^{k}(X), ⟨, ⟩_{P^{k}(X)}) for all k with 0 ≤ k ≤ n.

By (2.2), we see that, for all k with 0 ≤ k ≤ n,

sign(H^{k}(X), ⟨, ⟩_{H^{k}(X)}) = ∑_{i≥0}(−1)^{i} sign(P^{k−2i}(X), ⟨, ⟩_{P^{k−2i}(X)}).

Hence (2.3) implies the conditions in Proposition 2.10.

Conversely, assume the conditions in Proposition 2.10. We shall prove (2.3) by induction on k. The case k = 0 is trivial, since P^{0}(X) ≅ ℝ and L^{n} is positive. If (2.3) is proved for k < k_{0}, the condition of Proposition 2.10 for k = k_{0} implies

sign(H^{k_{0}}(X), ⟨, ⟩_{H^{k_{0}}(X)})

= ∑_{i≥0}(−1)^{i} dim_{ℝ} P^{k_{0}−2i}(X)

= dim_{ℝ} P^{k_{0}}(X) + ∑_{i≥1}(−1)^{i} sign(P^{k_{0}−2i}(X), ⟨, ⟩_{P^{k_{0}−2i}(X)}).

By (2.2), we have dim_{ℝ} P^{k_{0}}(X) = sign(P^{k_{0}}(X), ⟨, ⟩_{P^{k_{0}}(X)}). □

As a corollary, we have the following implications about the Hodge standard conjecture (Conjecture 2.6) by a “homotopy argument”. Corollary 2.12. (1) Let L, L′ be ample ℝ-divisors on X. If the hard Lef-

schetz conjecture (Conjecture 2.4) holds for (X, tL + (1 − t)L′) for all t ∈ ℝ with 0 ≤ t ≤ 1, then the Hodge standard conjecture (Conjecture 2.6) for (X, L), (X, L′) are equivalent to each other.

(2) In particular, if Conjecture 2.4 holds for (X, L) for all ample ℝ-divisors L on X, then Conjecture 2.6 for (X, L) for all ample ℝ-divisors L on X are equivalent to each other.

Proof. It is enough to prove the first assertion. Let ⟨, ⟩_{H^{k}(X)} (resp. ⟨, ⟩′_{H^{k}(X)}) be the pairing on H^{k}(X) defined by L (resp. L′). By Remark 2.5 and Proposition 2.10, it is enough to show that, for all k, ⟨, ⟩_{H^{k}(X)} and ⟨, ⟩′_{H^{k}(X)} have the same signatures. For t ∈ ℝ with 0 ≤ t ≤ 1, let ⟨, ⟩_{t} be the pairing on H^{k}(X) defined by tL + (1 − t)L′, which is an ample ℝ-divisor on X satisfying Conjecture 2.4 by assumption. We have ⟨x, y⟩_{0} = ⟨x, y⟩′_{H^{k}(X)}, ⟨x, y⟩_{1} = ⟨x, y⟩_{H^{k}(X)}, and ⟨, ⟩_{t} is nondegenerate for t ∈ ℝ with 0 ≤ t ≤ 1. Therefore, {⟨, ⟩_{t}}_{t} is a continuous family of nondegenerate pairings on the ℝ-vector space H^{k}(X). Hence, the signature of ⟨, ⟩_{t} is independent of t. By comparing the case t = 0 and t = 1, we have the assertion. □

2.5. Products of varieties. As in §2.1, let X (resp. Y) be a projective smooth variety over F satisfying Assumption 2.1, and L_{X} (resp. L_{Y}) an ample ℝ-divisor on X (resp. Y). Then, by the Künneth formula, we have an isomorphism :

H^{k}(X × Y) ≅ ⨁_{i=0}^{k} H^{i}(X) ⊗_{ℝ} H^{k−i}(Y).

Hence X × Y also satisfies Assumption 2.1. The condition that L_{X}, L_{Y} are ample is equivalent to the condition that pr_{1}^{∗}O(L_{X}) + pr_{2}^{∗}O(L_{Y}) is ample, where pr_{1} : X × Y → X and pr_{2} : X × Y → Y are projections ([Hart1], I, [Hart2], II).

Proposition 2.13. (1) If the hard Lefschetz conjecture (Conjecture 2.4) holds

for (X, L_{X}), (Y, L_{Y}), then it also holds for (X×Y, pr_{1}^{∗}O(L_{X})+pr_{2}^{∗}O(L_{Y})).

(2) Moreover, if the Hodge standard conjecture (Conjecture 2.6) holds for (X, L_{X}), (Y, L_{Y}), then it also holds for (X × Y, pr_{1}^{∗}O(L_{X}) + pr_{2}^{∗}O(L_{Y})).

Proof. Here we give a proof by using the representation theory of the Lie algebra 𝔰l(2) (for details, [De3], 1.6, Bourbaki Lie, VIII, §1). It is a consequence of Conjecture 2.4 that we can construct 𝔰l(2)-representations on H^{∗}(X), H^{∗}(Y) by using the Lefschetz operator L_{X}, L_{Y}. Then, the tensor product H^{∗}(X)⊗_{ℝ}H^{∗}(Y) has a natural 𝔰l(2)-action which corresponds to pr_{1}^{∗}O(L_{X}) + pr_{2}^{∗}O(L_{Y}). This proves the first assertion. The second assertion follows from a basic property of certain pairings defined on 𝔰l(2)-representations over ℝ (Bourbaki Lie, VIII, §1). □

Remark 2.14. Since H^{2}(X × Y) ≅ H^{2}(X) ⊕ H^{2}(Y), all ample ℝ-divisors on X ×Y is necessarily of the form pr_{1}^{∗}O(L_{X})+pr_{2}^{∗}O(L_{Y}) in H^{2}(X ×Y). Therefore, if Conjecture 2.4 (resp. Conjecture 2.6) holds for (X, L_{X}), (Y, L_{Y}) for all ample ℝ-divisors L_{X}, L_{Y} on X, Y, then it also holds for (X × Y, L_{X×Y}) for all ample ℝ-divisors L_{X×Y} on X × Y.

3. Cohomology of varieties obtained by blowing-ups

In this section, we study the Hodge standard conjecture (Conjecture 2.6) for varieties obtained by blowing-ups. After recalling basic facts about blowing-ups, we show that, under certain assumptions, Conjecture 2.6 holds for a blow-up for certain choice of an ample ℝ-divisor (for details of the computation, see [SGA5], VII, [Hart2], II, [Fu], 6.7). We use the same notation as in §2.1.

3.1. Setup. Let X be a projective smooth variety over F of dimension n, and Y_{1}, …, Y_{r} ⊂ X mutually disjoint smooth closed subvarieties of codimension d ≥ 2. Let i: Y = ∐_{k=1}^{r} Y_{k} ↪ X be the closed immersion. Let f: X′ → X be the blow-up of X along Y, Y′ ⊂ X′ the proper transform of Y, and j: Y′ ↪ X′, g: Y′ → Y natural morphisms. Assume that X, Y_{1}, …, Y_{r} satisfy Assumption 2.1 in §2.1. Then, it is easy to see that X′, Y′ also satisfy Assumption 2.1 in §2.1. This can be checked from the following computations of the cohomology of X′ and Y′.

We have the following cartesian diagram :

\begin{matrix}Y′ & ⟶^{j} & X′ \\ ^{g}↓ & & ^{f}↓ \\ Y & ⟶^{i} & X \end{matrix}

Since Y is a disjoint union of mutually disjoint smooth closed subvarieties, g: Y′ → Y has a structure of ℙ^{d−1}-bundle. More precisely, Y′ = ℙ(Ň_{Y/X}), where Ň_{Y/X} is the conormal bundle of Y in X. Let

ξ = c_{1}(O_{Y′}(1)) ∈ H^{2}(Y′)

be the first Chern class of O_{Y′}(1) of the ℙ^{d−1}-bundle g: Y′ → Y. The restriction of the line bundle O_{X′}(Y′) to Y′ itself is isomorphic to O_{Y′}(−1).

We define a vector bundle F of rank d−1 on Y′ by the kernel of a natural map g^{∗}(Ň_{Y/X}) → O_{Y′}(1). Then we have an exact sequence of vector bundles on Y′ :

0 ⟶ F ⟶ g^{∗}(Ň_{Y/X}) ⟶ O_{Y′}(1) ⟶ 0.

3.2. Cohomology of X′ and Y′. For all k, we have the following exact sequence of cohomology groups ([SGA5], VII, 8.5) :

(3.1) 0 ⟶ H^{k−2d}(Y) ⟶^{λ} H^{k−2}(Y′) ⊕ H^{k}(X) ⟶^{μ} H^{k}(X′) ⟶ 0,

where λ and μ are defined as follows :

λ(y) = (g^{∗}(y) ∪ c_{d−1}(F̌), −i_{∗}(y)), μ(y, x) = j_{∗}(y) + f^{∗}(x).

Note that F̌ is the dual vector bundle of F and c_{d−1}(F̌) ∈ H^{2d−2}(Y′) is the (d − 1)-th Chern class of F̌. Define λ′: H^{k−2}(Y′) ⊕ H^{k}(X) → H^{k−2d}(Y) by λ′(y, x) = g_{∗}(y), then we have

λ′ ∘ λ(y) = g_{∗}(g^{∗}(y) ∪ c_{d−1}(F̌)) = y ∪ g_{∗}(c_{d−1}(F̌)) = y

since g_{∗}(c_{d−1}(F̌)) = (−1)^{d−1}g_{∗}(c_{d−1}(F)) = g_{∗}(ξ^{d−1}) = 1 ([SGA5], VII, 8.4.2). Therefore, we have the following isomorphism of cohomology groups :

(3.2)  H^{k}(X′) ≅^{μ} Ker (g_{∗}: H^{k−2}(Y′) → H^{k−2d}(Y)) ⊕ H^{k}(X)

We shall compute the right hand side explicitly. For an integer m with 0 ≤ m ≤ d − 1, the map H^{k−2m}(Y) → H^{k}(Y′), y ↦ g^{∗}(y) ∪ ξ^{m} is injective because

g_{∗}((g^{∗}(y) ∪ ξ^{m}) ∪ ξ^{d−1−m}) = y ∪ g_{∗}(ξ^{d−1}) = y.

Let H^{k−2m}(Y)ξ^{m} ⊂ H^{k}(Y′) be the image of this map. For simplicity, we sometimes write y ξ^{m} instead of g^{∗}(y) ∪ ξ^{m}. Note that g_{∗}: H^{k}(Y′) → H^{k−2(d−1)}(Y) satisfies

g_{∗}(a ξ^{k}) = a ∪ g_{∗}(ξ^{k}) = \begin{cases}a & if k = d − 1 \\ 0 & otherwise. \end{cases}

On the other hand, we have an isomorphism ([SGA5], VII, 2.2.6):

H^{k}(Y′) ≅ ⨁_{m=0}^{d−1} H^{k−2m}(Y)ξ^{m}
= H^{k}(Y) ⊕ H^{k−2}(Y)ξ ⊕ ⋯ ⊕ H^{k−2(d−1)}(Y)ξ^{d−1}.

Therefore, by (3.2), we have the following explicit description of the cohomology groups of X′ :

(3.3) H^{k}(X′) ≅ H^{k−2}(Y) ⊕ H^{k−4}(Y)ξ ⊕ ⋯ ⊕ H^{k−2−2(d−2)}(Y)ξ^{d−2} ⊕ H^{k}(X).

The following property of the cohomology of X′ will be used later. Proposition 3.1. Let x′ ∈ H^{k}(X′) be a cohomology class on X′ whose restriction to Y′ is zero. Then x′ is of the form x′ = f^{∗}(x) for a unique x ∈ H^{k}(X).

Proof. The uniqueness is clear because f_{∗}(x′) = f_{∗}(f^{∗}(x)) = x. For the existence, we write x′ of the form x′ = x_{0} + x_{1} ξ + ⋯ + x_{d−2} ξ^{d−2} + x via (3.3). This means that there is a relation

x′ = j_{∗}(x_{0} + x_{1} ξ + ⋯ + x_{d−2} ξ^{d−2}) + f^{∗}(x).

It is enough to show x_{0} = x_{1} = ⋯ = x_{d−2} = 0. Since the restriction of O_{X′}(Y′) to Y′ is isomorphic O_{Y′}(−1), we have j^{∗}(j_{∗}(1_{Y′})) = −ξ. Therefore,

j^{∗}(x′) = j^{∗}j_{∗}(x_{0} + x_{1} ξ + ⋯ + x_{d−2} ξ^{d−2}) + j^{∗}f^{∗}(x)
= −x_{0} ξ − x_{1} ξ^{2} − ⋯ − x_{d−2} ξ^{d−1} + g^{∗}i^{∗}(x).

On the other hand, j^{∗}(x′) = 0 by assumption. Since H^{∗}(Y′) is a free H^{∗}(Y)- module with basis 1, ξ, …, ξ^{d−1}, we have x_{0} = x_{1} = ⋯ = x_{d−2} = i^{∗}(x) = 0. □

3.3. Multiplicative structure of H^{∗}(X′). The multiplicative structure of H^{∗}(X′) can be computed as follows. Define ∇^{k}(X′) by

∇^{k}(X′) = H^{k−2}(Y′) ⊕ H^{k}(X)

and ∪′: ∇^{k}(X′) × ∇^{m}(X′) → ∇^{k+m}(X′) by

(y′_{1}, x_{1}) ∪′ (y′_{2}, x_{2}) = ( − y′_{1} ∪ y′_{2} ∪ ξ + y′_{1} ∪ g^{∗}i^{∗}(x_{2}) + (−1)^{km}y′_{2} ∪ g^{∗}i^{∗}(x_{1}), x_{1} ∪ x_{2}).

Then the following diagram is commutative ([SGA5], VII, 8.6.3) :

∇^{k}(X′) × ∇^{m}(X′) ⟶^{μ×μ} H^{k}(X′) × H^{m}(X′)
∪′↓  ^{∪}↓
∇^{k+m}(X′)  ⟶^{μ}  H^{k+m}(X′)

where ∪ denotes the cup product on H^{∗}(X′). Since the horizontal maps μ, μ × μ are surjective, we can compute the multiplicative structure of H^{∗}(X′) in terms of the multiplicative structure of ∇^{∗}(X′) := ⨁_{k} ∇^{k}(X′).

Recall that the multiplicative structure of H^{∗}(Y′) is computed by Chern classes ([SGA5], VII, 3). Namely, we have the following isomorphism of H^{∗}(Y)-algebras

H^{∗}(Y′) ≅ H^{∗}(Y)[T]/(T^{d} + c_{1}(Ň_{Y/X}) T^{d−1} + c_{2}(Ň_{Y/X}) T^{d−2} + ⋯ + c_{d}(Ň_{Y/X})).

which sends ξ to T.

There is another relation on H^{∗}(X′) coming from the exact sequence (3.1). Namely, the image of 1_{Y} ∈ H^{0}(Y) by λ,

λ(1_{Y}) = (c_{d−1}(F̌), −i_{∗}(1_{Y})) ∈ H^{2d−2}(Y′) ⊕ H^{2d}(X),

maps to 0 in H^{2d}(X′). By definition, i_{∗}(1_{Y}) ∈ H^{2d}(X) is nothing but the cohomology class of Y. We denote it by [Y]. Since the coefficient of ξ^{d−1} in the expression of c_{d−1}(F̌) is 1 ([SGA5], VII, 8.4.2), we can replace ξ^{d−1} by (ξ^{d−1} −c_{d−1}(F̌), [Y]).

In conclusion, we can compute the cup product on H^{∗}(X′) in terms of the cohomology groups H^{∗}(X), H^{∗}(Y), the Chern classes of Ň_{Y/X}, and the cohomology class [Y] ∈ H^{2d}(X).

3.4. Ample divisors on X′. Proposition 3.2. Let notation be as in §3.1. Let L be an ample ℝ-divisor on X. There exists α ∈ ℝ_{>0} such that, for all ε ∈ ℝ, the ℝ-divisor L′ on X′ of the form

L′ = f^{∗}O(L) − εY′

is ample if and only if 0 < ε < α.

Proof. We use the following basic facts. Although some results quoted below are only stated for usual ample divisors in the literature (for example, [Hart1], [Hart2]), it is easy to prove them for ample ℝ-divisors.

(1) Take an effective curve C on X′ which passes through a point of Y′ and intersects with Y′ transversally. By the Nakai-Moishezon criterion of ampleness ([Hart1], I, §5), we have (f^{∗}O(L) − εY′) · C > 0. Since Y′ · C > 0, ε can’t be arbitrary large.

(2) If f^{∗}O(L) − εY′ is ample, then ε > 0. To see this, take an effective curve C contained in a fiber of g: Y′ → Y. Since f^{∗}O(L) · C = L · f_{∗}C = 0 and Y′ · C < 0, we have ε > 0 by the Nakai-Moishezon criterion of ampleness.

(3) We see that f^{∗}O(L) − εY′ is ample for sufficiently small ε ∈ ℝ_{>0}, because the restriction of O_{X′}(−Y′) to Y′ is isomorphic to O_{Y′}(1) ([Hart2], II, 7.10).

(4) The set I = {ε ∈ ℝ | f^{∗}O(L) − εY′ is ample} is an interval, because if D, D′ are ample ℝ-divisors, then tD + (1 − t)D′ is also ample for all t ∈ ℝ with 0 ≤ t ≤ 1.

(5) Moreover, I is an open interval because, for a ℝ-divisor D and an ample ℝ-divisor D′, tD + (1 − t)D′ is ample for sufficiently small t ∈ ℝ_{>0}.

By combining above facts, we conclude that I = {ε ∈ ℝ | 0 < ε < α} for some α ∈ ℝ_{>0}. □

Remark 3.3. Since all ℝ-divisors L′ on X′ is of the form L′ = f^{∗}f_{∗}O(L′) − εY′ for some ε ∈ ℝ, Proposition 3.2 determines the set of all ample ℝ-divisors on X′ in some sense. However, in practice, it seems difficult to compute the upper bound α in Proposition 3.2 explicitly.

3.5. Hodge standard conjecture for X′. Proposition 3.4. Let notation be as in §3.1 (Recall that X, Y, X′, Y′ satisfy Assumption 2.1 in §2.1). Let L be an ample ℝ-divisor on X. Assume that the hard Lefschetz conjecture (Conjecture 2.4) and the Hodge standard conjecture (Conjecture 2.6) hold for (X, L) and (Y, O(L)|_{Y} ), where O(L)|_{Y} denotes the restriction of O(L) to Y which is also ample. Then, there exists β ∈ ℝ_{>0} such that, for all ε ∈ ℝ with 0 < ε < β, the ℝ-divisor L′ on X′ of the form

L′ = f^{∗}O(L) − εY′

is an ample ℝ-divisor on X′ for which Conjecture 2.4 and Conjecture 2.6 hold.

Proof. First of all, by the isomorphism (3.3), we observe that H^{∗}(X′) is isomorphic to H^{∗}(Y × ℙ^{d−2}) ⊕ H^{∗}(X) as an ℝ-vector space, although the multiplicative structures are different. Nevertheless, if we choose appropriate bases of H^{∗}(X′) and H^{∗}(Y × ℙ^{d−2}) ⊕ H^{∗}(X), and take the limit ε → 0, the matrices representing the action of the Lefschetz operator and the cup product pairings on H^{∗}(X′) converge to those on H^{∗}(Y × ℙ^{d−2}) ⊕ H^{∗}(X).

Let us make the above observation precise. Let {x_{γ}} (resp. {y_{δ}}) be the basis of H^{∗}(X) (resp. H^{∗}(Y)) which is independent of ε. For ε ∈ ℝ_{>0}, we see that {ε^{−(d−2)/2+k} y_{δ} ξ^{k}} is a ε-dependent basis of H^{∗}(Y)ξ^{k} for 0 ≤ k ≤ d−2. By (3.3), we have the following isomorphism :

H^{∗}(X′) ≅ H^{∗}(Y) ⊕ H^{∗}(Y)ξ ⊕ ⋯ ⊕ H^{∗}(Y)ξ^{d−2} ⊕ H^{∗}(X).

such that the following set

(3.4) {ε^{−(d−2)/2} y_{δ}} ∪ {ε^{−(d−2)/2+1} y_{δ} ξ} ∪ ⋯ ∪ {ε^{−(d−2)/2+(d−2)} y_{δ} ξ^{d−2}} ∪ {x_{γ}}

is a basis of H^{∗}(X′). By using the basis (3.4), we have an isomorphism

(3.5)  H^{k}(X′) ≅ H^{k−2}(Y × ℙ^{d−2}) ⊕ H^{k}(X)

as an ℝ-vector space which sends ε^{−(d−2)/2+m} ξ^{m} to c_{1}(O_{ℙ^{d−2}}(1))^{m}. The reason we multiply ξ^{k} by ε^{−(d−2)/2+k} in (3.4) is as follows. If we take the cup product of two elements in (3.4), the coefficient of the top degree term ξ^{d−2} is a constant function in ε. If ξ^{k} for k > d − 2 appears in the cup product, we can replace ξ^{d−1} by (ξ^{d−1} − c_{d−1}(F̌), [Y]) in H^{∗}(X′) (see §3.3). However, since the order of ε in the coefficient of ξ^{d−1} is one, this term vanishes if we take the limit ε → 0. Therefore, by using the basis (3.4), we can consider the asymptotic behavior of the Lefschetz operator and the cup product pairings on H^{∗}(X′) as ε → 0.

We put H̃^{k}(X′) = H^{k−2}(Y × ℙ^{d−2}) ⊕ H^{k}(X) and define the map

(3.6)  L̃: H̃^{k}(X′) ⟶ H̃^{k+2}(X′)

as follows. Let pr_{1} : Y × ℙ^{d−2} → Y, pr_{2} : Y × ℙ^{d−2} → ℙ^{d−2} be projections. Then

L_{Y ×ℙ^{d−2}} = pr_{1}^{∗}(O(L)|_{Y} ) + pr_{2}^{∗}O_{ℙ^{d−2}}(1)

is an ample ℝ-divisor on Y × ℙ^{d−2}. We define the map L̃ as a direct sum of the Lefschetz operators L_{Y ×ℙ^{d−2}} and L. By Example 2.9, Proposition 2.13, the hard Lefschetz conjecture (Conjecture 2.4) and the Hodge standard conjecture (Conjecture 2.6) hold for (Y × ℙ^{d−2}, L_{Y ×ℙ^{d−2}}). Therefore, we see that L̃^{k} induces the following isomorphism :

L̃^{k}: H̃^{n−k}(X′) ⟶^{≅} H̃^{n+k}(X′).

Namely, L̃ satisfies an analogue of the hard Lefschetz conjecture (Conjecture 2.4). Moreover, an analogue of the Hodge standard conjecture (Conjecture 2.6) holds for L̃ in the following sense. Let P̃^{k}(X′) be the kernel of L̃^{n−k+1} on H̃^{k}(X′), which is isomorphic to P^{k−2}(Y × ℙ^{d−2}) ⊕ P^{k}(X). Then the following pairing on P̃^{k}(X′)

P̃^{k}(X′) × P̃^{k}(X′) ⟶ ℝ, (a, b) ↦ (−1)^{k/2} L̃^{n−k} a ˜∪ b

is positive definite, where ˜∪ is a difference of the cup products on H^{∗}(Y × ℙ^{d−2}) and H^{∗}(X) defined as follows :

˜∪: H̃^{∗}(X′) × H̃^{∗}(X′) ∋ ((y_{0}, x_{0}), (y_{1}, x_{1})) ↦ −y_{0} ∪ y_{1} + x_{0} ∪ x_{1}.

Note that the minus sign is inevitable for the Hodge standard conjecture (Conjecture 2.6) because, for (y, x) ∈ P̃^{k}(X′), the degree of y is smaller than the degree of x by 2.

Therefore, to prove Proposition 3.4, it is enough to prove the following claim.

Claim 3.5. If we take the limit ε → 0, the matrices representing the action of L′ = f^{∗}O(L) − εY′ with respect to the basis (3.4) converge to the matrices representing L̃ via (3.5). Similarly, for all k, the matrices representing the cup product pairing between H^{k}(X′) and H^{2n−k}(X′) converge to the matrices representing the pairing ˜∪ between H̃^{k}(X′) and H̃^{2n−k}(X′).

We shall compute the action of L′ = f^{∗}O(L) − εY′. Since [Y′] = j_{∗}(1_{Y′}) ∈ H^{2}(X′), we compute the cup product

(3.7)  (f^{∗}O(L) − εj_{∗}(1_{Y′})) ∪ (f^{∗}(x) + j_{∗}(y))

for x ∈ H^{k}(X), y ∈ H^{k−2}(Y′) with g_{∗}(y) = 0 (see (3.2)). Since (3.7) is equal to

f^{∗}O(L) ∪ f^{∗}(x) + f^{∗}O(L) ∪ j_{∗}(y) − εj_{∗}(1_{Y′}) ∪ f^{∗}(x) − εj_{∗}(1_{Y′}) ∪ j_{∗}(y),

we consider these four terms separately. Firstly, since the pullback preserves the cup product, we have f^{∗}O(L) ∪ f^{∗}(x) = f^{∗}(O(L) ∪ x) = f^{∗}(L ∪ x). Secondly, we have f^{∗}O(L)∪j_{∗}(y) = j_{∗}(j^{∗}f^{∗}O(L)∪y) = j_{∗}(g^{∗}i^{∗}O(L)∪y) = j_{∗}(g^{∗}(O(L)|_{Y})∪y) because f ∘j = i∘g. Similarly, we have −εj_{∗}(1_{Y′})∪f^{∗}(x) = −εj_{∗}(1_{Y′} ∪j^{∗}f^{∗}(x)) = −εj_{∗}(g^{∗}(x|_{Y})). Finally, by the self-intersection formula j^{∗}j_{∗}(1_{Y′}) = −ξ, we have −εj_{∗}(1_{Y′}) ∪ j_{∗}(y) = −εj_{∗}(j^{∗}j_{∗}(1_{Y′}) ∪ y) = j_{∗}(εξ ∪ y). Therefore, we conclude that (3.7) is equal to

f^{∗}(L ∪ x) + j_{∗}((g^{∗}(O(L)|_{Y}) + εξ) ∪ y) − εj_{∗}(g^{∗}(x|_{Y})).

It is clear that the first term f^{∗}(L ∪ x) corresponds to the action of L on H^{∗}(X) via (3.5). The second term j_{∗}((g^{∗}(O(L)|_{Y}) + εξ) ∪ y) almost corresponds to the action of L_{Y ×ℙ^{d−2}} on H^{∗}(Y × ℙ^{d−2}) via (3.5). The only difference is that ξ^{d−1} may appear in the cup product (g^{∗}(O(L)|_{Y}) + εξ) ∪ y. Since the order of ε in the coefficient of ξ^{d−1} is higher than others, if we replace ξ^{d−1} by (ξ^{d−1}−c_{d−1}(F̌), [Y]), this contribution vanishes if we take the limit ε → 0. The contribution from the third term −εj_{∗}(g^{∗}(x|_{Y})) vanishes if we take the limit ε → 0. Hence the first assertion of Claim 3.5 is proved.

Finally, we shall compute the cup product pairing on H^{∗}(X′). According to (3.2), we compute the cup product

(3.8)  (f^{∗}(x_{0}) + j_{∗}(y_{0})) ∪ (f^{∗}(x_{1}) + j_{∗}(y_{1}))

for x_{0} ∈ H^{k}(X), y_{0} ∈ H^{k−2}(Y′), x_{1} ∈ H^{2n−k}(X), y_{1} ∈ H^{2n−k−2}(Y′) with g_{∗}(y_{0}) = 0, g_{∗}(y_{1}) = 0. Since (3.8) is in H^{2n}(X′) = ℝ, and f_{∗}: H^{2n}(X′) → H^{2n}(X) is an isomorphism, it is enough to compute

f_{∗}((f^{∗}(x_{0}) + j_{∗}(y_{0})) ∪ (f^{∗}(x_{1}) + j_{∗}(y_{1}))) = f_{∗}(f^{∗}(x_{0}) ∪ f^{∗}(x_{1}))

+ f_{∗}(f^{∗}(x_{0}) ∪ j_{∗}(y_{1})) + f_{∗}(j_{∗}(y_{0}) ∪ f^{∗}(x_{1})) + f_{∗}(j_{∗}(y_{0}) ∪ j_{∗}(y_{1})).

For the first term, we have f_{∗}(f^{∗}(x_{0})∪f^{∗}(x_{1})) = f_{∗}(f^{∗}(x_{0}∪x_{1})) = x_{0}∪x_{1} since the pullback preserves the cup product and f_{∗}∘f^{∗} = id. For the second term, we have f_{∗}(f^{∗}(x_{0}) ∪ j_{∗}(y_{1})) = x_{0} ∪ f_{∗}(j_{∗}(y_{1})) = x_{0} ∪ i_{∗}(g_{∗}(y_{1})) = 0 because f ∘ j = i ∘ g and g_{∗}(y_{1}) = 0. Similarly, the third term f_{∗}(j_{∗}(y_{0}) ∪ f^{∗}(x_{1})) is zero because g_{∗}(y_{0}) = 0. For the last term, by the self-intersection formula j^{∗}j_{∗}(y_{0}) = −ξ ∪ y_{0}, we have f_{∗}(j_{∗}(y_{0}) ∪ j_{∗}(y_{1})) = f_{∗}(j_{∗}(j^{∗}j_{∗}(y_{0}) ∪ y_{1})) = i_{∗}(g_{∗}(−ξ ∪ y_{0} ∪ y_{1})). Since g_{∗}(ξ^{d−1}) = 1 and g_{∗}(ξ^{m}) = 0 for 0 ≤ m ≤ d − 2, if we take the limit ε → 0, the matrix representing the pairing

H^{k−2}(Y′) × H^{2n−k−2}(Y′) ∋ (y_{0}, y_{1}) ↦ i_{∗}(g_{∗}(−ξ ∪ y_{0} ∪ y_{1})) ∈ H^{2n}(X) = ℝ

with respect to the basis (3.4) converges to the minus of the matrix representing the cup product pairing

H^{k−2}(Y × ℙ^{d−2}) × H^{2n−k−2}(Y × ℙ^{d−2}) ⟶ H^{2n−4}(Y × ℙ^{d−2}) = ℝ

via (3.5). Therefore, the second assertion of Claim 3.5 is proved. Now the proof of Proposition 3.4 is completed. □

Corollary 3.6. Let notation and assumptions be as in Proposition 3.4.

(1) If the hard Lefschetz conjecture (Conjecture 2.4) holds for (X′, L′) for all ample ℝ-divisors L′ on X′ of the form L′ = f^{∗}O(L) − εY′ (ε ∈ ℝ_{>0}), then Hodge standard conjecture (Conjecture 2.6) holds for (X′, L′) for all ample ℝ-divisors L′ on X′ of the form L′ = f^{∗}O(L) − εY′.

(2) In particular, if the hard Lefschetz conjecture (Conjecture 2.4) holds for (X′, L′) for all ample ℝ-divisors L′ on X′, then the Hodge standard conjecture (Conjecture 2.6) holds for (X′, L′) for all ample ℝ-divisors L′ on X′.

Proof. By a “homotopy argument” (Corollary 2.12), the assertion easily follows from Proposition 3.4. □

Remark 3.7. Our proof of Proposition 3.4 is based on a consideration of the asymptotic behavior as ε → 0. Therefore, by our method, we can’t prove whether we can take β = α. In other words, here we can’t prove Conjecture 2.6 holds for all ample ℝ-divisors L′ on X′ of the form L′ = f^{∗}O(L) − εY′.

4. Some blowing-ups of projective spaces

In this section, we consider a projective smooth variety B^{n} of dimension n over \overline{𝔽}_{q} which is obtained by successive blowing-ups of ℙ_{\overline{𝔽}_{q}}^{n} along linear subvarieties. All varieties appearing in this section are defined over \overline{𝔽}_{q} and satisfy Assumption 2.1 in §2.1. Hence, for simplicity, we omit the subscript “\overline{𝔽}_{q}” and denote ℙ_{\overline{𝔽}_{q}}^{n} by ℙ^{n}, etc. This doesn’t cause any confusion. We use the same notation as in §2.1.

4.1. Construction of B^{n}. Let n ≥ 1 be an integer. We have a natural map φ: 𝔸^{n+1}\{0} → ℙ^{n}. A subvariety V ⊂ ℙ^{n} is called a linear subvariety if φ^{−1}(V)∪ {0} ⊂ 𝔸^{n+1} is a linear subspace. Let Gr_{d}(ℙ^{n}) be the Grassmann variety of linear subvarieties of dimension d in ℙ^{n}. Let Gr_{∗}(ℙ^{n}) be the disjoint union of Gr_{d}(ℙ^{n}) for all d. Gr_{d}(ℙ^{n})(𝔽_{q}) is the set of linear subvarieties of dimension d in ℙ^{n} defined over 𝔽_{q}.

We construct a birational map f: B^{n} → ℙ^{n} as follows. Firstly, we put Y_{0} := ℙ^{n}. Then, let Y_{1} be the blow-up of Y_{0} along the disjoint union of all 𝔽_{q}-rational points on ℙ^{n}. Let Y_{2} be the blow-up of Y_{1} along disjoint union of all proper transforms of lines in ℙ^{n}. Similarly, we construct projective smooth varieties Y_{0}, …, Y_{n−1} inductively as follows. Assume that Y_{k} was already constructed. Let Z_{k} ⊂ Y_{k} be the union of all proper transforms of linear subvarieties of dimension k in ℙ^{n} defined over 𝔽_{q}. These are disjoint because all intersections of them were already blown-up. Let g_{k}: Y_{k+1} → Y_{k} be the blow-up of Y_{k} along Z_{k}. Finally, we put B^{n} := Y_{n−1} and f := g_{0} ∘ ⋯ ∘ g_{n−2} : B^{n} → ℙ^{n}. We have the following sequence of blowing-ups.

B^{n} = Y_{n−1} ⟶^{g_{n−2}} Y_{n−2} ⟶^{g_{n−3}} ⋯ ⟶^{g_{1}} Y_{1} ⟶^{g_{0}} Y_{0} = ℙ^{n}

Note that there is a natural action of PGL_{n+1}(𝔽_{q}) on ℙ^{n}, and the above construction is equivariant with respect to PGL_{n+1}(𝔽_{q})-action. Hence we have a natural action of PGL_{n+1}(𝔽_{q}) on B^{n}.

4.2. Divisors on B^{n}. For V ∈ Gr_{k}(ℙ^{n})(𝔽_{q}), we define a smooth irreducible divisor D_{V} on B^{n} as follows. If k = n − 1, let D_{V} ⊂ B^{n} be the proper transform of V. If k < n − 1, the proper transform Ṽ ⊂ Y_{k} of V is a connected component of Z_{k}. Therefore, g_{k}^{−1}(Ṽ) ⊂ Y_{k+1} is a ℙ^{n−d−1}-bundle over Ṽ. Let D_{V} ⊂ B^{n} be the proper transform of g_{k}^{−1}(Ṽ).

By construction, an ℝ-divisor D on B^{n} is written uniquely as

D = f^{∗}O(H) + ∑_{\begin{matrix}V ∈Gr_{k}(ℙ^{n})(𝔽_{q}), \\ 0≤k≤n−2 \end{matrix}} a_{V} D_{V}  in H^{2}(B^{n}),

where H is an ℝ-divisor on ℙ^{n}, and a_{V} ∈ ℝ. We also see that, all ℝ-divisors D′ on B^{n} is written as

D′ = ∑_{V ∈Gr_{∗}(ℙ^{n})(𝔽_{q})}a′_{V} D_{V}  (a′_{V} ∈ ℝ) in H^{2}(B^{n}).

Note that this second expression is not unique.

For 0 ≤ k ≤ n − 1, we define a divisor D_{k} on B^{n} by

D_{k} := ∑_{V ∈Gr_{k}(ℙ^{n})(𝔽_{q})}D_{V} .

By construction, we see that the support of D_{k} is a disjoint union of smooth divisors on B^{n}.

In this section, we use the following terminology for ℝ-divisors on B^{n}.

Definition 4.1. Let D be an ℝ-divisor on B^{n}.

(1) D is called positive if D can be written as

D = ∑_{V ∈Gr_{∗}(ℙ^{n})(𝔽_{q})}a_{V} D_{V} (a_{V} ∈ ℝ_{>0}) in H^{2}(B^{n}).

(2) D is called PGL_{n+1}(𝔽_{q})-invariant if the class of D in H^{2}(B^{n}) is PGL_{n+1}(𝔽_{q})- invariant.

It is easy to see that D is PGL_{n+1}(𝔽_{q})-invariant if and only if D can be written as D = f^{∗}O(H) + ∑_{k=0}^{n−1} b_{k}D_{k} in H^{2}(B^{n}) for some H, b_{k} (for a similar criterion of positive divisors, see Proposition 4.4).

4.3. Intersections of divisors D_{V} on B^{n}. Here we study the combinatorial structure of intersections of divisors D_{V} on B^{n} (see also [Mus], proof of Theorem 4.1, I). Proposition 4.2. (1) For V, W ∈ Gr_{∗}(ℙ^{n})(𝔽_{q}), D_{V} ∩ D_{W} ≠ ∅ if and only if

V ⊂ W or W ⊂ V.

(2) For V ∈ Gr_{d}(ℙ^{n})(𝔽_{q}), we have a noncanonical isomorphism

D_{V} ≅ B^{d} × B^{n−d−1},

where B^{d} (resp. B^{n−d−1}) is a variety constructed by successive blowingups of ℙ^{d} (resp. ℙ^{n−d−1}) by the same way as B^{n}.

(3) Let x ∈ V be a point which doesn’t lie in any linear subvariety defined over 𝔽_{q} strictly contained in V. Let Ṽ ⊂ Y_{d} be the proper transform of V. Then g_{d}: g_{d}^{−1}(Ṽ) → Ṽ is a trivial ℙ^{n−d−1}-bundle. Moreover, Ṽ is isomorphic to B^{d}, and an isomorphism D_{V} ≅ B^{d} × B^{n−d−1} as in 2. is induced from a choice of x ∈ V and a trivialization g_{d}^{−1}(Ṽ) ≅ Ṽ × ℙ(Ň_{V/ℙ^{n},x}), where Ň_{V/ℙ^{n},x} is the fiber at x of the conormal bundle Ň_{V/ℙ^{n}}.

(4) For W ∈ Gr_{∗}(ℙ^{n})(𝔽_{q}) strictly contained in V, D_{W} intersects transversally with D_{V}, and the intersection D_{W} ∩ D_{V} is of the form

D_{W} ∩ D_{V} = D̃_{W} × B^{n−d−1} on D_{V} ,

where D̃_{W} is a divisor on B^{d} corresponding to the inclusion W ↪ V ≅ ℙ^{d}.

(5) For W′ ∈ Gr_{∗}(ℙ^{n})(𝔽_{q}) strictly containing V, D_{W′} intersects transversally with D_{V}, and the intersection D_{W′} ∩ D_{V} is of the form

D_{W′} ∩ D_{V} = B^{d} × ˜ D̃_{W′} on D_{V} ,

where ˜ D̃_{W′} is a divisor on B^{n−d−1} corresponding to the inclusion ℙ(Ň_{V/W′,x}) ↪ ℙ(Ň_{V/ℙ^{n},x}) ≅ ℙ^{n−d−1}.

Proof. Firstly, we shall prove the first assertion. If V ⊂ W or W ⊂ V, it is clear that D_{V} ∩D_{W} ≠ ∅. Assume that V ∩W is strictly smaller than V, W. Then, after blowing-up along the proper transform of V ∩ W in the construction of B^{n}, the proper transforms of V and W become disjoint. Therefore, D_{V} doesn’t intersect with D_{W}.

Recall that, for a vector bundle E of rank k over a smooth variety X, ℙ(E) is a trivial ℙ^{k−1}-bundle if and only if E⊗_{O_{X}}L is a trivial vector bundle for a line bundle L over X ([Hart2], II, Exercise 7.10). We call such E a twist of a trivial vector bundle over X. For V ∈ Gr_{d}(ℙ^{n})(𝔽_{q}), the normal bundle N_{V/ℙ^{n}} ≅ O_{V} (1)^{⊕n−d} is a twist of a trivial vector bundle over V. Hence the normal bundle N_{Ṽ /Y_{d}} is also a twist of a trivial vector bundle over Y_{d}, where Ṽ ⊂ Y_{d} is the proper transform of V ([Fu], B.6.10). Therefore, g_{d}^{−1}(Ṽ) is a trivial ℙ^{n−d−1}-bundle over Ṽ.

Here Ṽ is isomorphic to B^{d}. This follows from the fact that, for a sequence of regular embeddings Z ↪ Y ↪ X, the proper transform of Y in the blow-up of X along Z is isomorphic to the blow-up of Y along Z ([Fu], B.6.9).

We fix a point x ∈ V which doesn’t lie in any linear subvariety defined over 𝔽_{q} strictly contained in V, and a trivialization g_{d}^{−1}(Ṽ) ≅ Ṽ × ℙ(Ň_{V/ℙ^{n},x}), where Ň_{V/ℙ^{n},x} denotes the fiber at x of the conormal bundle Ň_{V/ℙ^{n}}. For W ∈ Gr_{∗}(ℙ^{n})(𝔽_{q}) strictly containing V, a natural inclusion V ⊂ W ⊂ ℙ^{n} induces N_{V/W} ⊂ N_{V/ℙ^{n}} and hence ℙ(Ň_{V/W}) ⊂ ℙ(Ň_{V/ℙ^{n}}). Thus, ℙ(Ň_{V/W ,x}) is a linear subvariety of ℙ(Ň_{V/ℙ^{n},x}). From this, we see that the blowing-up of g_{d}^{−1}(Ṽ) along the proper transform of a linear subvariety strictly containing V corresponds to the blowing-up along the proper transform of a linear subvariety of the second factor ℙ(Ň_{V/ℙ^{n},x}) of g_{d}^{−1}(Ṽ) ≅ Ṽ × ℙ(Ň_{V/ℙ^{n},x}). Therefore, we have an isomorphism D_{V} ≅ B^{d} ×B^{n−d−1}. The remaining assertions follow from the construction of this isomorphism. □

Corollary 4.3. Let H be an ℝ-divisor on ℙ^{n}, V ∈ Gr_{d}(ℙ^{n})(𝔽_{q}), and f: B^{n} → ℙ^{n} a natural map as in §4.1. Fix an isomorphism D_{V} ≅ B^{d} × B^{n−d−1} as in Proposition 4.2. Let pr_{1} : D_{V} → B^{d} be the projection to the first factor. Let f′: B^{d} → ℙ^{d} ≅ V be a map defined similarly as f. Then we have

(f^{∗}O(H))|_{D_{V}} = pr_{1}^{∗}(f′^{∗}(O(H)|_{V} )) in H^{2}(D_{V}).

Proof. This follows from the construction of an isomorphism D_{V} ≅ B^{d} × B^{n−d−1} in Proposition 4.2. □

4.4. PGL_{n+1}(𝔽_{q})-invariant ℝ-divisors on B^{n}. Proposition 4.4. (1) For V ∈ Gr_{n−1}(ℙ^{n})(𝔽_{q}), we have

f^{∗}O(V) = ∑_{\begin{matrix}W ∈Gr_{∗}(ℙ^{n})(𝔽_{q}), \\ W ⊂V \end{matrix}} D_{W}  in H^{2}(B^{n}).

(2) For an ℝ-divisor H on ℙ^{n}, f^{∗}O(H) is PGL_{n+1}(𝔽_{q})-invariant.

(3) Let D be a PGL_{n+1}(𝔽_{q})-invariant ℝ-divisor on B^{n} which is written as

D = α f^{∗}O_{ℙ^{n}}(1) + ∑_{d=0}^{n−1} a_{d}D_{d} (α, a_{d} ∈ ℝ) in H^{2}(B^{n}).

Then, D is positive if and only if α > 0 and

a_{d} + α \frac{|ℙ^{n−d−1}(𝔽_{q})|}{|ℙ^{n}(𝔽_{q})|} > 0 for all 0 ≤ d ≤ n − 1.

(4) Let D be a PGL_{n+1}(𝔽_{q})-invariant ℝ-divisor on B^{n}. Take V ∈ Gr_{d}(ℙ^{n})(𝔽_{q}), and fix an isomorphism D_{V} ≅ B^{d} × B^{n−d−1} as in Proposition 4.2. Let pr_{1} : D_{V} → B^{d}, pr_{2} : D_{V} → B^{n−d−1} be projections. Then, there exist a PGL_{d+1}(𝔽_{q})-invariant ℝ-divisor D′ on B^{d} and a PGL_{n−d}(𝔽_{q})-invariant ℝ-divisor D′′ on B^{n−d−1} such that

O(D)|_{D_{V}} = pr_{1}^{∗}O(D′) + pr_{2}^{∗}O(D′′) in H^{2}(D_{V}).

Proof. The first assertion follows from the fact that the multiplicity of V along W ∈ Gr_{∗}(ℙ^{n})(𝔽_{q}) with W ⊂ V is equal to 1.

The second assertion is obvious because PGL_{n+1}(𝔽_{q}) acts trivially on H^{2}(ℙ^{n}) and f^{∗} is PGL_{n+1}(𝔽_{q})-equivariant.

We shall compute f^{∗}O(H) explicitly. Since H^{2}(ℙ^{n}) is generated by the class of O_{ℙ^{n}}(1), we write H = α O_{ℙ^{n}}(1) in H^{2}(ℙ^{n}) for some α ∈ ℝ. Since |Gr_{n−1}(ℙ^{n})(𝔽_{q})| = |ℙ^{n}(𝔽_{q})|, and, for each W ∈ Gr_{d}(ℙ^{n})(𝔽_{q}), there exist |ℙ^{n−d−1}(𝔽_{q})| elements in Gr_{n−1}(ℙ^{n})(𝔽_{q}) containing W, we have the following equality in H^{2}(B^{n}) :

f^{∗}O(H) = \frac{α}{|ℙ^{n}(𝔽_{q})|} ∑_{V ∈Gr_{n−1}(ℙ^{n})(𝔽_{q})}( ∑_{\begin{matrix}W ∈Gr_{∗}(ℙ^{n})(𝔽_{q}), \\ W ⊂V \end{matrix}} D_{W} )

= \frac{α}{|ℙ^{n}(𝔽_{q})|} ∑_{d=0}^{n−1} |ℙ^{n−d−1}(𝔽_{q})| D_{d}.

Of course, from this expression, we can check that f^{∗}O(H) is PGL_{n+1}(𝔽_{q})- invariant.

The third assertion easily follows from the above computation because

D = ∑_{d=0}^{n−1} (a_{d} + α \frac{|ℙ^{n−d−1}(𝔽_{q})|}{|ℙ^{n}(𝔽_{q})|})D_{d} in H^{2}(B^{n}).

Finally, we shall prove the last assertion. To avoid self-intersection, we write D in the following form

D = α f^{∗}O_{ℙ^{n}}(1) + ∑_{0≤k≤n−1, k≠d} a_{k}D_{k} (α, a_{k} ∈ ℝ) in H^{2}(B^{n}).

It is enough to treat the case D = α f^{∗}O_{ℙ^{n}}(1) and D = D_{k} (k ≠ d) separately. The case D = α f^{∗}O_{ℙ^{n}}(1) follows from Corollary 4.3 and the second assertion. The case D = D_{k} (k ≠ d) follows from Proposition 4.2. □

4.5. Ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisors on B^{n}. Here we study ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisors on B^{n}. First of all, we note that there exists at least one ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisor on B^{n}. It follows from the construction of B^{n} in §4.1 and Proposition 3.2. Proposition 4.5. Let D be an ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisor on B^{n}. Then, D is positive (see Definition 4.1).

Proof. We prove the assertion by induction on n. The case n = 1 is obvious. Assume that Proposition 4.5 was already proved in dimension < n.

Let D be written uniquely as

D = α f^{∗}O_{ℙ^{n}}(1) + ∑_{d=0}^{n−2} a_{d}D_{d} (α, a_{d} ∈ ℝ) in H^{2}(B^{n}).

By Proposition 4.4, D is positive if and only if α > 0 and

a_{d} + α \frac{|ℙ^{n−d−1}(𝔽_{q})|}{|ℙ^{n}(𝔽_{q})|} > 0 for all 0 ≤ d ≤ n − 2.

Fix V ∈ Gr_{n−1}(ℙ^{n})(𝔽_{q}). By Proposition 4.2, D_{V} is isomorphic to B^{n−1}. We consider the restriction of O(D) to D_{V}. Let f′: D_{V} ≅ B^{n−1} → ℙ^{n−1} be a map defined similarly as f. Then, by Proposition 4.2 and Corollary 4.3, we have

O(D)|_{D_{V}} = α f′^{∗}O_{ℙ^{n−1}}(1) + ∑_{d=0}^{n−2} a_{d}D′_{d} in H^{2}(B^{n}),

where D′_{d} is a divisor on D_{V} ≅ B^{n−1} defined similarly as D_{d} on B^{n}.

By induction, O(D)|_{D_{V}} is positive. Therefore, by Proposition 4.4, we have α > 0 and

a_{d} + α \frac{|ℙ^{n−d−2}(𝔽_{q})|}{|ℙ^{n−1}(𝔽_{q})|} > 0 for all 0 ≤ d ≤ n − 2.

Therefore, it is enough to show the following inequality

\frac{|ℙ^{n−d−1}(𝔽_{q})|}{|ℙ^{n}(𝔽_{q})|} > \frac{|ℙ^{n−d−2}(𝔽_{q})|}{|ℙ^{n−1}(𝔽_{q})|}.

Since |ℙ^{k}(𝔽_{q})| = \frac{q^{k+1}−1}{q−1}, the above inequality is equivalent to

\frac{q^{n−d} − 1}{q^{n+1} − 1} > \frac{q^{n−d−1} − 1}{q^{n} − 1}

⟺ q^{2n−d} − q^{n} − q^{n−d} + 1 > q^{2n−d} − q^{n+1} − q^{n−d−1} + 1
⟺ q^{n+1} + q^{n−d−1} > q^{n} + q^{n−d}
⟺ q^{n}(q − 1) > q^{n−d−1}(q − 1).

Hence we prove Proposition 4.5. □

4.6. Hodge standard conjecture for B^{n} with ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisors. Lemma 4.6. For 0 ≤ k ≤ n − 1 and a ∈ H^{k}(B^{n}), if the restriction of a to D_{V} is zero for all V ∈ Gr_{∗}(ℙ^{n})(𝔽_{q}), then a = 0.

Proof. By applying Proposition 3.1 successively, we see that a is of the form a = f^{∗}a′ for some a′ ∈ H^{k}(ℙ^{n}). For V ∈ Gr_{n−1}(ℙ^{n})(𝔽_{q}), the restriction of a′ to V is zero because the restriction of a to D_{V} is zero. Since restriction map H^{k}(ℙ^{n}) → H^{k}(V) is an isomorphism, we conclude that a = 0. □

The following proposition is the main result of this section. Proposition 4.7. Let D be an ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisor on B^{n}. Then the hard Lefschetz conjecture (Conjecture 2.4) and the Hodge standard conjecture (Conjecture 2.6) hold for (B^{n}, D).

Proof. We prove the assertion by induction on n. The case n = 1 is obvious. Assume that Proposition 4.7 was already proved in dimension < n.

By the construction of B^{n} in §4.1 and Proposition 3.4, we see that there exists at least one ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisor on B^{n} for which Proposition 4.7 holds. If D, D′ are ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisors on B^{n}, then tD + (1 − t)D′ is also an ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisor on B^{n} for t ∈ ℝ with 0 ≤ t ≤ 1. Therefore, by a “homotopy argument” (Corollary 2.12), it is enough to show that the hard Lefschetz conjecture (Conjecture 2.4) holds for (B^{n}, D) for all ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisors D on B^{n}.

Assume that there exists an ample PGL_{n+1}(𝔽_{q})-invariant ℝ-divisor L on B^{n} and a nonzero cohomology class a ∈ H^{k}(B^{n}) for 0 ≤ k ≤ n − 1 such that

L^{n−k} ∪ a = 0.

For 0 ≤ d ≤ n − 1, take V ∈ Gr_{d}(ℙ^{n})(𝔽_{q}) and fix an isomorphism D_{V} ≅ B^{d} × B^{n−d−1} as in Proposition 4.2. Let pr_{1} : D_{V} → B^{d}, pr_{2} : D_{V} → B^{n−d−1} be projections. Then, by Proposition 4.4, the restriction O(L)|_{D_{V}} can be written as

O(L)|_{D_{V}} = pr_{1}^{∗}O(D′) + pr_{2}^{∗}O(D′′) in H^{2}(D_{V}),

where D′ (resp. D′′) is an ample PGL_{d+1}(𝔽_{q})-invariant (resp. PGL_{n−d}(𝔽_{q})- invariant) ℝ-divisor on B^{d} (resp. B^{n−d−1}). Therefore, by induction hypothesis and Proposition 2.13, the hard Lefschetz conjecture (Conjecture 2.4) and the Hodge standard conjecture (Conjecture 2.6) hold for (D_{V} , O(L)|_{D_{V}} ).

By restricting L^{n−k} ∪ a = 0 to D_{V}, we have (O(L)|_{D_{V}} )^{n−k} ∪ (a|_{D_{V}}) = 0. Since dim D_{V} = n − 1, this implies a|_{D_{V}} ∈ H^{k}(D_{V}) is in the primitive part P^{k}(D_{V}). Therefore, by the Hodge standard conjecture (Conjecture 2.6) for (D_{V} , O(L)|_{D_{V}} ), we have

(−1)^{k/2} · (O(L)|_{D_{V}} )^{n−k−1} ∪ (a|_{D_{V}}) ∪ (a|_{D_{V}}) ≥ 0

and the equality holds if and only if a|_{D_{V}} = 0.

By Proposition 4.5, L can be written as

L = ∑_{V ∈Gr_{∗}(ℙ^{n})(𝔽_{q})}a_{V} D_{V} (a_{V} ∈ ℝ_{>0}) in H^{2}(B^{n}).

Therefore, we have

(−1)^{k/2} · L^{n−k} ∪ a ∪ a

= ∑_{V ∈Gr_{∗}(ℙ^{n})(𝔽_{q})}a_{V} · D_{V} ∪ ((−1)^{k/2} · L^{n−k−1} ∪ a ∪ a)

= ∑_{V ∈Gr_{∗}(ℙ^{n})(𝔽_{q})}a_{V} ((−1)^{k/2} · (O(L)|_{D_{V}} )^{n−k−1} ∪ (a|_{D_{V}}) ∪ (a|_{D_{V}}))

= 0

because L^{n−k} ∪ a = 0. Since a_{V} > 0 for all V ∈ Gr_{∗}(ℙ^{n})(𝔽_{q}), we have a|_{D_{V}} = 0 for all V ∈ Gr_{∗}(ℙ^{n})(𝔽_{q}). By Lemma 4.6, we have a = 0. This is a contradiction. Hence the proof of Proposition 4.7 is completed. □

Remark 4.8. Mustafin, Kurihara showed that B^{d} is isomorphic to all irreducible components of the special fiber of the formal scheme model Ω̂_{O_{K}}^{d} over Spf O_{K} of the Drinfeld upper half space Ω̂_{K}^{d} of dimension d (for details, see §6, see also [Mus], [Ku]). Moreover, they also showed that

D = −(d + 1)f^{∗}O_{ℙ^{d}}(1) + ∑_{k=0}^{d−1}(d − k)D_{k}

is an ample divisor on B^{d}, and D coincides with the restriction of the “relative dualizing sheaf” ω_{Ω̂_{O_{K}}^{d} /O_{K}} to an irreducible component of the special fiber. These facts are crucial in our application to p-adic uniformization in §6. Remark 4.9. In our proof of Proposition 4.5, we heavily use the assumption that D is PGL_{n+1}(𝔽_{q})-invariant. Therefore, our proof doesn’t work for ample non-PGL_{n+1}(𝔽_{q})-invariant ℝ-divisors on B^{n}. It is easy to see that if all ample ℝ-divisors on B^{n} are positive, then Proposition 4.7 holds for them. However, the author doesn’t even know whether it is natural to expect this. He has neither evidence nor counter-example for ample non-PGL_{n+1}(𝔽_{q})-invariant ℝ-divisors.

5. Review of the weight spectral sequence of Rapoport-Zink

In this section, we recall the definitions and basic properties of monodromy filtration, weight filtration, and the weight spectral sequence of Rapoport-Zink. Since we work over a local field in this section, we slightly change the notation. As in §1, let K be a complete discrete valuation field with finite residue field 𝔽_{q} of characteristic p > 0, l be a prime number different from p. Let X be a proper smooth variety of dimension n over K, and V := H_{ét}^{w}(X\overline{K}, ℚ_{l}).

5.1. Monodromy filtration. Let I_{K} be the inertia group of K, which is a subgroup of Gal(\overline{K}/K) defined by the exact sequence :

1 ⟶ I_{K} ⟶ Gal(\overline{K}/K) ⟶ Gal(\overline{𝔽}_{q}/𝔽_{q}) ⟶ 1.

Gal(\overline{𝔽}_{q}/𝔽_{q}) acts on I_{K} by conjugation (τ: σ ↦ τστ^{−1}, τ ∈ Gal(\overline{𝔽}_{q}/𝔽_{q}), σ ∈ I_{K}). The pro-l-part of I_{K} is isomorphic to ℤ_{l}(1) as a Gal(\overline{𝔽}_{q}/𝔽_{q})-module by

t_{l}: I_{K} ∋ σ ↦ (\frac{σ(π^{1/l^{m}})}{π^{1/l^{m}}})_{m} ∈ \underleftarrow{lim} μ_{l^{m}} = ℤ_{l}(1),

where π is a uniformizer of K, and μ_{l^{m}} is the group of l^{m}-th roots of unity. It is known that t_{l} is independent of the choice of π and its l^{m}-th root π^{1/l^{m}} ([Se1]).

By Grothendieck’s monodromy theorem ([ST], Appendix), there exist r, s ≥ 1 such that (ρ(σ)^{r} − 1)^{s} = 0 for all σ ∈ I_{K}. Therefore, by replacing K by its finite extension, I_{K} acts on V through t_{l}: I_{K} → ℤ_{l}(1) and this action is unipotent. Then there is a unique nilpotent map of Gal(\overline{K}/K)-representations called the monodromy operator N: V (1) → V such that

ρ(σ) = exp (t_{l}(σ)N) for all σ ∈ I_{K}.

Here N is a nilpotent map means that N^{r}: V (r) → V is zero for some r ≥ 1. Definition 5.1. ([De3], I, 1.7.2) There exists a unique filtration M_{•} on V called the monodromy filtration characterized by the following properties.

(1) M_{•} is an increasing filtration ⋯ ⊂ M_{i−1}V ⊂ M_{i}V ⊂ M_{i+1}V ⊂ ⋯ of Gal(\overline{K}/K)-representations such that M_{i}V = 0 for sufficiently small i and M_{i}V = V for sufficiently large i.

(2) N(M_{i}V (1)) ⊂ M_{i−2}V for all i.

(3) By the second condition, we can define N: Gr_{i}^{M}V (1) → Gr_{i−2}^{M}V, where Gr_{i}^{M}V := M_{i}V/M_{i−1}V. Then, for each r ≥ 0, N^{r}: Gr_{r}^{M}V (r) → Gr_{−r}^{M}V is an isomorphism.

Remark 5.2. In the above definition, we replace K by its finite extension. We can easily see that M_{•} is stable under the action of Gal(\overline{K}/K) for the original K. Therefore, we can define the monodromy filtration M_{•} as a filtration of Gal(\overline{K}/K)-representations without replacing K by its finite extension.

5.2. Weight filtration. Let Fr_{q} ∈ Gal(\overline{𝔽}_{q}/𝔽_{q}) be the inverse of the q-th power map on \overline{𝔽}_{q} called the geometric Frobenius element. A Gal(\overline{𝔽}_{q}/𝔽_{q})-representation is said to have weight k if all eigenvalues of the action of Fr_{q} ∈ Gal(\overline{𝔽}_{q}/𝔽_{q}) are algebraic integers whose all complex conjugates have complex absolute value q^{k/2}. Definition 5.3. ([De1], [De3], I, 1.7.5) There exists a unique filtration W_{•} called the weight filtration on V characterized by the following properties (for existence, see §5.3).

(1) W_{•} is an increasing filtration ⋯ ⊂ W_{i−1}V ⊂ W_{i}V ⊂ W_{i+1}V ⊂ ⋯ of Gal(\overline{K}/K)-representations such that W_{i}V = 0 for sufficiently small i and W_{i}V = V for sufficiently large i.

(2) For a lift F̃r_{q} of Fr_{q} in Gal(\overline{K}/K), all eigenvalues of the action of F̃r_{q} on each Gr_{i}^{W} V := W_{i}V/W_{i−1}V are algebraic integers whose all complex conjugates have complex absolute value q^{i/2}.

Remark 5.4. By the second condition, the monodromy operator N: Gr_{i}^{W} V (1) → Gr_{i}^{W} V is zero for each i (see §5.1). Hence, by replacing K by its finite extension K′ with residue field 𝔽_{q^{r}}, I_{K′} acts on Gr_{i}^{W} V trivially, and Gal(\overline{𝔽}_{q^{r}}/𝔽_{q^{r}})- representation Gr_{i}^{W} V have weight i for each i.

5.3. Weight spectral sequence of Rapoport-Zink.

Definition 5.5. A proper scheme 𝔛 over O_{K} is called a proper semistable model of X over O_{K} if the following conditions are satisfied. 𝔛 has an open covering {U_{i}} consisting of finite open sets of 𝔛 such that, for each i, there exists r_{i} ≥ 1 and a commutative diagram

\xymatrix{U_{i} \ar[d] \ar[r]^{f_{i}} & Spec O_{K}[T_{0}, …, T_{n}]/(T_{1} ⋯ T_{r_{i}} − π) \ar@{-->}[dl] \\ Spec O_{K}}

where f_{i} is an étale morphism. Moreover, if every irreducible component of the special fiber of 𝔛 is smooth over F, O_{K} is called a proper strictly semistable model of X over O_{K}.

We recall the weight spectral sequence of Rapoport-Zink ([RZ1]). Assume that X has a proper strictly semistable model 𝔛 over O_{K}. Let X_{1}, …, X_{m} be the irreducible components of the special fiber of 𝔛, and

X^{(k)} := ∐_{1≤i_{1}<···<i_{k}≤m} X_{i_{1}} ∩ ⋯ ∩ X_{i_{k}}.

Then X^{(k)} is a proper smooth variety of dimension n − k + 1 over F. The weight spectral sequence of Rapoport-Zink is as follows :

\begin{matrix}E_{1}^{−r, w+r} = ⨁_{k≥max\{0,−r\}} H_{ét}^{w−r−2k}(X_{\overline{𝔽}_{q}}^{(2k+r+1)}, ℚ_{l}(−r − k)) \\ ⟹ H_{ét}^{w}(X\overline{K}, ℚ_{l}). \end{matrix}

This spectral sequence is Gal(\overline{K}/K)-equivariant. The map d_{1}^{i, j}: E_{1}^{i, j} → E_{1}^{i+1, j} can be described by restriction morphisms and Gysin morphisms explicitly (see [RZ1], 2.10 for details).

The action of the monodromy operator N on H_{ét}^{w}(X\overline{K}, ℚ_{l}) in §5.1 is induced from a natural map N: E_{1}^{i, j}(1) → E_{1}^{i+2, j−2} satisfying

N^{r}: E_{1}^{−r, w+r}(r) ⟶^{≅} E_{1}^{r, w−r}

for all r, w. We can describe N: E_{1}^{i, j}(1) → E_{1}^{i+2, j−2} explicitly ([RZ1], 2.10). Hence we can define N without replacing K by its finite extension (see Remark 5.2).

I_{K} acts on each E_{1}^{i, j} trivially and Gal(\overline{𝔽}_{q}/𝔽_{q}) acts on them. By the Weil conjecture ([De2], [De3]), H_{ét}^{w−r−2k}(X_{\overline{𝔽}_{q}}^{(2k+r+1)}, ℚ_{l}(−r −k)) has weight (w −r −2k)−

2(−r−k) = w+r (see §5.2). Hence E_{1}^{i, j} has weight j. Therefore, the filtration on H_{ét}^{w}(X\overline{K}, ℚ_{l}) induced by the weight spectral sequence is the weight filtration W_{•} in Definition 5.3. This proves the existence of W_{•} in Definition 5.3 (for general X, we may use de Jong’s alteration [dJ]). Moreover, since d_{r}^{i, j}: E_{r}^{i, j} → E_{r}^{i+r, j−r+1} is a map between Gal(\overline{𝔽}_{q}/𝔽_{q})-representations with different weights for r ≥ 2, d_{r}^{i, j} = 0 for r ≥ 2. Therefore, the weight spectral sequence degenerates at E_{2}.

By combining above facts, we see that the weight-monodromy conjecture (Conjecture 1.1) is equivalent to the following conjecture on the weight spectral sequence of Rapoport-Zink. Conjecture 5.6 ([RZ1], [Il1], [Il2]). Let X be a proper smooth variety over K which has a proper strictly semistable model 𝔛 over O_{K}. Let E_{1}^{−r, w+r} ⇒ H_{ét}^{w}(X\overline{K}, ℚ_{l}) be the weight spectral sequence of Rapoport-Zink. Then N^{r} induces an isomorphism

N^{r}: E_{2}^{−r, w+r}(r) ⟶^{≅} E_{2}^{r, w−r}

on E_{2}-terms for all r, w.

5.4. An argument of Steenbrink, M. Saito. Proposition 5.7. Let X be a proper smooth variety over K which has a proper strictly semistable model 𝔛 over O_{K}. Let X_{1}, …, X_{m} be the irreducible components of the special fiber of 𝔛. Assume that 𝔛 is projective over O_{K} with an ample line bundle L. Assume further that, for 1 ≤ i_{1} < ⋯ < i_{k} ≤ m, each irreducible component Y of X_{i_{1}} ∩ ⋯ ∩ X_{i_{k}} satisfies Assumption 2.1 in §2.1, and the Hodge standard conjecture (Conjecture 2.6) holds for (Y, L|_{Y}). Then, Conjecture 5.6 holds for 𝔛, and Conjecture 1.1 holds for X.

The idea of the proof here is essentially the same as in [SaM], 4.2.5, where M. Saito used polarized Hodge structures to prove a Hodge analogue of Conjecture 5.6. The only difference is that we use ℚ-structures on l-adic cohomology and the Hodge standard conjecture instead of the polarized Hodge structures (A similar argument can be found in [It2]).

Since the presentation of M. Saito in [SaM] is too sophisticated, it isn’t very clear to non-specialists that their argument can be applied to the situation in Proposition 5.7 even in positive or mixed characteristic. So we reproduce their argument here with slight modification suitable for Proposition 5.7 for reader’s convenience. Therefore, those who are familiar with their argument may skip to the next section. In the followings, the proof of Claim 5.13 is the only part where we crucially use the assumptions of Proposition 5.7. In [SaM], M. Saito proved a corresponding statement by using polarized Hodge structures.

First of all, we recall the structure of the E_{1}-terms of the weight spectral sequence of Rapoport-Zink. Let

d_{1}^{−r, w+r} = ∑_{k≥max{0,−r}}(ρ + τ): E_{1}^{−r, w+r} ⟶ E_{1}^{−r+1, w+r}

denote the differential on E_{1}-terms, where

ρ: H_{ét}^{s}(X_{\overline{𝔽}_{q}}^{(t)}, ℚ_{l}) ⟶ H_{ét}^{s}(X_{\overline{𝔽}_{q}}^{(t+1)}, ℚ_{l})

is a linear combination of restriction morphisms for some s, t (ρ is (−1)^{r+k}θ in [RZ1], 2.10), and

τ: H_{ét}^{s}(X_{\overline{𝔽}_{q}}^{(t)}, ℚ_{l}) ⟶ H_{ét}^{s+2}(X_{\overline{𝔽}_{q}}^{(t−1)}, ℚ_{l}(1))

is a linear combination of Gysin morphisms for some s, t (τ is (−1)^{k}d′_{1} in [RZ1], 2.10). From the construction, ρ increases the index k by 1, and τ preserves the index k. Moreover, ρ, τ satisfy ρ ∘ ρ = 0, τ ∘ τ = 0, τ ∘ ρ + ρ ∘ τ = 0. Hence ρ, τ also satisfy

ρ ∘ τ ∘ ρ = −ρ ∘ ρ ∘ τ = 0, τ ∘ ρ ∘ τ = −τ ∘ τ ∘ ρ = 0.

Let k be an integer with 1 ≤ k ≤ n. To distinguish ρ, τ for different degree, we use the following notation.

ρ_{i}^{(k)} : H_{ét}^{i}(X_{\overline{𝔽}_{q}}^{(k)}, ℚ_{l}) ⟶ H_{ét}^{i}(X_{\overline{𝔽}_{q}}^{(k+1)}, ℚ_{l}),
τ_{i}^{(k+1)} : H_{ét}^{i}(X_{\overline{𝔽}_{q}}^{(k+1)}, ℚ_{l}) ⟶ H_{ét}^{i+2}(X_{\overline{𝔽}_{q}}^{(k)}, ℚ_{l}(1)).

In the followings, to simply the notation, we fix an isomorphism ℚ_{l} ≅ ℚ_{l}(1) to ignore the Tate twists, and simply write H_{ét}^{i}(X^{(k)}) := H_{ét}^{i}(X_{\overline{𝔽}_{q}}^{(k)}, ℚ_{l}), where we add a subscript “ét” not to confuse with the notation H^{i}(X) in §2.1.

Let L: H_{ét}^{i}(X^{(k)}) → H_{ét}^{i+2}(X^{(k)}) be the Lefschetz operator defined by L. Since L commutes with ρ, τ, we use the same letter L for different i, k, which doesn’t cause any confusion. Let

H_{ét}^{i}(X^{(k)}) × H_{ét}^{2(n−k+1)−i}(X^{(k)}) → ℚ_{l}, H_{ét}^{i}(X^{(k+1)}) × H_{ét}^{2(n−k)−i}(X^{(k+1)}) → ℚ_{l}

be the sum of the cup product pairings. For simplicity, we denote them by ∪. Then, for a ∈ H_{ét}^{i}(X^{(k)}), b ∈ H_{ét}^{2(n−k)−i}(X^{(k+1)}), we have

(5.1)  a ∪ τ_{2(n−k)−i}^{(k+1)}(b) = ρ_{i}^{(k)}(a) ∪ b.

Namely, ρ_{i}^{(k)} and τ_{2(n−k)−i}^{(k+1)} are dual to each other with respect to ∪.

Claim 5.8. In the following sequence of ρ, τ

H_{ét}^{i−2}(X^{(k+1)}) ⟶^{τ_{i−2}^{(k+1)}} H_{ét}^{i}(X^{(k)}) ⟶^{ρ_{i}^{(k)}} H_{ét}^{i}(X^{(k+1)}) ⟶^{τ_{i}^{(k+1)}} H_{ét}^{i+2}(X^{(k)}),

we have (Ker τ_{i}^{(k+1)}) ∩ (Im ρ_{i}^{(k)}) = Im (ρ_{i}^{(k)} ∘ τ_{i−2}^{(k+1)}). Moreover, in the following sequence of ρ, τ

H_{ét}^{i}(X^{(k)}) ⟶^{ρ_{i}^{(k)}} H_{ét}^{i}(X^{(k+1)}) ⟶^{τ_{i}^{(k+1)}} H_{ét}^{i+2}(X^{(k)}) ⟶^{ρ_{i+2}^{(k)}} H_{ét}^{i+2}(X^{(k+1)}),

we have (Ker ρ_{i+2}^{(k)}) ∩ (Im τ_{i}^{(k+1)}) = Im (τ_{i}^{(k+1)} ∘ ρ_{i}^{(k)}).

By assuming the above claim, it is easy to prove Conjecture 5.6.

Proof of Proposition 5.7 assuming Claim 5.8. For r ≥ 1, we write E_{1}^{−r, w+r}, E_{1}^{r, w−r} explicitly as follows (see §5.3)

E_{1}^{−r, w+r} = ⨁_{k≥0} H_{ét}^{w−r−2k}(X^{(2k+r+1)}), E_{1}^{r, w−r} = ⨁_{k≥r} H_{ét}^{w+r−2k}(X^{(2k−r+1)}).

Put k′ = k − r and rewrite E_{1}^{r, w−r} as

E_{1}^{r, w−r} = ⨁_{k′≥0} H_{ét}^{w−r−2k′}(X^{(2k′+r+1)}).

From this expression, there is a natural isomorphism between E_{1}^{−r, w+r} and E_{1}^{r, w−r}, which is nothing but an isomorphism induced by N^{r} in §5.3. Now, we look at the following part of the weight spectral sequence of Rapoport-Zink :

(5.2) \xymatrix{E_{1}^{−r−1, w+r} \ar[r] & E_{1}^{−r, w+r} \ar[r] \ar[drr]^{N^{r}} & E_{1}^{−r+1, w+r} \\  &  & E_{1}^{r−1, w−r} \ar[r] & E_{1}^{r, w−r} \ar[r] & E_{1}^{r+1, w−r}.}

Since we have

E_{1}^{−r−1, w+r} = ⨁_{k≥0} H_{ét}^{w−r−2k−2}(X^{(2k+r+2)}),
E_{1}^{−r+1, w+r} = ⨁_{k≥0} H_{ét}^{w−r−2k+2}(X^{(2k+r)}),
E_{1}^{r−1, w−r} = ⨁_{k≥r−1} H_{ét}^{w+r−2k−2}(X^{(2k−r+2)}) = ⨁_{k′≥−1} H_{ét}^{w−r−2k′−2}(X^{(2k′+r+2)}),
E_{1}^{r+1, w−r} = ⨁_{k≥r+1} H_{ét}^{w+r−2k+2}(X^{(2k−r)}) = ⨁_{k′≥1} H_{ét}^{w−r−2k′+2}(X^{(2k′+r)}),

we put

A = ⨁_{k≥0} H_{ét}^{w−r−2k−2}(X^{(2k+r+2)}),
B = ⨁_{k≥1} H_{ét}^{w−r−2k}(X^{(2k+r+1)}), B′ = H_{ét}^{w−r}(X^{(r+1)}),
C = ⨁_{k≥1} H_{ét}^{w−r−2k+2}(X^{(2k+r)}), D = H_{ét}^{w−r+2}(X^{(r)}), E = H_{ét}^{w−r}(X^{(r)}).

Then, (5.2) becomes

(5.3)  \xymatrix{A \ar[r]^{d+d′} & B ⊕ B′ \ar[r]^{(e+0, f+f′)} \ar[dr] & C ⊕ D \\  & A ⊕ E \ar[r]_{(d+d′, 0+g)} & B ⊕ B′ \ar[r]_{(e,f)} & C}

for some d: A → B, d′: A → B′, e: B → C, f: B′ → C, f′: B′ → D, g: E → B′, where B ⊕B′ → B ⊕B′ is the identity map. Recall that ρ increases the index k by 1, and τ preserves the index k.

It is enough to show that the identity map B ⊕ B′ → B ⊕ B′ induces an isomorphism between the cohomologies the first and second rows of (5.3).

We shall prove the injectivity. Take an element (b, b′) ∈ B ⊕ B′ such that e(b) + f(b′) = 0, f′(b′) = 0. Assume that (b + b′) ∈ Im (d + d′, 0 + g). Namely, there exists (x, y) ∈ A ⊕ E such that d(x) = b, d′(x) + g(y) = b′. Then we have

e(b) + f(b′) = e ∘ d(x) + f ∘ d′(x) + f ∘ g(y) = 0
f′(b′) = f′ ∘ d′(x) + f′ ∘ g(y) = 0.

From the definition, we see that d′ = τ_{w−r−2}^{(r+2)}, f′ = τ_{w−r}^{(r+1)}, g = ρ_{w−r}^{(r)}. Hence f′ ∘ d′ = τ_{w−r}^{(r+1)} ∘ τ_{w−r−2}^{(r+2)} = 0. Therefore, we have

f′ ∘ g(y) = τ_{w−r}^{(r+1)} ∘ ρ_{w−r}^{(r)}(y) = 0.

By the first assertion of Claim 5.8, there exists z ∈ H_{ét}^{w−r−2}(X^{(r+1)}) such that

g(y) = ρ_{w−r}^{(r)}(y) = ρ_{w−r}^{(r)} ∘ τ_{w−r−2}^{(r+1)}(z).

Since ρ_{w−r}^{(r)} ∘ τ_{w−r−2}^{(r+1)}(z) = −τ_{w−r−2}^{(r+2)} ∘ ρ_{w−r−2}^{(r+1)}(z), we have

b′ = d′(x) + g(y) = τ_{w−r−2}^{(r+2)}(x) − τ_{w−r−2}^{(r+2)} ∘ ρ_{w−r−2}^{(r+1)}(z) = τ_{w−r−2}^{(r+2)}(x − ρ_{w−r−2}^{(r+1)}(z)).

On the other hand, the restriction of d: A → B to the k = 0 part H_{ét}^{w−r−2}(X^{(r+2)}) is ρ_{w−r−2}^{(r+2)}. Then, we have

d (x − ρ_{w−r−2}^{(r+1)}(z)) = d(x) − ρ_{w−r−2}^{(r+2)} ∘ ρ_{w−r−2}^{(r+1)}(z) = d(x) = b,

since ρ_{w−r−2}^{(r+2)} ∘ ρ_{w−r−2}^{(r+1)} = 0. Hence (b, b′) is the image of x − ρ_{w−r−2}^{(r+1)}(z) by d + d′, i.e. (b, b′) ∈ Im (d + d′). This proves the injectivity.

Finally, we shall prove the surjectivity. Take an element (b, b′) ∈ B ⊕ B′ with e(b)+f(b′) = 0. From the construction, we see that f = ρ_{w−r}^{(r+1)} and the restriction of e: B → C to the k = 1 part H_{ét}^{w−r−2}(X^{(r+3)}) is τ_{w−r−2}^{(r+3)}. Therefore, the equality e(b) + f(b′) = 0 implies τ_{w−r−2}^{(r+3)}(b̃) + ρ_{w−r}^{(r+1)}(b′) = 0 in the k = 1 part H_{ét}^{w−r}(X^{(r+2)}) of C, where b̃ is the k = 1 part of b. Hence we have

ρ_{w−r+2}^{(r)} ∘ τ_{w−r}^{(r+1)}(b′) = −τ_{w−r}^{(r+2)} ∘ ρ_{w−r}^{(r+1)}(b′) = τ_{w−r}^{(r+2)} ∘ τ_{w−r−2}^{(r+3)}(b̃) = 0,

since ρ_{w−r+2}^{(r)} ∘ τ_{w−r}^{(r+1)} = −τ_{w−r}^{(r+2)} ∘ ρ_{w−r}^{(r+1)} and τ_{w−r}^{(r+2)} ∘ τ_{w−r−2}^{(r+3)} = 0. By the second assertion of Claim 5.8, there exists a ∈ H_{ét}^{w−r}(X^{(r)}) = E such that

τ_{w−r}^{(r+1)}(b′) = τ_{w−r}^{(r+1)} ∘ ρ_{w−r}^{(r)}(a).

Consider the element (b, b′ − g(a)) ∈ B ⊕ B′ which is in the same class as (b, b′) in the cohomology of the second row. Then, we have

f′(b′ − g(a)) = τ_{w−r}^{(r+1)}(b′) − τ_{w−r}^{(r+1)} ∘ ρ_{w−r}^{(r)}(a) = 0,

hence (b, b′ − g(a)) ∈ Ker (e + 0, f + f). This proves the surjectivity. □

Therefore, it is enough to prove Claim 5.8. First, recall the primitive decomposition of H_{ét}^{i}(X^{(k)}) :

H_{ét}^{i}(X^{(k)}) = ⨁_{j≥0} L^{j}P_{ét}^{i−2j}(X^{(k)}),

where P_{ét}^{i−2j}(X^{(k)}) is the primitive part of H_{ét}^{i−2j}(X^{(k)}) as in §2.2. Recall that dim X^{(k)} = n−k+1. The idea of our proof of Claim 5.8 is to analyze the primitive decomposition carefully. Since ρ, τ don’t preserve the primitive decomposition, for 0 ≤ i ≤ dim X^{(k+1)} = n − k, we put

P^{i} Im^{0}ρ_{i}^{(k)} := (Im ρ_{i}^{(k)}) ∩ P_{ét}^{i}(X^{(k+1)}),

and P^{i} Im^{0}ρ_{i}^{(k)} = 0 otherwise. Then, we define

Im^{0}ρ_{i}^{(k)} := ⨁_{j≥0} L^{j}P^{i−2j} Im^{0}ρ_{i−2j}, Im^{1}ρ_{i}^{(k)} := Im ρ_{i}^{(k)}/Im^{0}ρ_{i}^{(k)}.

Roughly speaking, Im^{0}ρ_{i}^{(k)} is a part which has a decomposition according to the primitive decomposition of H_{ét}^{i}(X^{(k+1)}), and Im^{1}ρ_{i}^{(k)} is a part which doesn’t have any decomposition compatible with the primitive decomposition of H_{ét}^{i}(X^{(k+1)}). Similarly for Im τ_{i}^{(k+1)}, for 0 ≤ i ≤ dim X^{(k)} − 2 = n − k − 1, we put

P^{i+2} Im^{0}τ_{i}^{(k+1)} := Im τ_{i}^{(k+1)} ∩ P_{ét}^{i+2}(X^{(k)})

and P^{i+2} Im^{0}τ_{i}^{(k+1)} = 0 otherwise. Then, we define

Im^{0}τ_{i}^{(k+1)} := ⨁_{j≥0} L^{j}P^{i+2−2j} Im^{0}τ_{i−2j}^{(k+1)}, Im^{1}τ_{i}^{(k+1)} := Im τ_{i}^{(k+1)}/Im^{0}τ_{i}^{(k+1)}.

Claim 5.9 (Hard Lefschetz theorem for Im^{0}). We have the primitive decomposition for Im^{0} in the following sense :

Im^{0}ρ_{i}^{(k)} = ⨁_{j≥0} (L^{j}P_{ét}^{i−2j}(X^{(k+1)}) ∩ Im^{0}ρ_{i}^{(k)}),
Im^{0}τ_{i}^{(k+1)} = ⨁_{j≥0} (L^{j}P_{ét}^{i−2j}(X^{(k)}) ∩ Im^{0}τ_{i}^{(k+1)}).

Hence we have an analogue of the hard Lefschetz theorem for Im^{0} in the following form :

L^{dim X^{(k+1)}−i} = L^{(n−k)−i}: Im^{0}ρ_{i}^{(k)}  ⟶^{≅} Im^{0}ρ_{2(n−k)−i}^{(k)},
L^{dim X^{(k)}−(i+2)} = L^{(n−k+1)−(i+2)} : Im^{0}τ_{i}^{(k+1)} ⟶^{≅} Im^{0}τ_{2(n−k+1)−i−4}^{(k+1)}.

Proof of Claim 5.9. These immediately follows from the definition of Im^{0}ρ_{i}^{(k)}, Im^{0}τ_{i}^{(k+1)}. □

Claim 5.10 (Hard Lefschetz theorem for Im^{1}). We also have an analogue of the hard Lefschetz theorem for Im^{1} in the following sense :

L^{dim X^{(k+1)}+1−i} = L^{(n−k)+1−i}: Im^{1}ρ_{i}^{(k)}  ⟶^{≅} Im^{1}ρ_{2((n−k)+1)−i}^{(k)},
L^{dim X^{(k)}+1−(i+2)} = L^{(n−k+1)+1−(i+2)} : Im^{1}τ_{i}^{(k+1)} ⟶^{≅} Im^{1}τ_{2((n−k+1)+1)−(i+4)}^{(k+1)}.

Proof of Claim 5.10. The surjectivity follows from the hard Lefschetz theorem. The injectivity follows from the surjectivity of the maps

L^{(n−k)+1−i}: Im^{0}ρ_{i}^{(k)}  ⟶ Im^{0}ρ_{2((n−k)+1)−i}^{(k)}
L^{(n−k+1)+1−(i+2)} : Im^{0}τ_{i}^{(k+1)} ⟶ Im^{0}τ_{2((n−k+1)+1)−(i+4)}^{(k+1)},

which follows from Claim 5.9. □

Remark 5.11. The primitive decomposition of Im^{0}ρ_{i}^{(k)} (resp. Im^{0}τ_{i}^{(k+1)}) is the same as that of H_{ét}^{i}(X^{(k+1)}) (resp. H_{ét}^{i+2}(X^{(k)})). On the other hand, the primitive decomposition of Im^{1}ρ_{i}^{(k)} (resp. Im^{1}τ_{i}^{(k+1)}) is the same as that of H_{ét}^{i}(X^{(k)}) (resp. H_{ét}^{i}(X^{(k+1)})). Hence the center of symmetry of Im^{1} differs by 1 from that of Im^{0}. Claim 5.12. For all i, we have the following equality of dimensions :

dim_{ℚ_{l}} Im^{0}ρ_{i}^{(k)} = dim_{ℚ_{l}} Im^{1}τ_{i}^{(k+1)}, dim_{ℚ_{l}} Im^{1}ρ_{i+2}^{(k)} = dim_{ℚ_{l}} Im^{0}τ_{i}^{(k+1)}.

Proof of Claim 5.12. Since ρ_{i}^{(k)} and τ_{2(n−k)−i}^{(k+1)} are dual to each other (see (5.1)),

we have dim_{ℚ_{l}} Im ρ_{i}^{(k)} = dim_{ℚ_{l}} Im τ_{2(n−k)−i}^{(k+1)}. Hence we have

(5.4) dim_{ℚ_{l}} Im^{0}ρ_{i}^{(k)} + dim_{ℚ_{l}} Im^{1}ρ_{i}^{(k)} = dim_{ℚ_{l}} Im^{0}τ_{2(n−k)−i}^{(k+1)} + dim_{ℚ_{l}} Im^{1}τ_{2(n−k)−i}^{(k+1)}.

Since dim_{ℚ_{l}} Im^{0}τ_{2(n−k)−i}^{(k+1)} = dim_{ℚ_{l}} Im^{0}τ_{i−2}^{(k+1)} by Claim 5.9, and dim_{ℚ_{l}} Im^{1}τ_{2(n−k)−i}^{(k+1)} =

dim_{ℚ_{l}} Im^{1}τ_{i}^{(k+1)} by Claim 5.10, we have

dim_{ℚ_{l}} Im^{0}ρ_{i}^{(k)} + dim_{ℚ_{l}} Im^{1}ρ_{i}^{(k)} = dim_{ℚ_{l}} Im^{0}τ_{i−2}^{(k+1)} + dim_{ℚ_{l}} Im^{1}τ_{i}^{(k+1)}.

Equivalently, we have

(5.5) dim_{ℚ_{l}} Im^{0}ρ_{i}^{(k)} − dim_{ℚ_{l}} Im^{1}τ_{i}^{(k+1)} = −( dim_{ℚ_{l}} Im^{1}ρ_{i}^{(k)} − dim_{ℚ_{l}} Im^{0}τ_{i−2}^{(k+1)}).

On the other hand, we have dim_{ℚ_{l}} Im^{0}ρ_{i}^{(k)} = dim_{ℚ_{l}} Im^{0}ρ_{2(n−k)−i}^{(k)} by Claim 5.9,

and dim_{ℚ_{l}} Im^{1}ρ_{i}^{(k)} = dim_{ℚ_{l}} Im^{1}ρ_{2(n−k)−i+2}^{(k)} by Claim 5.10. Hence, by (5.4), we have

dim_{ℚ_{l}} Im^{0}ρ_{2(n−k)−i}^{(k)} + dim_{ℚ_{l}} Im^{1}ρ_{2(n−k)−i+2}^{(k)}
= dim_{ℚ_{l}} Im^{0}τ_{2(n−k)−i}^{(k+1)} + dim_{ℚ_{l}} Im^{1}τ_{2(n−k)−i}^{(k+1)}.

From this, by replacing 2(n − k) − i by i, we have

(5.6) dim_{ℚ_{l}} Im^{0}ρ_{i}^{(k)} − dim_{ℚ_{l}} Im^{1}τ_{i}^{(k+1)} = −( dim_{ℚ_{l}} Im^{1}ρ_{i+2}^{(k)} − dim_{ℚ_{l}} Im^{0}τ_{i}^{(k+1)}).

By combining (5.5) and (5.6), we have

dim_{ℚ_{l}} Im^{1}ρ_{i}^{(k)} − dim_{ℚ_{l}} Im^{0}τ_{i−2}^{(k+1)} = dim_{ℚ_{l}} Im^{1}ρ_{i+2}^{(k)} − dim_{ℚ_{l}} Im^{0}τ_{i}^{(k+1)}.

Hence the difference dim_{ℚ_{l}} Im^{1}ρ_{i+2}^{(k)} − dim_{ℚ_{l}} Im^{0}τ_{i}^{(k+1)} is independent of i. Since this is zero for sufficiently large (or small) i, we have

dim_{ℚ_{l}} Im^{1}ρ_{i+2}^{(k)} − dim_{ℚ_{l}} Im^{0}τ_{i}^{(k+1)} = 0 for all i,

which proves the second assertion. Hence we also have the first assertion by (5.5) or (5.6). □

We call the pairing on H_{ét}^{i}(X^{(k)}) defined by

H_{ét}^{i}(X^{(k)}) × H_{ét}^{i}(X^{(k)}) → ℚ_{l}, (a, b) ↦ L^{n−k+1−i}a ∪ b

the sum of the Lefschetz pairings. We denote it simply by ⟨, ⟩. The pairing ⟨, ⟩ is nondegenerate by the hard Lefschetz theorem (§2.2). However, the following claim is far from trivial, and we use the assumptions of Proposition 5.7 to prove it. Claim 5.13. The restriction of ⟨, ⟩ on H_{ét}^{i}(X^{(k+1)}) to Im^{0}ρ_{i}^{(k)} is nondegenerate. Similarly, The restriction of ⟨, ⟩ on H_{ét}^{i+2}(X^{(k)}) to Im^{0}τ_{i}^{(k+1)} is also nondegenerate.

Proof of Claim 5.13. Since the decomposition of Im^{0}ρ_{i}^{(k)} in Claim 5.9 is orthogonal with respect to ⟨, ⟩, it is enough to show that the restriction of ⟨, ⟩ to L^{j}P^{i−2j} Im^{0}ρ_{i}^{(k)} is nondegenerate for each j. By the assumptions of Proposition 5.7, we observe that all cohomology groups of X^{(k)}, X^{(k+1)}, maps ρ, τ, L, and pairings ⟨, ⟩ have natural ℚ-structures (see §2.1). In particular, the inclusion

L^{j}P^{i−2j} Im^{0}ρ_{i}^{(k)} ⊂ L^{j}P_{ét}^{i−2j}(X^{(k+1)})

has a ℚ-structure. Namely, let V ⊂ L^{j}P^{i−2j} Im^{0}ρ_{i}^{(k)} and W ⊂ L^{j}P_{ét}^{i−2j}(X^{(k+1)}) be the ℚ-vector subspaces generated by algebraic cycles, then we have

V ⊗_{ℚ} ℚ_{l} = L^{j}P^{i−2j} Im^{0}ρ_{i}^{(k)}, W ⊗_{ℚ} ℚ_{l} = L^{j}P_{ét}^{i−2j}(X^{(k+1)}).

It is enough to show that the restriction of ⟨, ⟩ to V is nondegenerate. We may assume i is even because there is no odd degree cohomology. By assumption, the Hodge standard conjecture holds for all irreducible components of X^{(k+1)}. Hence the restriction of ⟨, ⟩ to W is positive (resp. negative) definite if (i−2j)/2 is even (resp. odd). Therefore, the restriction of ⟨, ⟩ to V is nondegenerate because it is definite. □

Claim 5.14. The composition of the following maps is an isomorphism :

Im^{0}ρ_{i}^{(k)} ⟶^{τ_{i}^{(k+1)}} Im τ_{i}^{(k+1)} ⟶ Im^{1}τ_{i}^{(k+1)} = Im τ_{i}^{(k+1)}/Im^{0}τ_{i}^{(k+1)}.

Similarly, the composition of the following maps is also an isomorphism :

Im^{0}τ_{i}^{(k+1)} ⟶^{ρ_{i+2}^{(k)}} Im ρ_{i+2}^{(k)} ⟶ Im^{1}ρ_{i+2}^{(k)} = Im ρ_{i+2}^{(k)}/Im^{0}ρ_{i+2}^{(k)}.

Proof of Claim 5.14. By Claim 5.12, we have only to show that the compositions are injective. Assume that the composition of the first row is not injective and take a nonzero x ∈ Im^{0}ρ_{i}^{(k)} such that τ_{i}^{(k+1)}(x) ∈ Im^{0}τ_{i}^{(k+1)}. If τ_{i}^{(k+1)}(x) ≠ 0, by Claim 5.13, there exists τ_{i}^{(k+1)}(y) ∈ Im^{0}τ_{i}^{(k+1)} such that

L^{2(n−k+1)−(i+2)} τ_{i}^{(k+1)}(x) ∪ τ_{i}^{(k+1)}(y) ≠ 0.

However, since x = ρ_{i}^{(k)}(x′) for some x′ ∈ H_{ét}^{i}(X^{(k)}), by (5.1), we have

0 ≠ L^{(n−k+1)−(i+2)} τ_{i}^{(k+1)}(x) ∪ τ_{i}^{(k+1)}(y)
= L^{(n−k+1)−(i+2)}ρ_{i+2}^{(k)} ∘ τ_{i}^{(k+1)} ∘ ρ_{i}^{(k)}(x′) ∪ y
= 0.

Recall that ρ_{i+2}^{(k)} ∘ τ_{i}^{(k+1)} ∘ ρ_{i}^{(k)} = 0. This is a contradiction. Hence we have τ_{i}^{(k+1)}(x) = 0. By Claim 5.13 again, there exists ρ_{i}^{(k)}(z) ∈ Im^{0}ρ_{i}^{(k)} such that

L^{n−k−i} x ∪ ρ_{i}^{(k)}(z) ≠ 0.

However, by (5.1), we have

0 ≠ L^{n−k−i} x ∪ ρ_{i}^{(k)}(z) = L^{n−k−i} τ_{i}^{(k+1)}(x) ∪ z = 0

since τ_{i}^{(k+1)}(x) = 0. This is a contradiction. Hence the composition of the first row is injective. Similarly, we see that the composition of the second row is also injective. □

Claim 5.15. By Claim 5.14, the surjections Im τ_{i}^{(k+1)} → Im^{1}τ_{i}^{(k+1)} and Im ρ_{i+2}^{(k)} → Im^{1}ρ_{i+2}^{(k)} have canonical splittings. Therefore, we have the following decompositions of Im τ_{i}^{(k+1)} and Im ρ_{i+2}^{(k)} :

(5.7)  Im τ_{i}^{(k+1)} = Im^{0}τ_{i}^{(k+1)} ⊕ Im^{1}τ_{i}^{(k+1)}, Im ρ_{i+2}^{(k)} = Im^{0}ρ_{i+2}^{(k)} ⊕ Im^{1}ρ_{i+2}^{(k)}.

These are orthogonal decompositions with respect to ⟨, ⟩.

Proof of Claim 5.15. The proof is easy and immediate. Take (a, b) ∈ Im^{0}τ_{i}^{(k+1)} ⊕ Im^{1}τ_{i}^{(k+1)}. Then, by the first assertion of Claim 5.14, we can write a = τ_{i}^{(k+1)}(a′) and b = τ_{i}^{(k+1)} ∘ ρ_{i}^{(k)}(b′) for some a′, b′. Then, by (5.1), we have

L^{n−k+1−(i+2)} a ∪ b = L^{n−k+1−(i+2)} τ_{i}^{(k+1)}(a′) ∪ (τ_{i}^{(k+1)} ∘ ρ_{i}^{(k)}(b′))
= L^{n−k+1−(i+2)} a′ ∪ (ρ_{i+2}^{(k+1)} ∘ τ_{i}^{(k+1)} ∘ ρ_{i}^{(k)}(b′))
= 0,

since ρ_{i+2}^{(k+1)} ∘ τ_{i}^{(k+1)} ∘ ρ_{i}^{(k)} = 0. Similarly, we see that the decomposition Im ρ_{i+2}^{(k)} = Im^{0}ρ_{i+2}^{(k)} ⊕ Im^{1}ρ_{i+2}^{(k)} is also orthogonal with respect to ⟨, ⟩. □

Finally, by combining above results, we shall prove Claim 5.8.

Proof of Claim 5.8. First, we shall prove the first assertion. Since τ_{i}^{(k+1)} ∘ ρ_{i}^{(k)} ∘ τ_{i−2}^{(k+1)} = 0, the inclusion

(Ker τ_{i}^{(k+1)}) ∩ (Im ρ_{i}^{(k)}) ⊃ Im (ρ_{i}^{(k)} ∘ τ_{i−2}^{(k+1)})

is obvious. To prove the opposite inclusion, we take x ∈ (Ker τ_{i}^{(k+1)}) ∩ (Im ρ_{i}^{(k)}). Then, for all y ∈ H_{ét}^{i}(X^{(k)}), by (5.1), we have

L^{n−k−i} x ∪ ρ_{i}^{(k)}(y) = L^{n−k−i} τ_{i}^{(k+1)}(x) ∪ y = 0

since τ_{i}^{(k+1)}(x) = 0. By Claim 5.15, we write x = x_{0} + x_{1} for x_{0} ∈ Im^{0}ρ_{i}^{(k)} and x_{1} ∈ Im^{1}ρ_{i}^{(k)}. If x_{0} ≠ 0, by Claim 5.13 and Claim 5.15, there exists y′ ∈ Im^{0}ρ_{i}^{(k)} such that

L^{n−k−i} x ∪ y′ = L^{n−k−i} x_{0} ∪ y′ + L^{n−k−i} x_{1} ∪ y′
= L^{n−k−i} x_{0} ∪ y′
≠ 0,

which is absurd. Hence x_{0} = 0. Then, we have

x = x_{1} ∈ Im^{1}ρ_{i}^{(k)} = ρ_{i}^{k}(Im τ_{i−2}^{(k+1)}) ⊂ Im (ρ_{i}^{(k)} ∘ τ_{i−2}^{(k+1)}).

This proves the first assertion. Similarly, we can prove the second assertion by using Claim 5.13 and Claim 5.15. Therefore, the proof of Claim 5.8 and hence Proposition 5.7 is completed. □

6. Application to p-adically uniformized varieties

In this section, we give a proof of the main theorem (Theorem 1.2) of this paper. We also give some immediate applications.

6.1. Drinfeld upper half spaces and p-adic uniformization. Let K be a complete discrete valuation field with finite residue field 𝔽_{q} of characteristic p > 0. Let Ω̂_{K}^{d} be the Drinfeld upper half space of dimension d over K, which is a rigid analytic space obtained by removing all K-rational hyperplanes from ℙ_{K}^{d}. We have a natural action of PGL_{d+1}(K) on Ω̂_{K}^{d}.

As a formal scheme, Ω̂_{K}^{d} can be constructed as follows. Take the projective space ℙ_{O_{K}}^{d} over O_{K}. Then, take successive blowing-ups of ℙ_{O_{K}}^{d} along all linear subvarieties in the special fiber ℙ_{𝔽_{q}}^{d} as in §4.1. By continuing this process for all exceptional divisors appearing in the blowing-ups, we obtain a formal scheme Ω̂_{O_{K}}^{d} locally of finite type over Spf O_{K}. By construction, the rigid analytic space associated with Ω̂_{O_{K}}^{d} is isomorphic to Ω̂_{K}^{d}. In other words, Ω̂_{K}^{d} is the “generic fiber” in the sense of Raynaud ([Ray]) of the formal scheme Ω̂_{O_{K}}^{d} over Spf O_{K}.

Let Γ be a cocompact torsion free discrete subgroup of PGL_{d+1}(K). We have a natural action of Γ on Ω̂_{O_{K}}^{d}, and this action is discontinuous with respect to Zariski topology. Then, we take a quotient 𝔛̂_{Γ} := Γ\Ω̂_{O_{K}}^{d} as a formal scheme. Mustafin, Kurihara showed that the relative dualizing sheaf ω_{𝔛̂_{Γ}/O_{K}} is invertible and ample. Therefore, 𝔛̂_{Γ} can be algebraized to a projective scheme 𝔛_{Γ} over O_{K}. The generic fiber X_{Γ} := 𝔛_{Γ} ⊗_{O_{K}} K is a projective smooth variety over K whose associated rigid analytic space is the rigid analytic quotient Γ\Ω̂_{K}^{d}. By construction, 𝔛_{Γ} is a proper semistable model of X_{Γ} over O_{K} whose special fiber is described by the cell complex Γ\𝔗, where 𝔗 denotes the Bruhat-Tits building of PGL_{d+1}(K) (for details, see [Mus], Theorem 4.1, [Ku], Theorem 2.2.5).

6.2. Proof of the main theorem (Theorem 1.2). As in §6.1, let K be a complete discrete valuation field with finite residue field 𝔽_{q} of characteristic p > 0, Γ ⊂ PGL_{d+1}(K) a cocompact torsion free discrete subgroup, and 𝔛_{Γ} an algebraization of Γ\Ω̂_{O_{K}}^{d} with generic fiber X_{Γ} as in §6.1.

The following lemma seems well-known. Lemma 6.1. Let X, Y be proper smooth variety of dimension d over K, and f: X → Y a proper surjective generically finite morphism of degree deg f (f is called an alteration in [dJ]). If the weight-monodromy conjecture (Conjecture 1.1) holds for X, then Conjecture 1.1 holds also for Y.

Proof. Since f is finite of degree deg f on some open dense subscheme U ⊂ X, f^{∗}: H_{ét}^{k}(Y\overline{K}, ℚ_{l}) → H_{ét}^{k}(X\overline{K}, ℚ_{l}) and f_{∗}: H_{ét}^{k}(X\overline{K}, ℚ_{l}) → H_{ét}^{k}(Y\overline{K}, ℚ_{l}) satisfy f_{∗} ∘ f^{∗} = multiplication by deg f. Since f_{∗}, f^{∗} commute with Gal(\overline{K}/K)-action, H_{ét}^{k}(Y\overline{K}, ℚ_{l}) is a direct summand of H_{ét}^{k}(X\overline{K}, ℚ_{l}) as a Gal(\overline{K}/K)-representation. Therefore, the restriction of the monodromy filtration (resp. weight filtration) on H_{ét}^{k}(X\overline{K}, ℚ_{l}) to H_{ét}^{k}(Y\overline{K}, ℚ_{l}) coincides with the monodromy filtration (resp. weight filtration) on H_{ét}^{k}(Y\overline{K}, ℚ_{l}) itself. Hence the assertion follows. □

Now, we shall prove the main theorem (Theorem 1.2) of this paper.

Proof of Theorem 1.2. First of all, it is known that the intersection of all finite index subgroups Γ′ ⊂ Γ is equal to the identity ([Ga], Theorem 2.7, Proof, (b)). From this, by looking at the action of Γ on the Bruhat-Tits building 𝔗 of PGL_{d+1}(K) ([Mus], [Ku]), it is easy to see that 𝔛_{Γ′} is strictly semistable for some finite index subgroup Γ′ ⊂ Γ. Since we have a finite étale covering X_{Γ′} → X_{Γ}, by Lemma 6.1, we may assume that 𝔛_{Γ} is a strictly semistable model of X_{Γ} over O_{K}.

Under this assumption, from the construction of 𝔛_{Γ} in §6.1, we see that all irreducible components X_{1}, …, X_{m} of the special fiber of 𝔛_{Γ} are isomorphic to the variety B^{d} constructed in §4.1. By Proposition 4.2, 4, 5, for i ≠ j, each irreducible component of X_{i} ∩ X_{j} is isomorphic to a divisor of the form D_{V} on B^{d} in §4.2. Moreover, by induction, we see that, for 1 ≤ i_{1} < ⋯ < i_{k} ≤ m, each irreducible component Y of X_{i_{1}} ∩ ⋯ ∩ X_{i_{k}} is isomorphic to the product

Y ≅ B^{n_{1}} × ⋯ × B^{n_{k}}

for some n_{1}, …, n_{k} ≥ 1 with n_{1} + ⋯ + n_{k} = d − k + 1.

Let L be the relative dualizing sheaf ω_{𝔛_{Γ}/O_{K}}, which is invertible and ample by Mustafin, Kurihara ([Mus], [Ku]). Take i with 1 ≤ i ≤ m and fix an isomorphism X_{i} ≅ B^{d}. Then, by an explicit calculation in [Mus], [Ku], the restriction of L to each X_{i} is isomorphic to

−(n + 1)f^{∗}O_{ℙ^{d}}(1) + ∑_{d=0}^{n−1}(n − d)D_{d}

(for notation, see §4.1, §4.2, see also Remark 4.8), which is an ample PGL_{d+1}(𝔽_{q})- invariant divisor on X_{i} ≅ B^{d}. Therefore, by Proposition 4.7, the Hodge standard conjecture (Conjecture 2.6) holds for (X_{i}, L|_{X_{i}}).

Moreover, for 1 ≤ i_{1} < ⋯ < i_{k} ≤ m, take an irreducible component Y of X_{i_{1}} ∩ ⋯ ∩ X_{i_{k}}. Then, Y is isomorphic to B^{n_{1}} × ⋯ × B^{n_{k}} for some n_{1}, …, n_{k} ≥ 1 with n_{1} +⋯+n_{k} = d−k +1. By the construction of B^{n} in §4.1 and the Künneth formula, Y satisfies Assumption 2.1 in §2.1 (see also §2.5). Let pr_{j}: Y → B^{n_{j}} be the projection to the i-th factor for j = 1, …, k. Then, by applying Proposition 4.2, 4, 5 and Proposition 4.4, 4 inductively, we see that the restriction of L to Y is of the form

L|_{Y} = pr_{1}^{∗}O(L_{1}) + ⋯ + pr_{k}^{∗}O(L_{k}) in H^{2}(Y)

where L_{j} is an ample PGL_{n_{j}+1}(𝔽_{q})-invariant divisor on B^{n_{j}} for j = 1, …, k. By Proposition 4.7, the Hodge standard conjecture (Conjecture 2.6) holds for (B^{n_{j}}, L_{j}) for j = 1, …, k. By applying Proposition 2.13 inductively, we conclude that the Hodge standard conjecture (Conjecture 2.6) also holds for (Y, L|_{Y}).

Therefore, 𝔛_{Γ} satisfies all assumptions in Proposition 5.7. Hence the weightmonodromy conjecture (Conjecture 1.1) holds for X_{Γ}, and the proof of Theorem 1.2 is now completed. □

Remark 6.2. Theorem 1.2 is new only in mixed characteristic. However, we note that Theorem 1.2 in characteristic p > 0 doesn’t automatically follows from Deligne’s results in [De3] because 𝔛_{Γ} doesn’t automatically come from a family of varieties over a curve over a finite field (This was explained by Illusie in [Il2], 8.7). We need an approximation argument to reduce the general characteristic p > 0 case to Deligne’s case ([Te], [It1]).

6.3. Schneider-Stuhler’s conjecture on the filtration F^{•}. In [SS], Schneider-Stuhler computed the cohomology of Ω̂_{K}^{d} and its quotient Γ\Ω̂_{K}^{d} by methods from rigid analytic geometry and representation theory. Their results are valid for cohomology theories satisfying certain axioms, and they showed that de Rham cohomology and rigid analytic l-adic cohomology satisfy these axioms. Precisely speaking, at that time, they used unpublished results of Gabber to check the axioms for rigid analytic l-adic cohomology. After that, these axioms were established by Berkovich ([Be]), de Jong - van der Put ([dJvdP]). By comparison theorem between rigid analytic and usual l-adic cohomology, Schneider-Stuhler’s computation is valid for l-adic cohomology of the variety X_{Γ}. Here we combine our results with the results of Schneider-Stuhler in [SS], and prove Schneider-Stuhler’s conjecture on the filtration F^{•} for l-adic cohomology (However, for de Rham cohomology, see Remark 6.15).

First of all, we recall the results of Schneider-Stuhler (for details, see [SS], Theorem 4). As in §6.1, let K be a complete discrete valuation field with finite residue field 𝔽_{q} of characteristic p > 0. For a cocompact torsion free discrete subgroup Γ ⊂ PGL_{d+1}(K), let

Ind_{Γ} := C^{∞}(PGL_{d+1}(K)/Γ, ℂ)

be the PGL_{d+1}(K)-representation induced from the trivial character on Γ. Let μ(Γ) be the multiplicity of the Steinberg representation in Ind_{Γ}. In [SS], §5, Schneider-Stuhler explicitly computed the E_{2}-terms of a Hochschild-Serre type spectral sequence :

(6.1)  E_{2}^{r,s} = H^{r}(Γ, H_{ét}^{s}(Ω̂_{K}^{d} ⊗_{K} \overline{K}, ℚ_{l})) ⟹ H_{ét}^{r+s}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l})

([SS], §5, Proposition 2), and proved

(6.2)  H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l}) ≅ \begin{cases}ℚ_{l}(−\frac{k}{2}) & if k is even, 0 ≤ k ≤ 2d, k ≠ d \\ 0 & if k is odd, k ≠ d. \end{cases}

For the middle degree k = d, they proved that the covering spectral sequence (6.1) defines a decreasing filtration F^{•} on V := H_{ét}^{d}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l}) such that

V = F^{0}V ⊃ F^{1}V ⊃ ⋯ ⊃ F^{d}V ⊃ F^{d+1}V = 0,

(6.3)  F^{r}V/F^{r+1}V ≅ \begin{cases}ℚ_{l}(r − d)^{⊕μ(Γ)} & if 0 ≤ r ≤ d, r ≠ \frac{d}{2} \\ ℚ_{l}(−\frac{d}{2})^{⊕(μ(Γ)+1)} & if r = \frac{d}{2} \\ 0 & otherwise, \end{cases}

and conjectured that F^{•} essentially coincides with the monodromy filtration (see [SS], introduction and a remark following Theorem 5). Theorem 6.3 (Schneider-Stuhler’s conjecture for l-adic cohomology). Let M_{•} be the monodromy filtration on V (Definition 5.1). Define an increasing filtration F′_{•} on V by F′_{i}V = F^{−⌊i/2⌋}V. Then, we have

M_{i} V = F′_{i−d} V for all i.

Proof. We compute the graded quotients of F′_{•} explicitly. If i is even, we have

F′_{i} V/F′_{i−1} V = F^{−i/2}V/F^{−i/2+1}V

= \begin{cases}ℚ_{l}(−\frac{i}{2} − d)^{⊕μ(Γ)} & if − 2d ≤ i ≤ 0, i ≠ −d \\ ℚ_{l}(−\frac{d}{2})^{⊕(μ(Γ)+1)} & if i = −d \\ 0 & otherwise. \end{cases}

If i is odd, we have ⌊i/2⌋ = ⌊(i − 1)/2⌋ hence F′_{i} V/F′_{i−1} V = 0. Therefore, each F′_{i} V/F′_{i−1} V has weight i + 2d as a Gal(\overline{𝔽}_{q}/𝔽_{q})-representation. Now the assertion follows from Theorem 1.2. □

6.4. Application to the local zeta function of X_{Γ}. Let notation be as in §6.3. Let us recall the definition of the local zeta functions (for details, see [Se2], [Rap]). For a continuous Gal(\overline{K}/K)-representation V over ℚ_{l}, the local L-function of V is defined by

L(s, V) := det (1 − q^{−s} · Fr_{q}; V^{I_{K}})^{−1}

(for notation, see §5.1, §5.2). For a variety X over K, the local zeta function of X is defined by

ζ(s, X) := ∏_{k=0}^{2 dim X} L(s, H_{ét}^{k}(X\overline{K}, ℚ_{l}))^{(−1)^{k}}.

Let us compute the local zeta function of X_{Γ}. For an even integer k with 0 ≤ k ≤ 2d, k ≠ d, by (6.2), we easily see that

L(s, H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l})) = det (1 − q^{−s} · Fr_{q}; ℚ_{l}(−\frac{k}{2}))^{−1} = \frac{1}{1 − q^{k/2−s}} .

On the other hand, for the middle degree k = d, we can’t compute the local Lfunction only from the results of Schneider-Stuhler (6.3) because (6.3) only gives us the semisimplification of the Gal(\overline{K}/K)-representation H_{ét}^{d}(X_{Γ}⊗_{K} \overline{K}, ℚ_{l}). We need the weight-monodromy conjecture to recover the local L-function from the semisimplification (see [Rap], §2).

Now, we shall use Theorem 6.3. By Theorem 6.3, we see that the inertia fixed part is given by

H_{ét}^{d}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l})^{I_{K}} = \begin{cases}ℚ_{l}^{⊕μ(Γ)} ⊕ ℚ_{l}(−\frac{d}{2}) & if d is even \\ ℚ_{l}^{⊕μ(Γ)} & if d is odd \end{cases}

(see also Definition 5.1, 3). Therefore, we have

L(s, H_{ét}^{d}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l})) = \begin{cases}\frac{1}{(1 − q^{−s})^{μ(Γ)}} · \frac{1}{1 − q^{d/2−s}} & if d is even \\ \frac{1}{(1 − q^{−s})^{μ(Γ)}} & if d is odd. \end{cases}

By combining above results, we have the following theorem. Theorem 6.4 (Local zeta function of X_{Γ}). Let Γ ⊂ PGL_{d+1}(K) be a cocompact torsion free discrete subgroup. Then, the local zeta function of X_{Γ} is given by

ζ(s, X_{Γ}) = (1 − q^{−s})^{μ(Γ)·(−1)^{d+1}} · ∏_{k=0}^{d} \frac{1}{1 − q^{k−s}} ,

where μ(Γ) is the multiplicity of the Steinberg representation in Ind_{Γ} as in §6.3. Remark 6.5. As a consequence, we also see that the local zeta function ζ(s, X_{Γ}) is independent of l ≠ p, which is a nontrivial result. In §6.6, we consider the case l = p by using p-adic cohomology.

6.5. Application to mod l Galois representations. Let notation be as in §6.3 and in §5.3. For l ≠ p, we have the weight spectral sequence of Rapoport-Zink with coefficients in 𝔽_{l}

′E_{1}^{−r, w+r} = ⨁_{k≥max\{0,−r\}} H_{ét}^{w−r−2k}(X_{\overline{𝔽}_{q}}^{(2k+r+1)}, 𝔽_{l}(−r − k))

⟹ H_{ét}^{w}(X_{Γ} ⊗_{K} \overline{K}, 𝔽_{l}),

similarly as in §5.3. We also have a natural monodromy operator N: ′E_{1}^{i, j}(1) → ′E_{1}^{i+2, j−2}. Here we study the above spectral sequence for all but finitely many l. Note that, in general, we don’t even know the E_{2}-degeneracy because there doesn’t exist good theory of weights for étale cohomology with coefficients in 𝔽_{l}.

Fix an isomorphism ℤ_{l} ≅ ℤ_{l}(1), and let E_{1}^{−r, w+r} ⇒ H_{ét}^{w}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l}) be the weight spectral sequence for l-adic cohomology as in §5.3. The crucial observation in the proof of Theorem 1.2 in §6.2 is the existence of ℚ-structures on E_{1}^{i, j}, d_{1}^{i, j}, N. In fact, we can say little more. As we already saw in §6.2, all varieties appearing in the weight spectral sequence are products of the variety B^{n} in §4.1. Therefore, from the construction of B^{n}, we see that the cohomology of B^{n} has ℤ-structures as well as ℚ-structures. Namely, we have an isomorphism N^{k}(B^{n}) ⊗_{ℤ} 𝔽_{l} ≅ H_{ét}^{2k}(B^{n}, 𝔽_{l}(k)) for all k (see Lemma 2.3 in §2.1). Let Ẽ_{1}^{−r, w+r} be the free sub-ℤ-module of E_{1}^{−r, w+r} generated by algebraic cycles over ℤ. Since d_{1}^{i, j}, N are defined geometrically, there exist natural maps d̃_{1}^{i, j}, Ñ on Ẽ_{1}^{−r, w+r} which give ℤ-structures on d_{1}^{i, j}, N. These are independent of l by Lemma 2.3. Hence we have the data {Ẽ_{1}^{−r, w+r}, d̃_{1}^{i, j}, Ñ} such that, for all l ≠ p and for Λ = ℚ_{l} or 𝔽_{l}, the data

{Ẽ_{1}^{−r, w+r} ⊗_{ℤ} Λ, d̃_{1}^{i, j} ⊗_{ℤ} Λ, Ñ ⊗_{ℤ} Λ}

coincide with the E_{1}-terms of the weight spectral sequence with coefficients in Λ. Lemma 6.6. For all but finitely many l, the mod l weight spectral sequence ′E_{1}^{−r, w+r} ⇒ H_{ét}^{w}(X_{Γ}⊗_{K} \overline{K}, 𝔽_{l}) degenerates at E_{2}, and N^{r} induces an isomorphism

N^{r}: ′E_{2}^{−r, w+r} ⟶^{≅} ′E_{2}^{r, w−r}

on E_{2}-terms for all r, w (Recall that we fix an isomorphism ℤ_{l} ≅ ℤ_{l}(1) here).

Proof. First of all, we have an inequality :

∑_{i, j} dim_{𝔽_{l}} ′E_{2}^{i, j} ≥ ∑_{k} dim_{𝔽_{l}} H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, 𝔽_{l}),

and the equality holds if and only if the mod l weight spectral sequence degenerates at E_{2}. Since the E_{2}-degeneracy is known for l-adic cohomology (see §5.3), we have

∑_{i, j} dim_{ℚ_{l}} E_{2}^{i, j} = ∑_{k} dim_{ℚ_{l}} H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l}),

where E_{1}^{−r, w+r} ⇒ H_{ét}^{w}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l}) is the l-adic weight spectral sequence. On the other hand, in general, we have the inequality :

dim_{𝔽_{l}} H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, 𝔽_{l}) ≥ dim_{ℚ_{l}} H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l}),

and the equality holds if and only if the cohomology with coefficients in ℤ_{l} is torsion free. Hence it is enough to show the equality dim_{𝔽_{l}} ′E_{2}^{i, j} = dim_{ℚ_{l}} E_{2}^{i, j} for all but finitely many l.

To show this, we use the ℤ-structures on the E_{1}-terms. We put

Ẽ_{2}^{i, j} := Ker (d̃_{1}^{i, j})/Im (d̃_{1}^{i−1, j}).

Since ℚ_{l} is flat over ℤ, we have Ẽ_{2}^{i, j} ⊗_{ℤ} ℚ_{l} = E_{2}^{i, j}. In general, the natural map Ẽ_{2}^{i, j} ⊗_{ℤ} 𝔽_{l} → ′E_{2}^{i, j} isn’t an isomorphism. However, since there are only finitely many nonzero terms at E_{1} and each term is finitely generated over ℤ, we have an isomorphism Ẽ_{2}^{i, j} ⊗_{ℤ} 𝔽_{l} ≅ ′E_{2}^{i, j} and the equality

dim_{𝔽_{l}} (Ẽ_{2}^{i, j} ⊗_{ℤ} 𝔽_{l}) = dim_{ℚ_{l}} (Ẽ_{2}^{i, j} ⊗_{ℤ} ℚ_{l})

for all but finitely many l. Hence we have the desired equality dim_{𝔽_{l}} ′E_{2}^{i, j} = dim_{ℚ_{l}} E_{2}^{i, j} for all but finitely many l.

For the monodromy operator, by the same reason as above, it is enough to show that the map

Ñ^{r} ⊗_{ℤ} 𝔽_{l}: Ẽ_{2}^{−r, w+r} ⊗_{ℤ} 𝔽_{l} ⟶ Ẽ_{2}^{r, w−r} ⊗_{ℤ} 𝔽_{l}

is an isomorphism for all but finitely many l. By Theorem 1.2 (see also Conjecture 5.6, §6.2), Ñ^{r} ⊗_{ℤ} ℚ_{l} is an isomorphism for all l ≠ p. Since all Ẽ_{2}^{i,j} are finitely generated over ℤ, we have the assertion. □

In conclusion, by Lemma 6.6, we have the following theorem for mod l Galois representations of X_{Γ} for all but finitely many l (see also an argument in §6.4). In the following statement, we don’t ignore the Tate twists. Theorem 6.7. Let Γ ⊂ PGL_{d+1}(K) be a cocompact torsion free discrete subgroup. We put V := H_{ét}^{d}(X_{Γ} ⊗_{K} \overline{K}, 𝔽_{l}). Then, for all but finitely many l, we have

dim_{𝔽_{l}} V = (d + 1) · μ(Γ) + ε, dim_{𝔽_{l}} V^{I_{K}} = μ(Γ) + ε,

where ε = 0 (resp. ε = 1) if d is odd (resp. even), and μ(Γ) is the multiplicity of the Steinberg representation in Ind_{Γ} as in §6.3. More precisely, we have

V^{I_{K}} = \begin{cases}𝔽_{l}^{⊕μ(Γ)} ⊕ 𝔽_{l}(−\frac{d}{2}) & if d is even \\ 𝔽_{l}^{⊕μ(Γ)} & if d is odd. \end{cases}

In particular, the characteristic polynomial of Fr_{q} acting on V^{I_{K}} is the reduction modulo l of the characteristic polynomial of Fr_{q} acting on H_{ét}^{d}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l})^{I_{K}}. Moreover, if μ(Γ) ≥ 1, then the mod l Gal(\overline{K}/K)-representation V is ramified. Remark 6.8. The author doesn’t know whether Theorem 6.7 holds for small l. This seems an interesting problem from the viewpoint of the theory of mod l automorphic forms and mod l Galois representations.

6.6. A p-adic analogue. We recall some results in p-adic Hodge theory to state an analogue of the weight-monodromy conjecture for p-adic étale cohomology.

Let K be a finite extension of ℚ_{p} with residue field 𝔽_{q}, W (𝔽_{q}) the ring of Witt vectors with coefficients in 𝔽_{q}, K_{0} the field of fractions of W (𝔽_{q}), and X a proper smooth variety over K. For simplicity, assume that there is a proper strictly semistable model 𝔛 of X over O_{K} (Definition 5.5).

For V := H_{ét}^{w}(X\overline{K}, ℚ_{p}), by the C_{st}-conjecture proved by Tsuji, there is a canonical isomorphism

(6.4) D_{st}(V) := (V ⊗_{ℚ_{p}} B_{st})^{Gal(\overline{K}/K)} ≅ H_{log-crys}^{w}(𝔛_{𝔽_{q}}^{×}/W (𝔽_{q})^{×}) ⊗_{W (𝔽_{q})} K_{0}

where B_{st} is Fontaine’s ring of p-adic periods, Gal(\overline{K}/K) acts on V ⊗_{ℚ_{p}} B_{st} diagonally, and H_{log-crys}^{w}(𝔛_{𝔽_{q}}^{×}/W (𝔽_{q})^{×}) is the log crystalline cohomology of the special fiber 𝔛_{𝔽_{q}} = 𝔛 ⊗_{O_{K}} 𝔽_{q} endowed with a natural log structure ([Fo], [HK], [Ts]). It is known that dim_{K_{0}} D_{st}(V) = dim_{ℚ_{p}} V. Moreover, it is also known that and the monodromy operator N and the Frobenius φ act on both hand sides, and (6.4) is compatible with their actions. By using N, φ, we can define the monodromy filtration M_{•} and the weight filtration W_{•} on D_{st}(V) by the same way as in Definition 5.1, Definition 5.3. Then, we have the following p-adic analogue of the weight-monodromy conjecture. Conjecture 6.9 (p-adic weight-monodromy conjecture).

M_{i}D_{st}(V) = W_{i+w}D_{st}(V) for all i.

Let notation be as in §5.3. Mokrane constructed a p-adic analogue of the weight spectral sequence of Rapoport-Zink of the following form

E_{1}^{−r, w+r} = ⨁_{k≥max{0,−r}} H_{crys}^{w−r−2k}(X^{(2k+r+1)}/W (𝔽_{q}))(−r − k)

⟹ H_{log-crys}^{w}(𝔛_{𝔽_{q}}^{×}/W (𝔽_{q})^{×}),

where H_{crys}^{∗} denotes the crystalline cohomology ([Mo], §3.23, Théorème 3.32). This spectral sequence has similar properties as the l-adic case (compare with §5.3). By the Weil conjecture for crystalline cohomology proved by Katz-Messing ([KM]), this spectral sequence degenerates at E_{2} modulo torsion, and defines the weight filtration W_{•} on H_{log-crys}^{w}(𝔛_{𝔽_{q}}^{×}/W (𝔽_{q})^{×}). Moreover, there is a monodromy operator N satisfying the same properties as the l-adic case (N is written as ν in [Mo], §3.33).

Therefore, by (6.4), Conjecture 6.9 is equivalent to the following conjecture on the weight spectral sequence of Mokrane (compare with Conjecture 5.6). Conjecture 6.10 ([Mo], Conjecture 3.27, §3.33). N^{r} induces an isomorphism

N^{r}: E_{2}^{−r, w+r}(r) ⊗_{W (𝔽_{q})} K_{0} ⟶^{≅} E_{2}^{r, w−r} ⊗_{W (𝔽_{q})} K_{0}

for all r, w. Theorem 6.11 (p-adic weight-monodromy conjecture for X_{Γ}). Let Γ ⊂ PGL_{d+1}(K) be a cocompact torsion free discrete subgroup. Then, a p-adic analogue of the weight-monodromy conjecture (Conjecture 6.9, Conjecture 6.10) holds for X_{Γ}.

Proof. The proof is basically the same as the l-adic case. Instead of l-adic cohomology, we use the cycle map for crystalline cohomology ([GM], [Gr]). □

Application to the p-adic local zeta function of X_{Γ}. Recall that a finite dimensional continuous Gal(\overline{K}/K)-representation V over ℚ_{p} is called semistable if dim_{K_{0}} D_{st}(V) = dim_{ℚ_{p}} V ([Ts]). For semistable V, the local L-function of V is defined by

L_{p-adic}(s, V) := det (1 − q^{−s} · φ; D_{st}(V)^{N=0})^{−1},

where D_{st}(V)^{N=0} denotes the kernel of N acting on D_{st}(V). In fact, we can define L(s, V) for potentially semistable V, but we omit it here ([Fo]). For a variety X over K which has a proper strictly semistable model over O_{K}, H_{ét}^{k}(X\overline{K}, ℚ_{p}) is semistable for all k by [Ts]. We define the p-adic local zeta function ζ_{p-adic}(s, X) of X as follows :

ζ_{p-adic}(s, X) := ∏_{k=0}^{2 dim X} L_{p-adic}(s, H_{ét}^{k}(X\overline{K}, ℚ_{p}))^{(−1)^{i}}.

Here we put the subscript “p-adic” to distinguish it from the l-adic case in §6.4. Theorem 6.12 (p-adic local zeta function of X_{Γ}). Let Γ ⊂ PGL_{d+1}(K) be a cocompact torsion free discrete subgroup. Then, for all k, L_{p-adic}(s, H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{p})) is the same as the usual local zeta function defined by l-adic cohomology for all l ≠ p, which is computed explicitly in §6.4. In particular, we have the following equality of local zeta functions of X_{Γ} :

ζ_{p-adic}(s, X_{Γ}) = ζ(s, X_{Γ}) = (1 − q^{−s})^{μ(Γ)·(−1)^{d+1}} · ∏_{k=0}^{d} \frac{1}{1 − q^{k−s}} ,

where μ(Γ) is the multiplicity of the Steinberg representation in Ind_{Γ} as in §6.3.

Proof. We give a proof by comparing the l-adic and the p-adic local zeta functions because Schneider-Stuhler’s computation doesn’t work directly for p-adic étale cohomology (This argument is inspired by [SaT]). Fix a prime number l ≠ p. Since the weight-monodromy conjecture holds for both l-adic and p-adic cohomology (Theorem 1.2, Theorem 6.11), it is enough to prove the equality :

(6.5)  Tr(Fr_{q}^{m}; H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l})) = Tr(φ^{m}; D_{st}(H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{p})))

for all k, m ≥ 0 (see [Rap], Lemma 2.12). For k ≠ d, since the dimensions of l-adic and p-adic étale cohomology are the same, we see that, for odd k ≠ d, H_{ét}^{k}(X_{Γ}⊗_{K} \overline{K}, ℚ_{p}) = 0, and for even k ≠ d, H_{ét}^{k}(X_{Γ}⊗_{K} \overline{K}, ℚ_{p}(\frac{k}{2})) is of dimension 1 and generated by algebraic cycles. Therefore, (6.5) holds for k ≠ d. By Ochiai’s result ([O], Theorem D), we know that the alternating sum of (6.5) for all k holds. Hence (6.5) also holds for k = d. □

Remark 6.13. For strictly semistable 𝔛_{Γ}, we can also prove Theorem 6.12 by comparing the l-adic and the p-adic weight spectral sequences as follows. As in the proof of Lemma 6.6 in §6.5, there are finitely generated ℤ-modules Ẽ_{2}^{i, j} with operator Ñ such that, up to Tate twists, {Ẽ_{2}^{i, j} ⊗_{ℤ}ℚ_{l}, Ñ ⊗_{ℤ}ℚ_{l}} coincide with the E_{2}-terms of the l-adic weight spectral sequence. By the same reason, {Ẽ_{2}^{i, j} ⊗_{ℤ} W (𝔽_{q}), Ñ ⊗_{ℤ} W (𝔽_{q})} coincide with the E_{2}-terms of the p-adic weight spectral sequence up to torsion and Tate twists. Since the weight-monodromy conjecture holds for both l-adic and p-adic cohomology (Theorem 1.2, Theorem 6.11), we conclude that the characteristic polynomial of Fr_{q}, φ on each graded quotient of the monodromy filtration on H_{ét}^{k}(X_{Γ} ⊗_{K} \overline{K}, ℚ_{l}), H_{log-crys}^{k}(𝔛_{𝔽_{q}}^{×}/W (𝔽_{q})^{×}) are the same for all k. Hence we have Theorem 6.12. Remark 6.14. The proof above is somewhat indirect compared with the l-adic case in §6.3. The reason why we have to make such a detour is that Schneider-Stuhler’s result on de Rham cohomology isn’t enough to compute the p-adic local zeta functions because we don’t have natural actions of N, φ on de Rham cohomology (see also Remark 6.15 below). It seems a very challenging problem to generalize Schneider-Stuhler’s computation for appropriately modified p-adic cohomology theory with N, φ for rigid analytic spaces, and compute p-adic local zeta functions without using l-adic cohomology. Remark 6.15. We say few words about Schneider-Stuhler’s conjecture for de Rham cohomology (for l-adic cohomology, see Theorem 6.3). In [SS], §5, Schneider-Stuhler defined a decreasing filtration F^{•} on de Rham cohomology H_{DR}^{d}(X_{Γ}) having similar properties as in the l-adic case (§6.3, see also [SS], Theorem 5). They conjectured that F^{•} essentially coincides with the monodromy filtration defined by Hyodo ([Hy]). We can’t prove this here because we don’t have natural actions of N, φ on de Rham cohomology. The problem is that Hyodo’s monodromy filtration is actually defined on the log crystalline cohomology, which is isomorphic to de Rham cohomology by forgetting N, φ ([HK]). Hence we have the same problem as in Remark 6.14 above. Schneider-Stuhler also conjectured that F^{•} on de Rham cohomology is opposite to the Hodge-de Rham filtration. Recently, this conjecture was proved by Iovita-Spiess in [IS].

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Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

  E-mail address: itote2@ms.u-tokyo.ac.jp

Max-Planck-Institut Für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

  E-mail address: tetsushi@mpim-bonn.mpg.de