A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures

Vladimir G. Berkovich

Department of Mathematics, Weizmann Institute, 76100 Rehovot, ISRAEL

vladimir.berkovich@weizmann.ac.il

To Yuri Ivanovich Manin on the occasion of his 70th birthday

Introduction

Let X be a proper scheme over the field F of functions meromorphic in an open neighborhood of zero in the complex plane C. The scheme X gives rise to a proper morphism of complex analytic spaces X^{h} → D^{∗} = D\{0}, where D is an open disc with center at zero (see §3). It is well known that, after shrinking the disc D (and replacing X^{h} by its preimage), the cohomology groups H^{i}(X_{t}^{h}, Z) of the fiber X_{t}^{h} at a point t ∈ D^{∗} form a local system of finitely generated abelian groups, and that the corresponding action of the fundamental group π_{1}(D^{∗}) = π_{1}(D^{∗}, t) on H^{i}(X_{t}^{h}, Z) is quasi-unipotent. Furthermore, the mixed Hodge structures on the above groups define a variation of mixed Hodge structures on D^{∗}. Let \overline{D}^{∗} → D^{∗} be a universal covering of D^{∗}, and \overline{X}^{h} = X^{h} ×_{D^{∗}} \overline{D}^{∗}. Then the cohomology group H^{i}(\overline{X}^{h}, Z) admits a mixed Hodge structure, which is the limit (in a certain sense) of the above variation of mixed Hodge structures on D^{∗} (see [GNPP, Exp. IV, Theorem 7.4]). One of the purposes of this paper is to describe the weight zero subspace W_{0}H^{i}(\overline{X}^{h}, Q) in terms of non-Archimedean analytic geometry.

Let K be the completion of the discrete valuation field F, and fix a corresponding multiplicative valuation on it. The scheme X gives rise to a proper K-analytic space X^{an} = (X ⊗_{F} K)^{an} in the sense of [Ber1] and [Ber2]. Recall that, as a topological space, X^{an} is compact and locally arcwise connected, and the topological dimension of X^{an} is equal to the dimension of X. If X is smooth, then X^{an} is even locally contractible. Furthermore, let \overline{X}^{an} = (X ⊗_{F} K̂^{a})^{an}, where K̂^{a} is the completion of the algebraic closure K^{a} of K, which corresponds to the universal covering \overline{D}^{∗} → D^{∗}. Recall that the cohomology groups H^{i}(\overline{X}^{an}, Z) of the underlying topological space of \overline{X}^{an} are finitely generated, and there is a finite extension K′′ of K in K^{a} such that they coincide with H^{i}((X ⊗_{F} K′)^{an}, Z) for any finite extension K′ of K′′ in K^{a} (see [Ber5, 10.1]).

In §3, we construct a topological space X^{An} and a surjective continuous map λ : X^{An} → [0, 1] for which there are an open embedding λ^{−1}(]0, 1]) ↪ X^{h}×]0, 1], which is a homotopy equivalence, and a homeomorphism λ^{−1}(0) →^{∼} X^{an}×]0, r[, where r is the radius of the disc D. We show that the induced maps H^{i}(X^{An}, Z) → H^{i}(X^{an}×]0, r[, Z) = H^{i}(X^{an}, Z) are isomorphisms for all i ≥ 0. In this way, we get a homomorphism H^{i}(X^{an}, Z) → H^{i}(X^{h}, Z), whose composition with the canonical map H^{i}(X^{h}, Z) → H^{i}(\overline{X}^{h}, Z) gives a homomorphism H^{i}(X^{an}, Z) → H^{i}(\overline{X}^{h}, Z). The same construction applied to finite extensions of F in F^{a} gives rise to a homomorphism of π_{1}(D^{∗})-modules H^{i}(\overline{X}^{an}, Z) → H^{i}(\overline{X}^{h}, Z). Theorem 5.1 states that the latter gives rise to a functorial isomorphism π_{1}(D^{∗})-modules

H^{i}(\overline{X}^{an}, Q) →^{∼} W_{0}H^{i}(\overline{X}^{h}, Q) .

If X is projective and smooth, one can describe the group W_{0}H^{i}(\overline{X}^{h}, Q) as follows. By the local monodromy theorem, the action of (T^{m} − 1)^{i+1} on H^{i}(X_{t}^{h}, Z) is zero (for some m ≥ 1), where T is the canonical generator of π_{1}(D^{∗}). If we fix a point of \overline{D}^{∗} over t, there is an induced isomorphism H^{i}(\overline{X}^{h}, Z) →^{∼} H^{i}(X_{t}^{h}, Z) which gives rise to an isomorphism between W_{0}H^{i}(\overline{X}^{h}, Q) and the maximal unipotent monodromy subspace of H^{i}(X_{t}^{h}, Q), i.e., (T^{m} − 1)^{i}H^{i}(X_{t}^{h}, Q). Thus, in the case considered, there is a functorial isomorphism of π_{1}(D^{∗})-modules

H^{i}(\overline{X}^{an}, Q) →^{∼} (T^{m} − 1)^{i}H^{i}(X_{t}^{h}, Q) .

The mixed Hodge theory (see [St, p. 247], [Ill, p. 29]) provides an upper bound on the dimension of the space on the right hand side, which implies the following bound on that of the left hand side

dim_{Q} H^{i}(\overline{X}^{an}, Q) ≤ dim_{F} H^{i}(X , O_{X}) .

The equality is achieved for a totally degenerate family of abelian varieties (see [Ber1, §6]), and for a totally degenerate family of Calabi-Yau varieties (in the strong sense). In the latter example, X^{an} has rational cohomology of the sphere of dimension dim(X), and is simply connected (see Remark 4.4(ii)).

In fact X^{An} is the underlying topological space of an analytic space over a commutative Banach ring. The idea of such an object was introduced in [Ber1, §1.5], and developed there in detail in the case when the Banach ring is a non-Archimedean field. The spaces considered in this paper are defined over the field of complex numbers C provided with the following Banach norm: ||a|| = max{|a|_{∞}, |a|_{0}} for a ∈ C, where | |_{∞} is the usual Archimedean valuation, and | |_{0} is the trivial valuation (i.e., |a|_{0} = 1 for a ≠ 0). One has [0, 1] →^{∼} M(C, || ||). Namely, a nonzero point ρ ∈]0, 1] corresponds to the Archimedean valuation | |_{∞}^{ρ}, and the zero 0 corresponds to the trivial valuation | |_{0}. The above map λ is a canonical map X^{An} → M(C, || ||) = [0, 1]. The preimage λ^{−1}(ρ) of ρ ∈]0, 1] is the restriction of the complex analytic space X^{h} to the smaller open disc D(r^{\frac{1}{ρ}}), and λ^{−1}(0) is a non-Archimedean analytic space over the field C provided with the trivial valuation | |_{0}. Thus, the space X^{An} incorporates both complex analytic and non-Archimedean analytic spaces, and the result on a non-Archimedean interpretation of the weight zero subspaces is evidence that analytic spaces over (C, || ||) are worth studying.

In §1, we recall a construction from [Ber1, §1] that associates with an algebraic variety over a commutative Banach ring k the underlying topological space of a k-analytic space. We do not develop a theory of k-analytic spaces, but restrict ourselves with establishing basic properties necessary for this paper. In §2, we specify our study for the field C provided with the above Banach norm || ||, and prove a particular case of the main result from §4. Let O_{C,0} be the local ring of functions analytic in an open neighborhood of zero in C. In §3, we associate with an algebraic variety X over O_{C,0} analytic spaces of three types: a complex analytic space X^{h}, a (C, || ||)-analytic space X^{An}, and a (C, | |_{0})-analytic space X_{0}^{An}. All three spaces are provided with a morphism to the corresponding open discs, and are closely interrelated. The construction gives rise to a commutative diagram of maps between topological spaces. In §4, we prove our main result (Theorem 4.1) which states that, if X is proper over O_{C,0}, the homomorphisms between integral cohomology groups induced by certain maps from that diagram are isomorphisms. Essential ingredients of the proof are C. H. Clemens’s results from [Cle] and similar results from [Ber5]. If X is strictly semi-stable over O_{C,0}, the former provide a strong deformation retraction of X^{h} to its fiber X_{s}^{h} at zero, and the latter provide a similar homotopy description of the non-Archimedean space X_{0}^{An}. In §5, we prove Theorem 5.1 which was already formulated.

We want to emphasize that the above result is an analog of the description of the weight zero subspaces of l-adic étale cohomology groups of algebraic varieties defined over a local field in terms of cohomology groups of the associated non-Archimedean spaces (see [Ber6]). All of these results are evidence for the fact that the underlying topological space of the non-Archimedean analytic space associated with an algebraic variety somehow represents the weight zero part of the mixed motive of the variety.

I am very grateful to Johannes Nicaise for pointing out a mistake in a previous version of Theorem 4.1, and to the referee for many corrections and suggestions that significantly improved the paper.

1 Topological spaces associated with algebraic varieties over a commutative Banach ring

Let k be a commutative Banach ring, i.e., a commutative ring provided with a Banach norm || || and complete with respect to it. For an affine scheme X = Spec(A) of finite type over k, let X^{An} denote the set of all nonzero multiplicative seminorms | | : A → R_{+} on the ring A whose restriction to k is bounded with respect to the norm || ||. The set X^{An} is provided with the weakest topology with respect to which all real valued functions of the form | | ↦ |f|, f ∈ A, are continuous. For a point x ∈ X^{An}, the corresponding multiplicative seminorm | |_{x} on A gives rise to a multiplicative norm on the integral domain A/Ker(| |_{x}) and, therefore, extends to a multiplicative norm on its field of fractions. The completion of the latter is denoted by H(x), and the image of an element f ∈ A under the corresponding character A → H(x) is denoted by f(x). (In particular, |f|_{x} = |f(x)| for all f ∈ A.) If A = k ≠ 0, the space X^{An} is the spectrum M(k) of k, which is a nonempty compact space, by [Ber1, 1.2.1]. If A = k[T_{1}, …, T_{n}], the space X^{An} is denoted by A^{n} (the n-dimensional affine space over k). Notice that the correspondence X ↦ X^{An} is functorial in X.

A continuous map of topological spaces φ : Y → X is said to be Hausdorff if, for any pair of different points y_{1}, y_{2} ∈ Y with φ(y_{1}) = φ(y_{2}), there exist open neighborhoods V_{1} of y_{1} and V_{2} of y_{2} with V_{1} ∩ V_{2} = ∅ (i.e., the image of Y in Y ×_{X} Y is closed). Furthermore, let X be a topological space such that each point of it has a compact neighborhood. A continuous map φ : Y → X is said to be compact, if the preimage of a compact subset of X is a compact subset of Y (i.e., φ is proper in the usual sense, but we use the terminology of [Ber2]). Such a map is Hausdorff, it takes closed subsets of Y to closed subsets of X, and each point of Y has a compact neighborhood.

Lemma 1.1. (i) The space X^{An} is locally compact and countable at infinity; (ii)given a closed (resp. an open) immersion φ : Y → X, the map φ^{an} :

Y^{An} → X^{An} induces a homeomorphism of Y^{An} with a closed (resp. open) subset of X^{An};

(iii)given morphisms φ : Y → X and Z → X, the canonical map

(Y ×_{X} Z)^{An} → Y^{An} ×_{X^{An}} Z^{An}

is compact.

Proof. (ii) If φ is a closed immersion, the required fact is trivial. If φ is an open immersion, it suffices to consider the case of a principal open subset Y = Spec(A_{f}) for an element f ∈ A. It is clear that the map φ^{An} is injective and its image is an open subset of X^{An}. A fundamental system of open sets in Y^{An} is formed by finite intersections of sets of the form U = \{y ∈ Y^{An}||\frac{g}{f^{n}} (y)| < r\} and V = \{y ∈ Y^{An}||\frac{g}{f^{n}} (y)| > r\}, where g ∈ A, n ≥ 0 and r > 0. It suffices therefore to verify that the sets U and V are open in X^{An}. Given a point y ∈ U (resp. V), there exist ε, δ > 0 such that \frac{|g(y)|+δ}{|f^{n}(y)|−ε} < r (resp. \frac{|g(y)|−δ}{|f^{n}(y)|+ε} > r). Then the set {z ∈ X^{An}||g(z)| < |g(y)| + δ, |f^{n}(z)| > |f^{n}(y)| − ε} (resp. {z ∈ X^{An}||g(z)| > |g(y)| − δ, |f^{n}(z)| < |f^{n}(y)| + ε}) is an open neighborhood of the point y in X^{An}, which is contained in U (resp. V).

(i) By (ii), it suffices to consider the case of the affine space A^{n}, which is associated with the ring of polynomials k[T] = k[T_{1}, …, T_{n}]. One has A^{n} = ⋃_{r} E(r), where the union is taken over tuples of positive numbers r = (r_{1}, …, r_{n}) and E(r) is the closed polydisc of radius r with center at zero {x ∈ A^{n}||T_{i}(x)| ≤ r_{i} for all 1 ≤ i ≤ n}. The latter is a compact space. Indeed, let k⟨r^{−1}T ⟩ = k⟨r_{1}^{−1}T_{1}, …, r_{n}^{−1}T_{n}⟩ denote the commutative Banach ring of all power series f = ∑_{ν} a_{ν}T^{ν} over k such that ||f|| = ∑_{ν} ||a_{ν}||r^{ν} < ∞. By [Ber1, Theorem 1.2.1], the spectrum M(k⟨r^{−1}T ⟩) is a nonempty compact space, and the canonical homomorphism k[T] → k⟨r^{−1}T ⟩ induces a homeomorphism M(k⟨r^{−1}T ⟩) →^{∼} E(r).

(iii) Let X = Spec(A), Y = Spec(B) and Z = Spec(C). By (ii), it suffices to consider the case when A = k[T_{1}, …, T_{n}], B = A[U_{1}, …, U_{p}] and C = A[V_{1}, …, V_{q}], i.e., it suffices to verify that the corresponding map A^{n+p+q} → A^{n+p}×_{A^{n}}A^{n+q} is compact. This is clear since the preimage of E(r′)×_{A^{n}}E(r′′) with r′ = (r_{1}, …, r_{n}, s_{1}, …, s_{p}) and r′′ = (r_{1}, …, r_{n}, t_{1}, …, t_{q}) is the polydisc E(r) with r = (r_{1}, …, r_{n}, s_{1}, …, s_{p}, t_{1}, …, t_{q}). ⊓ ⊔

Let now X be a scheme of finite type over k. By Lemma 1.1(ii), one can glue the spaces U^{An} for open affine subschemes U ⊂ X to get a topological space X^{An} in which all U^{An} are open subspaces. Here is an equivalent description of the space X^{An}. For a bounded character χ : k → K to a valuation field K (i.e., a field complete with respect to a valuation), let X (K)^{χ} denote the set of all K-points of X which induce the character χ on k. Furthermore, let X̃^{An} be the disjoint union of the sets X (K)^{χ} taken over bounded characters χ : k → K to a valuation field K. Two points x′ ∈ X (K′)^{χ′} and x′′ ∈ X (K′′)^{χ′′} are said to be equivalent if there exist a bounded character χ : k → K, a point x ∈ X (K)^{χ}, and isometric embeddings K → K′ and K → K′′ which are compatible with the characters χ′ and χ′′, taking x to the points x′ and x′′, respectively. It is really an equivalence relation, and the space X^{An} is the set of equivalence classes in X̃^{An}.

The correspondence X ↦ X^{An} is functorial in X, and the properties (ii) and (iii) of Lemma 1.1 are straightforwardly extended to arbitrary schemes of finite type over k.

Lemma 1.2. Let φ : Y → X be a morphism of schemes of finite type over k, and let φ^{An} be the induced map Y^{An} → X^{An}. Then

(i) if φ is separated, then the map φ^{An} is Hausdorff; (ii)if φ is projective, then the map φ^{An} is compact; (iii)if φ is proper and either the ring k is Noetherian, or Y has a finite number

of irreducible components, then the map φ^{An} is compact.

The assumptions in (iii) guarantee application of Chow’s Lemma (see [EGAII, 5.6.1]).

Proof. We may assume that the scheme X = Spec(A) is affine.

(i) Since φ is separated, the diagonal map Y^{An} → (Y ×_{X} Y)^{An} has a closed image. Since the image of the latter in Y^{An} ×_{X^{An}} Y^{An} is closed, it follows that the map φ^{An} is Hausdorff.

(ii) It suffices to consider the case when Y = Proj(A[T_{0}, …, T_{n}]) is the projective space over A. In this case, Y = ⋃_{i=0}^{n} Y_{i}, where Y_{i} = Spec(A[\frac{T_{0}}{T_{i}} , …, \frac{T_{0}}{T_{i}}]). If E_{i} = \{y ∈ Y_{i}^{An}||\frac{T_{0}}{T_{i}} (y)| ≤ 1 for all 0 ≤ j ≤ n\}, then the map E_{i} → X^{An} is compact, and one has Y^{An} = ⋃_{i=0}^{n} E_{i}. It follows that φ^{An} is a compact map.

(iii) Chow’s Lemma reduces the situation to the case considered in (ii). ⊓ ⊔

2 The case of the Banach ring (C, || ||)

We now consider the case when k is the field of complex numbers C provided with the following Banach norm: ||a|| = max{|a|_{∞}, |a|_{0}} for all a ∈ C. Notice that there is a homeomorphism [0, 1] →^{∼} M(C, || ||) : ρ ↦ p_{ρ}, where the point p_{0} corresponds to the trivial norm | |_{0}, and each point p_{ρ} with ρ > 0 corresponds to the Archimedean norm | |_{∞}^{ρ}. Indeed, if | | is a valuation which is different from the above ones, then it is nontrivial and not equivalent to | |_{∞}. It follows that there exists a complex number a ∈ C with |a|_{∞} < 1 and |a| > 1, i.e., |a| > ||a||, and the valuation | | is not bounded with respect to the Banach norm || ||.

For every scheme X of finite type over C, there is a canonical surjective map λ = λ_{X} : X^{An} → M(C, || ||) = [0, 1]. If ρ ∈]0, 1], then H(p_{ρ}) is the field C provided with the Archimedean valuation | |_{∞}^{ρ}. The fiber λ^{−1}(1) is the complex analytic space X^{h} associated with X, by complex GAGA ([Serre]). The fiber λ^{−1}(0) is the non-Archimedean (C, | |_{0})-analytic space X^{an} associated with X, by non-Archimedean GAGA ([Ber1]).

Lemma 2.1. There is a functorial homeomorphism λ^{−1}(]0, 1]) →^{∼} X^{h}×]0, 1] : x ↦ (y, ρ), which commutes with the projections onto ]0, 1] and, in the case of affine X = Spec(A), is defined by ρ = λ(x) and |f(y)|_{∞} = |f(x)|^{\frac{1}{ρ}}, f ∈ A.

Proof. Assume first that X = Spec(A) is affine. The map considered is evidently continuous. It has an inverse map X^{h}×]0, 1] → λ^{−1}(]0, 1]) : (y, ρ) ↦ y_{ρ}, defined by |f(y_{ρ})| = |f(y)|_{∞}^{ρ} for f ∈ A, and, therefore, it is bijective. The inverse map is continuous since the topology on X^{h}×]0, 1] coincides with the weakest one with respect to which all functions of the form X^{h}×]0, 1] → R_{+} : (y, ρ) ↦ |f(y)|^{ρ} for f ∈ A are continuous. It is trivial that the homeomorphisms are functorial in X, and they extend to the class of all schemes of finite type over C. ⊓ ⊔ Corollary 2.2. If X is connected, then the topological space X^{An} is also connected.

Proof. Any C-point of X defines a section M(C, || ||) = [0, 1] → X^{An} of the canonical map λ : X^{An} → [0, 1], and so the required fact follows from the corresponding facts in complex GAGA ([Serre]) and non-Archimedean GAGA ([Ber1, 3.5.3]). ⊓ ⊔

Proposition 2.3. If X is proper, then H^{q}(X^{An}, Z) →^{∼} H^{q}(X^{an}, Z) for all q ≥ 0.

Proof. Since the space X^{An} is compact, it suffices to show that the cohomology groups with compact support H_{c}^{q}(X^{h}×]0, 1], Z) are zero for all q ≥ 0. For this we use the Leray spectral sequence

E_{2}^{p,q} = H_{c}^{p}(]0, 1], R^{q}λ_{∗}Z) ⟹ H_{c}^{p+q}(X^{h}×]0, 1], Z) .

The sheaves R^{q}λ_{∗}Z are constant and, therefore, E_{2}^{p,q} = 0 for all p, q ≥ 0, and the required fact follows. ⊓ ⊔

By Proposition 2.3, if X is proper, there is a homomorphism

H^{q}(X^{an}, Z) ⟶ H^{q}(X^{h}×]0, 1], Z) = H^{q}(X^{h}, Z) .

Corollary 2.4. If X is proper, the above homomorphism gives rise to an isomorphism

H^{q}(X^{an}, Q) →^{∼} W_{0}H^{q}(X^{h}, Q)

Proof. By the construction from [Del3, §6.2] and Hironaka’s theorem on resolution of singularities, there exists a proper hypercovering X_{•} → X such that each X_{n} is smooth. By [SGA4, Exp. V bis], it gives rise to a homomorphism of spectral sequences

′E_{1}^{p,q} = H^{q}(X_{p}^{an}, Q) ⟹ H^{p+q}(X^{an}, Q)
↓  ↓
′′E_{1}^{p,q} = H^{q}(X_{p}^{h}, Q) ⟹ H^{p+q}(X^{h}, Q)

By [Ber5, §5], the connected components of each X_{p}^{an} are contractible. This implies that ′E_{1}^{p,q} = 0 for all q ≥ 1 and, therefore, the first spectral sequence gives rise to isomorphisms ′E_{2}^{p,0} →^{∼} H^{p}(X^{an}, Q). On the other hand, by [Del2, 3.2.15(ii)], the mixed Hodge structure on H^{q}(X_{p}^{h}, Q) has the property that W_{i} = 0 for i < q. Since the functor H ↦ W_{0}H on the category of rational mixed Hodge structures H with W_{i}H = 0 for i < 0 is exact ([Del2, 2.3.5(iv)]), the latter implies that W_{0}(′′E_{1}^{p,q}) = 0 for all q ≥ 1 and, therefore, the second spectral sequence gives rise to an isomorphism W_{0}(′′E_{2}^{p,0}) →^{∼} W_{0}H^{p}(X^{h}, Q). The required fact now follows from Corollary 2.2. ⊓ ⊔ Remarks 2.5. (i) It would be interesting to know if Proposition 2.3. is true for an arbitrary separated scheme X of finite type over F. If this is true, then the similar induced homomorphism H^{q}(X^{an}, Z) → H^{q}(X^{h}, Z) gives rise to an isomorphism analogous to that of Corollary 2.4. That such an isomorphism exists is shown in [Ber6, Theorem 1.1(c)] using the same reasoning as that used in the proof of Corollary 2.4 (see also Remark 5.3).

(ii) It is very likely that the map X^{an} → X^{An} is a homotopy equivalence at least in the case when X is a proper scheme over F with the property that, for every n ≥ 1, a nonempty intersection of n irreducible components is smooth and of codimension n − 1 (see also Remark 4.4(iii)).

3 Topological spaces associated with algebraic varieties over the ring O_{C,0}

Let X be a scheme of finite type over the local ring O_{C,0}. We are going to associate with X (the underlying topological spaces of) analytic spaces of three types. The first one is a classical object. This is a complex analytic space X^{h} over an open disc D(r) in C of radius r (with center at zero). The second one is a (C, || ||)-analytic space X^{An} over an open disc D(r) in A^{1}. And the third one is a non-Archimedean (C, | |_{0})-analytic space X_{0}^{An} over an open disc D_{0}(r) in the (C, | |_{0})-analytic affine line A_{0}^{1}. The first two objects are related to two representations of the ring O_{C,0} in the form of a filtered inductive limit of the same commutative rings, provided with two different commutative Banach ring structures. The third object is the analytic space associated with the base change of X under the homomorphism O_{C,0} → C[[T]] = O_{A_{0}^{1},0}, and is a particular case of an object introduced in [Ber3, §3].

For r > 0, let C⟨r^{−1}T ⟩ denote the commutative Banach algebra of formal power series f = ∑_{i=0}^{∞} a_{i}T^{i} over C absolutely convergent at the closed disc E(r) = {x ∈ C||T (x)| ≤ r} and provided with the norm ||f|| = ∑_{i=0}^{∞} |a_{i}|_{∞}r^{i}. The canonical homomorphism C[T] → C⟨r^{−1}T ⟩ induces a homeomorphism M(C⟨r^{−1}T ⟩) →^{∼} E(r), and one has O_{C,0} = lim_{⟶} C⟨r^{−1}T ⟩ for r tending to zero. By [EGAIV, §8], for any scheme X of finite type over O_{C,0}, there exist r > 0 and a scheme X′ of finite type over C⟨r^{−1}T ⟩ whose base change with respect to the canonical homomorphism C⟨r^{−1}T ⟩ → O_{C,0} is X. By the construction of §1, there is an associated topological space X^{′h}, and we denote by X^{h} the preimage of the open disc D(r) = {x ∈ C||T (x)| < r} with respect to the canonical map X^{′h} → E(r). The morphism X^{h} → D(r) does not depend, up to a change of r, on the choice of X′, and the construction is functorial in X (see Remark 3.3).

Furthermore, for r > 0, let C⟨⟨r^{−1}T ⟩⟩ denote the commutative Banach ring of formal power series f = ∑_{i=0}^{∞} a_{i}T^{i} over C such that ||f|| = ∑_{i=0}^{∞} ||a_{i}||r^{i} < ∞. The canonical homomorphism C[T] → C⟨⟨r^{−1}T ⟩⟩ gives rise to a homeomorphism M(C⟨⟨r^{−1}T ⟩⟩) →^{∼} E(r) = {x ∈ A^{1}||T (x)| ≤ r}. If

r ≥ 1, then C[T] →^{∼} C⟨⟨r^{−1}T ⟩⟩, and, if r < 1, then C⟨⟨r^{−1}T ⟩⟩ →^{∼} C⟨r^{−1}T ⟩

(as C-subalgebras of C[[T]]). One has O_{C,0} = lim_{⟶} C⟨⟨r^{−1}T ⟩⟩ for r tending to zero. By [EGAIV, §8] again, for any scheme X of finite type over O_{C,0}, there exist 0 < r < 1 and a scheme X′ of finite type over C⟨⟨r^{−1}T ⟩⟩ whose base change with respect to the canonical homomorphism C⟨⟨r^{−1}T ⟩⟩ → O_{C,0} is X. By the construction of §1, there is an associated topological space X^{′An}, and we denote by X^{An} the preimage of D(r) = {x ∈ A^{1}||T (x)| < r} with respect to the canonical map X^{′An} → E(r). The map φ : X^{An} → D(r) does not depend, up to a change of r, on the choice of X′, and the construction is functorial in X (see again Remark 3.3).

Finally, for r > 0, let C{r^{−1}T } denote the commutative Banach ring of formal power series f = ∑_{i=0}^{∞} a_{i}T^{i} over C convergent at the closed disc E_{0}(r) =

{x ∈ A_{0}^{1}||T (x)| ≤ r} and provided with the norm ||f|| = max_{i≥0}{|a_{i}|_{0}r^{i}}.

The canonical homomorphism C[T] → C{r^{−1}T } gives rise to a homeomorphism M(C{r^{−1}T }) →^{∼} E_{0}(r). If r ≥ 1, then C[T] →^{∼} C{r^{−1}T }, and, if r < 1, then C{r^{−1}T } →^{∼} C[[T]]. One has O_{A_{0}^{1},0} = C[[T]] = C{r^{−1}T } for every 0 < r < 1. Thus, given a scheme X of finite type over O_{C,0}, we set X_{0} = X ⊗_{O_{C,0}} C[[T]] and, for 0 < r < 1, we set X′_{0} = X_{0} ⊗_{C[[T]]} C{r^{−1}T }. There is an associated topological space X_{0}^{′An}, and we denote by X_{0}^{An} the preimage of D_{0}(r) = {x ∈ A_{0}^{1}||T (x)| < r} with respect to the canonical map X_{0}^{′An} → E_{0}(r).

Recall that F denotes the fraction field of O_{C,0}, and K denotes the completion of F with respect to a fixed valuation, which is determined by its value at T. Let ε be the latter value. The K-analytic space associated with a scheme X of finite type over F is denoted by X^{an} (instead of (X ⊗_{F} K)^{an}).

Lemma 3.1. Let X be a scheme of finite type over O_{C,0}, X_{η} = X ⊗_{O_{C,0}} F its generic fiber, and X_{s} = X ⊗_{O_{C,0}} C its closed fiber. Let X^{An} be the associated (C, || ||)-analytic space over D(r) with 0 < r < 1, and let λ, λ_{η} and λ_{s} be the canonical maps to M((C, || ||) = [0, 1] from X^{An}, X_{η}^{An} and X_{s}^{An}, respectively. Then

(i) there is a homeomorphism

λ^{−1}(]0, 1]) →^{∼} \{(x, ρ) ∈ X^{h}×]0, 1]||T (x)|_{∞} < r^{\frac{1}{ρ}} \} : x′ ↦ (x, ρ) ,

which commutes with the projections onto ]0, 1] and, in the case of the affine X = Spec(A), is defined by ρ = λ(x′) and |f(x)|_{∞} = |f(x′)|^{\frac{1}{ρ}}, f ∈ A;

(ii)λ^{−1}(0) →^{∼} X_{0}^{An};

(iii)there is a homeomorphism

λ_{η}^{−1}(0) →^{∼} X_{η}^{an}×]0, r[: x′ ↦ (x, ρ) ,

which, in the case of the affine X = Spec(A), is defined by ρ = |T (x′)| and |f(x)| = |f(x′)|^{log_{ρ}(ε)}, f ∈ A; (iv) X^{An}\X_{η}^{An} = X_{s}^{An}, where the right hand side is the (C, || ||)-analytic space

associated with X_{s} in the sense of §2.

Proof. In (i), the converse map (x, ρ) ↦ x_{ρ} is defined by |f(x_{ρ})| = |f(x)|_{∞}^{ρ}, f ∈ A, and, in (iii), the converse map X_{η}^{an}×]0, r[→^{∼} λ^{−1}(0)\φ^{−1}(0) : (x, ρ) ↦ P_{x,ρ} is defined by |f(P_{x,ρ})| = |f(x)|^{log_{ε}(ρ)}, f ∈ A. The statements (ii) and (iv) are trivial. ⊓ ⊔

Corollary 3.2. The open embedding λ^{−1}(]0, 1]) ↪ X^{h}×]0, 1] is a homotopy equivalence.

Proof. The formula ((x, ρ), t) ↦ (x, max(ρ, t)) defines a strong deformation retraction of λ^{−1}(]0, 1]) and X^{h}×]0, 1] to X^{h} × {1}. ⊓ ⊔

Remarks 3.3. (i) The spaces X^{h}, X^{An} and X_{0}^{An} are in fact pro-objects (i.e., filtered projective systems of objects) of the corresponding categories of analytic spaces (see [Ber3, §2]). The functoriality of their constructions means that they give rise to functors from the category of schemes of finite type over O_{C,0} to the corresponding categories of pro-objects.

(ii) Suppose that X is a scheme of finite type over O_{C,0} for which the

canonical morphism to Spec(O_{C,0}) is a composition X →^{φ} Spec(O_{C,0}) →^{ψ} Spec(O_{C,0}), where ψ is induced by the homomorphism O_{C,0} → O_{C,0} : T ↦ T^{n} for n ≥ 1. Let Y denote the same scheme X but considered over Spec(O_{C,0}) with respect to the morphism φ. Then there is a canonical homeomorphism of topological spaces Y_{0}^{An} →^{∼} X_{0}^{An} : y ↦ x which, in the case of affine X = Spec(A), is defined by |f(x)| = |f(y)|^{n} for f ∈ A. It induces homeomorphisms

Y_{η}^{An}×]0, r^{\frac{1}{n}} [→^{∼} X_{η}^{An}×]0, r[: (y, ρ) ↦ (x, ρ^{n}) and Y_{s}^{An} →^{∼} X_{s}^{An} : y ↦ x (see

Lemma 3.1(i) and (iii)), defined by |f(x)| = |f(y)|^{n} for f ∈ A.

4 The main result

Let X be a scheme of finite type over O_{C,0}. By the previous subsection, for some 0 < r < 1 there is a commutative diagram in which hook and down arrows are open embeddings, left and up arrows are closed embeddings, and all squares are cartesian:

X_{η}^{h}×]0, 1] ↩ λ_{η}^{−1}(]0, 1]) ↪^{∗} X_{η}^{An} ⟵ λ_{η}^{−1}(0) ⟶^{∼} X_{η}^{an}×]0, r[
↓∗ ↓∗ ↓ ↓ ↓
X^{h}×]0, 1] ↩ λ^{−1}(]0, 1]) ↪^{∗} X^{An} ⟵ λ^{−1}(0) ⟶^{∼}  X_{0}^{An}
↑ ↑ ↑ ↑ ↑
X_{s}^{h}×]0, 1] ⟵^{∼} λ_{s}^{−1}(]0, 1]) ↪^{∗} X_{s}^{An} ⟵ λ_{s}^{−1}(0) ⟶^{∼}  X_{s}^{an}

Theorem 4.1. Assume that X is proper (resp. proper and strictly semistable) over O_{C,0}. Then for a sufficiently small r all of the horizontal (resp. vertical) arrows of the diagram, except those marked by ∗, induce an isomorphism between integral cohomology groups of the corresponding topological spaces.

The following lemma is a version of Grothendieck’s Proposition 3.10.2 from [Gro]. If F is a sheaf on a topological space X, Φ is a family of supports in X, and Y is a subspace of X, then H_{Φ∩Y}^{q} (Y, F) denotes the cohomology groups with coefficients in the pullback of F at Y and with supports in Φ ∩ Y = {A ∩ Y |A ∈ Φ}.

Lemma 4.2. Let X be a paracompact locally compact topological space, X_{1} ⊂ X_{2} ⊂ … an increasing sequence of closed subsets such that the union of their topological interiors in X coincides with X, and Φ a paracompactifying family of supports in X such that A ∈ Φ if and only if A ∩ X_{i} ∈ Φ ∩ X_{i} for all i ≥ 1. Let F be an abelian sheaf on X, and let q ≥ 1. Assume that for each i ≥ 1 the image H_{Φ∩X_{i+1}}^{q−1}(X_{i+1}, F) in H_{Φ∩X_{i}}^{q−1}(X_{i}, F) under the restriction homomorphism coincides with the image of H_{Φ∩X_{i+2}}^{q−1}(X_{i+2}, F). Then there is a canonical isomorphism

H_{Φ}^{q}(X, F) →^{∼} lim_{⟵} H_{Φ∩X_{i}}^{q}(X_{i}, F) .

Remark 4.3. An analog of [Gro, Proposition 3.10.2] for étale cohomology groups of non-Archimedean analytic spaces is [Ber2, Proposition 6.3.12]. In the formulation of the latter only the assumption that X is a union of all X_{i}’s was made. This is not enough, and one has to make the stronger assumption that X is a union of the topological interiors of all X_{i}’s (it guarantees that F (X) →^{∼} lim_{⟵} F (X_{i}) for any sheaf F on X).

Proof. First of all, by the above remark and the assumptions on X_{i}’s and Φ, for any abelian sheaf G on X one has Γ_{Φ}(X, G) →^{∼} lim_{⟵} Γ_{Φ∩X_{i}}(X_{i}, G). We claim that, given an injective abelian sheaf J on X and a closed subset Y ⊂ X, the canonical map Γ_{Φ}(X, J) → Γ_{Φ∩Y} (Y, J) is surjective. Indeed, let g be an element from Γ_{Φ∩Y} (Y, J) and let B be its support. By [God, Ch. II, Theorem 3.3.1], g is the restriction of a section g′ of J over an open neighborhood U of B in X. Furthermore, let A ∈ Φ be such that B = A ∩ Y, and let A′ ∈ Φ be a neighborhood of A in X. Shrinking U, we may assume that U ⊂ A′. Since J is injective, the map Γ (X, J) → Γ (U ∐(X\A′), J) = Γ (U, J) ⊕ Γ (X\A′, J) is surjective. It follows that there exists an element f ∈ Γ (X, J) whose restriction to U is g′ and the restriction to X\A′ is zero. Since the support of f lies in A′, one has f ∈ Γ_{Φ}(X, J), and the claim follows.

The claim implies that the pullback of J at any closed subset Y ⊂ X is a (Φ ∩ Y)-soft sheaf on Y, i.e., for any A ∈ Φ ∩ Y, the canonical map Γ_{Φ∩Y} (Y, J) → Γ (A, J) is surjective (see [God, Ch. II, §3.5]). Thus, given an injective resolution 0 → F → J^{0} → J^{1} → … of F, there is a commutative diagram

0 → Γ_{Φ}(X, J^{0}) → Γ_{Φ}(X, J^{1}) → Γ_{Φ}(X, J^{2}) → …
↓ ↓ ↓
0 → Γ_{Φ∩X_{i}}(X_{i}, J^{0}) → Γ_{Φ∩X_{i}}(X_{i}, J^{1}) → Γ_{Φ∩X_{i}}(X_{i}, J^{2}) → …

in which the first and second rows give rise to the groups H_{Φ}^{q}(X, F) and H_{Φ∩X_{i}}^{q}(X_{i}, F), respectively, and the vertical arrows are surjections. The injectivity of the map considered is verified by a simple diagram search in the same way as in the proof of [Ber2, Proposition 6.3.12], and verification of its surjectivity is even easier (and because of that it was omitted in loc.cit) and goes as follows.

Let \overline{β}_{i} ∈ H_{Φ∩X_{i}}^{q}(X_{i}, F), i ≥ 1, be a compatible system. Assume that, for i ≥ 1, we constructed elements β_{j} ∈ Γ_{Φ∩X_{j}}(X_{j}, J^{q}) each from the class of \overline{β}_{j}, 1 ≤ j ≤ i, with β_{j+1}|_{X_{j}} = β_{j} for 1 ≤ j ≤ i−1, and let β′_{i+1} be an element from

the class of \overline{β}_{i+1}. Then β′_{i+1}|_{X_{i}} = β_{i} + dγ_{i} for some γ_{i} ∈ Γ_{Φ∩X_{i}}(X_{i}, J^{q−1}). If α ∈ Γ_{Φ}(X, J^{q−1}) is such that α|_{X_{i}} = γ_{i}, then for the element β_{i+1} = β′_{i+1} − dα|_{X_{i+1}}, we have β_{i+1}|_{X_{i}} = β_{i}. By the remark at the beginning of the proof, there exists an element β ∈ Γ_{Φ}(X, J^{q}) such that β|_{X_{i}} = β_{i} for all i ≥ 1. Then dβ = 0, and the surjectivity follows. ⊓ ⊔

Proof of Theorem 4.1. Step 1. First of all, the isomorphism H^{q}(X_{s}^{An}, Z) →^{∼} H^{q}(λ_{s}^{−1}(0), Z) = H^{q}(X_{s}^{an}, Z) follows from Proposition 2.3. The isomorphisms

H^{q}(X^{h}×]0, 1], Z) →^{∼} H^{q}(λ^{−1}(]0, 1]), Z) and

H^{q}(X_{η}^{h}×]0, 1], Z) →^{∼} H^{q}(λ_{η}^{−1}(]0, 1]), Z)

follow from Corollary 3.2.

Step 2. To get the isomorphisms

H^{q}(X^{An}, Z) →^{∼} H^{q}(λ^{−1}(0), Z) and H^{q}(X_{η}^{An}, Z) →^{∼} H^{q}(λ_{η}^{−1}(0), Z) ,

we assume that r is sufficiently small so that the groups H^{q}(X_{t}^{h}, Z), t ∈ D^{∗}(r), form a local system for all q ≥ 0 and, therefore, R^{q}ψ_{η∗}(Z_{X_{η}^{h}}) are locally constant quasi-unipotent sheaves of finitely generated abelian groups for all q ≥ 0, where ψ is the canonical morphism X^{h} → D(r). Let X̃^{h} and X̃_{η}^{h} denote the images of λ^{−1}(]0, 1]) and λ_{η}^{−1}(]0, 1]) in X^{h}×]0, 1] and X_{η}^{h}×]0, 1], respectively. (If X = Spec(O_{C,0}), they will be denoted by D̃(r) and D̃^{∗}(r).) It suffices to show that H_{Φ}^{q}(X̃^{h}, Z) = 0 and H_{Φ_{η}}^{q}(X̃_{η}^{h}, Z) = 0 for all q ≥ 0,

where Φ and Φ_{η} are families of supports in X̃^{h} and X̃_{η}^{h} consisting of the closed subsets which are also closed in X^{An} and X_{η}^{An}, respectively.

Consider the following commutative diagrams in which all squares are cartesian

D̃(r) ↪ D(r) ⟵ [0, r[  D̃^{∗}(r) ↪ D^{∗}(r) ⟵  ]0, r[
↑ φ̃  ↑ φ  ↑  ↑ φ̃_{η}  ↑ φ_{η}  ↑
X̃^{h} ↪ X^{An} ⟵ X_{0}^{An}  X̃_{η}^{h} ↪ X_{η}^{An} ⟵ X_{η}^{an}×]0, r[

Since all of the vertical maps are compact, there are spectral sequences (with initial terms E_{2}^{p,q})

H_{Φ}^{p}(D̃(r), R^{q}φ̃_{∗}Z_{X^{h}}) ⟹ H_{Φ}^{p+q}(X̃^{h}, Z) and

H_{Φ_{η}}^{p}(D̃^{∗}(r), R^{q}φ̃_{η∗}Z_{X_{η}^{h}}) ⟹ H_{Φ_{η}}^{p+q}(X̃_{η}^{h}, Z) ,

where Φ and Φ_{η} in the E_{2}^{p,q} terms denote the similar families of supports in D̃(r) and D̃^{∗}(r), respectively. Thus, it suffices to verify the following fact. Let L be an abelian sheaf on D(r) whose restriction to D^{∗}(r) is locally constant and quasi-unipotent, and let π denote the canonical projection D̃(r) → D(r). Then (∗) H_{Φ}^{p}(D̃(r), π^{∗}L) = 0 and (∗_{η}) H_{Φ_{η}}^{p}(D̃^{∗}(r), π^{∗}L) = 0 for all p ≥ 0. Both equalities are proved in the same way using Lemma 4.2 as follows.

The equality (∗). The space D(r) is a union of the closed discs E(ρ) = {x ∈ D(r)||T (x)| ≤ ρ\} with ρ < r. Let Ẽ(ρ) = E(ρ)∩D̃(r) = \{(y, t)||T (y)|_{∞} ≤ ρ^{\frac{1}{t}} \}. Then D̃(r) is a union of all Ẽ(ρ) with ρ < r, and, if ρ < ρ′, then E(ρ) and Ẽ(ρ) are contained in the topological interiors of E(ρ′) and Ẽ(ρ′) in D(r) and D̃(r), respectively. It follows easily that a closed subset B ⊂ D̃(r) is closed in D(r) if and only if B ∩ Ẽ(ρ) is closed in E(ρ) for all ρ < r. Since the spaces E(ρ) are compact, from Lemma 4.2 it follows that to prove the equality (∗), it suffices to show that H_{c}^{q}(Ẽ(ρ), π_{ρ}^{∗}L) = 0 for all q ≥ 0, where π_{ρ} is the canonical projection Ẽ(ρ) → E(ρ).

One has π_{ρ}^{−1}(0) →^{∼}]0, 1] and, for y ≠ 0, π_{ρ}^{−1}(y) →^{∼} [t_{y}, 1], where 0 < t_{y} ≤ 1

is such that |T (y)|_{∞} = ρ^{\frac{1}{ty}}. It follows that (R^{q}π_{ρ!}(π_{ρ}^{∗}L))_{y} is zero, if q ≥ 1 or q = 0 and y = 0, and coincides with L_{y}, if q = 0 and y ≠ 0. This means that R^{q}π_{ρ!}(π_{ρ}^{∗}L) is zero for q ≥ 1, and coincides with j_{ρ!}(j_{ρ}^{∗}L) for q = 0, where j_{ρ} is the canonical open embedding E^{∗}(ρ) ↪ E(ρ). The Leray spectral sequence E_{2}^{p,q} = H_{c}^{p}(E(ρ), R^{q}π_{ρ!}(π_{ρ}^{∗}L)) ⟹ H_{c}^{p+q}(Ẽ(ρ), π_{ρ}^{∗}L) implies that H_{c}^{q}(Ẽ(ρ), π_{ρ}^{∗}L) = H_{c}^{q}(E^{∗}(ρ), L) for all q ≥ 0. Thus, the equality (∗) is a consequence of the following simple fact: H_{c}^{q}(E^{∗}(ρ), L) = 0, q ≥ 0, for any abelian quasi-unipotent sheaf L on E^{∗}(ρ).

If L is constant, the above fact follows from the long exact sequence of cohomology with compact supports associated with the maps

E^{∗}(ρ) ↪^{j_{ρ}} E(ρ) ⟵ {0} .

It follows easily that the same is true for any unipotent abelian sheaf L. Assume now that L is quasi-unipotent. Then there exists n ≥ 1 such that the pullback of L under the n-power map φ : E^{∗}(ρ^{\frac{1}{n}}) → E^{∗}(ρ) : z ↦ z^{n} is unipotent. By the previous case, H_{c}^{q}(E^{∗}(ρ^{\frac{1}{n}}), φ^{∗}L) = 0 for all q ≥ 0. The spectral sequence E_{2}^{p,q} = H^{p}(Z/nZ, H_{c}^{q}(E^{∗}(ρ^{\frac{1}{n}}), φ^{∗}L)) ⟹ H_{c}^{p+q}(E^{∗}(ρ), L) implies required fact for such L.

The equality (∗_{η}) (see also Remark 4.4(i)). The space D^{∗}(r) is a union of the closed annuli A_{ρ} = \{x ∈ D(r)|ρ ≤ |T (x)| ≤ r − ρ\} with 0 < ρ < \frac{r}{2}. Let Ã_{ρ} = A_{ρ} ∩ D̃^{∗}(r) = \{(y, t)|ρ^{\frac{1}{t}} ≤ |T (y)|_{∞} ≤ (r − ρ)^{\frac{1}{t}} \}. Then D̃^{∗}(r) is a union of Ã_{ρ} with 0 < ρ < \frac{r}{2} and, for ρ < ρ′, A_{ρ} and Ã_{ρ} lie in the topological interiors of A_{ρ′} and Ã_{ρ′} in D^{∗}(r) and D̃^{∗}(r), respectively. It follows that a closed subset B ⊂ D̃^{∗}(r) is closed in D^{∗}(r) if and only if B ∩Ã_{ρ} is closed in A_{ρ} for all 0 < ρ < \frac{r}{2}. Since the spaces A_{ρ} are compact, from Lemma 4.2 it follows that to prove the equality (∗_{η}) it suffices to show that H_{c}^{q}(Ã_{ρ}, π_{ρ}^{∗}L) = 0 for all q ≥ 0, where π_{ρ} is the canonical projection Ã_{ρ} → E^{∗}(r − ρ).

Notice that, in comparison with the previous case, the preimage of any point of E^{∗}(ρ) under the latter map is always a closed interval or a point. It follows that R^{q}π_{ρ∗}(π_{ρ}^{∗}L) is zero, if q ≥ 1, and coincides with L|_{E^{∗}(r−ρ)}, if

q = 0. The Leray spectral sequence of the map π_{ρ} implies that H_{c}^{q}(Ã_{ρ}, π_{ρ}^{∗}L) = H_{c}^{q}(E^{∗}(r − ρ), L) for all q ≥ 0, and the equality (∗_{η}) follows from the fact we already verified.

Step 3. It remains to show that, if X is proper and strictly semi-stable over O_{C,0}, then the unmarked vertical arrows in the extreme left and right columns induce isomorphisms of cohomology groups. In this case, X_{s}^{h} is even a strong deformation retract of X^{h}, by the results of C. H. Clemens (see [Cle, §6]), and both maps X_{η}^{an}×]0, r[→ X_{0}^{An} and X_{s}^{an} → X_{0}^{An} are homotopy equivalences, by results from [Ber5], we are going to explain.

Consider a more general situation. Let k be an arbitrary field (instead of C) provided with the trivial valuation. The ring of formal power series k[[z]] coincides with the ring O_{A^{1},0} of formal power series convergent in an open neighborhood of zero in the affine line A^{1} over k as well as with the ring O(D(1)) of those power series which are convergent in the open disc D(1) (of radius one with center at zero). The formal spectrum 𝔛 = Spf(k[[z]]) is a special formal scheme over k^{∘} = k in the sense of [Ber4], and its generic fiber 𝔛_{η} coincides with D(1). Notice that there is a canonical homeomorphism [0, 1[→^{∼} D(1): ρ ↦ P_{ρ}, where P_{ρ} is defined by |z(P_{ρ})| = ρ.

Let X be a scheme of finite type over k[[z]]. For any number 0 < r < 1, the ring k[[z]] coincides with the k-affinoid algebra k{r^{−1}z}, the algebra of analytic functions on the closed disc E(r) ⊂ A^{1} (which is canonically homeomorphic to [0, r]), and so there is an associated k-analytic space Y^{an}(r). If r < r′, X^{an}(r) is identified with a closed analytic subdomain of X^{an}(r′), we set X^{an} = ∪X^{an}(r). There is a canonical surjective morphism φ : X^{an} → D(1) →^{∼} [0, 1[. The fiber φ^{−1}(ρ) at ρ ∈ [0, 1[ is identified with the H(P_{ρ})-analytic space X_{ρ}^{an} associated with the scheme X ⊗_{k[[z]]} H(P_{ρ}). The formal completion X̂ of X along its closed fiber X_{s} is a special formal scheme, and there is a canonical morphism of strictly k-analytic spaces X̂_{η} → X^{an} whose composition with the above morphism φ is induced by the canonical morphism of formal schemes X̂ → 𝔛. If X is separated and of finite type over k[[z]], X̂_{η} is identified with a closed strictly analytic subdomain of X^{an}. If X is proper over k[[z]], then X̂_{η} →^{∼} X^{an}. If X is semi-stable over k[[z]], then so is X̂.

Assume now that 𝔜 be a semi-stable formal scheme over 𝔛 = Spf(k[[z]]) (or, more generally, poly-stable in the sense of [Ber5]). For ρ ∈ [0, 1[, we set 𝔜_{ρ} = 𝔜 ×𝔛 Spf(H(P_{ρ})^{∘}). It is a semi-stable formal scheme of H(P_{ρ}), and there are canonical isomorphisms 𝔜_{ρ,η} →^{∼} 𝔜_{η,ρ} and 𝔜_{ρ,s} →^{∼} 𝔜_{s}. In [Ber5, §5], we constructed a closed subset S(𝔜̂_{ρ}) (the skeleton of 𝔜_{ρ}), and a strong deformation retraction Φ_{ρ} : 𝔜_{ρ,η} ×[0, 1] → 𝔜_{ρ,η} of 𝔜_{ρ,η} to the skeleton S(𝔜̂_{ρ}). We denote by S(𝔜/𝔛) the union of S(𝔜̂_{ρ}) over all ρ ∈ [0, 1[, and by Φ the mapping 𝔜_{η} × [0, 1] → 𝔜_{η} that coincides with Φ_{ρ} at each fiber of φ. In [Ber5, §4], we also associated with the closed fiber 𝔜_{s} of 𝔜 a simplicial set C(𝔜_{s}) which has a geometric realization |C(𝔜_{s})|. Thus, to prove the claim, it suffices to verify the following two facts:

(a) the mapping Φ : 𝔜_{η} × [0, 1] → 𝔜_{η} is continuous and compact, and (b) there is a homeomorphism |C(𝔜_{s})| × 𝔛_{η} → S(𝔜/𝔛) which commutes

with the canonical projections to [0, 1[.

(a) follows from the proof of [Ber5, Theorem 7.1]. In the formulation of the latter the formal scheme 𝔛 was in fact assumed to be locally finitely presented over the ring of integers of the ground field (in our case k^{∘} = k), but its proof only uses the fact that the morphism 𝔜 → 𝔛 is poly-stable and works in the case when 𝔛 is an arbitrary special formal scheme.

(b) By the properties of the skeleton established in [Ber5, §5], the situation is easily reduced to the case when 𝔜 = Spf(B), where

B = k[[z]]{T_{0}, …, T_{m}}/(T_{0} · . . . T_{n} − z) , 0 ≤ n ≤ m .

If n = 0, then S(𝔜/𝔛) →^{∼} 𝔛_{η} →^{∼} [0, 1[, |C(𝔜_{s})| is a point, and (b) follows. Assume that n ≥ 1. Then S(𝔜/𝔛) is identified with the set {(P_{ρ}, r_{0}, …, r_{n}) ∈ 𝔛_{η} × [0, 1]^{n+1}|r_{0} · . . . · r_{n} = ρ}, and |C(𝔜_{s})| is identified with the set {(u_{0}, …, u_{n}) ∈ [0, 1]^{n+1}|u_{0}+…+u_{n} = 1}. The required map |C(𝔜_{s})|×𝔛_{η} → S(𝔜/𝔛) takes a point ((u_{0}, …, u_{n}), ρ) to (P_{ρ}, (ρ^{u_{0}}, …, ρ^{u_{n}})). ⊓ ⊔

Remarks 4.4. (i) The equality (∗_{η}) can be established in a different way. Namely, we claim that there is a strong deformation retraction of D^{∗}(r) to the subset λ_{η}^{−1}(0) (identified with ]0, r[). Indeed, let P_{ρ} denote the point of λ_{η}^{−1}(0) that corresponds to ρ ∈]0, r[ (it is a unique point from λ_{η}^{−1}(0) with |T (P_{ρ})| = ρ). Then the required strong deformation retraction Ψ : D^{∗}(r)×[0, 1] → D^{∗}(r) (with Ψ(x, 1) = x and Ψ(x, 0) ∈ λ_{η}^{−1}(0)) is defined as follows:

(1) if (ρe^{iφ}, s) ∈ D̃^{∗}(r), then Ψ((ρe^{iφ}, s), t) = (ρ^{\frac{1}{t}} e^{iφ}, st) ∈ D̃^{∗}(r) for t ∈]0, 1];

(2) Ψ((ρe^{iφ}, s), 0) = P_{ρ^{s}};

(3) if ρ ∈]0, r[, then Ψ(P_{ρ}, t) = P_{ρ} for all t ∈ [0, 1].

The claim implies that, if the sheaf L is constant, then H^{p}(D^{∗}(r), π^{∗}L) →^{∼} H^{p}(]0, r[, π^{∗}L) = 0 and, therefore, H_{Φ_{η}}^{p}(D̃^{∗}(r), π^{∗}L) = 0 for all p ≥ 0. Thus, the equality (∗_{η}) is true for constant L, and is easily extended to arbitrary quasi-unipotent sheaves L.

(ii) Let X be a connected projective scheme over F that admits a strictly semi-stable reduction over O_{C,0}. Then the fundamental group of X^{an} is isomorphic to a quotient of the fundamental group of the fiber X_{t}^{h}, t ∈ D^{∗}(r), and, in particular, if the latter is simply connected, then so is X^{an}. Indeed, let Y be a projective strictly semi-stable scheme over O_{C,0} with Y_{η} = X. The canonical morphism Y → Spec(O_{C,0}) has a section Spec(O_{C,0}) → Y (with the image in the smooth locus of that morphism), and, therefore, for some 0 < r < 1, the canonical morphism Y^{h} → D(r) has a section D(r) → Y^{h}. It follows the canonical surjection π_{1}(X^{h}) → π_{1}(D^{∗}) has a section whose image lies in the kernel of the canonical homomorphism π_{1}(X^{h}) → π_{1}(Y^{h}) and, therefore, the image of π_{1}(X_{t}^{h}) in π_{1}(Y^{h}) coincides with that of π_{1}(X^{h}). But the canonical homomorphism π_{1}(X^{h}) → π_{1}(Y^{h}) is surjective since the preimage of X^{h} in a universal covering of Y^{h} is connected (it is the complement of a Zariski closed subset of a connected smooth complex analytic space). Thus, π_{1}(Y^{h}) is a quotient of π_{1}(X_{t}^{h}). Furthermore, by the result of C. Clemens ([Cle]) used in the proof of Theorem 4.1, Y_{s}^{h} is a strong deformation retract of Y^{h}, i.e., π_{1}(Y_{s}^{h}) is a quotient of π_{1}(X_{t}^{h}). If C is the simplicial set associated with the scheme Y_{s}, there is a surjective homomorphism from π_{1}(Y_{s}^{h}) to the fundamental group of the geometric realization |C| of C. It remains to notice that, by [Ber5, Theorem 5.2], X^{an} is homotopy equivalent to |C|. (I am due to O. Gabber for the above reasoning.)

(iii) Assume that X is proper and strictly semi-stable over O_{C,0}. It is very likely that all of the maps in the diagram from the beginning of this section, except those marked by ∗, are in fact homotopy equivalences.

(iv) It would be interesting to know whether Theorem 4.1 is true for not necessarily proper schemes.

5 An interpretation of the weight zero subspaces

Let X be a proper scheme over F, and let 0 < r < 1 be small enough so that the isomorphisms H^{q}(X^{an}×]0, r[, Z) = H^{q}(X^{an}, Z) →^{∼} H^{q}(X^{An}, Z) from Theorem 4.1 take place. They give rise to homomorphisms H^{q}(X^{an}, Z) → H^{q}(X^{h}, Z), q ≥ 0. Let \overline{D}^{∗}(r) → D^{∗}(r) be a universal covering of D^{∗}(r). The fundamental group π_{1}(D^{∗}) = π_{1}(D^{∗}(r), t) (which does not depend on the choice of r and a point t ∈ D^{∗}(r)) acts on \overline{D}^{∗}(r) and, therefore, it acts on \overline{X}^{h} = X^{h} ×_{D^{∗}(r)} \overline{D}^{∗}(r). Furthermore, let F^{a} be the field of functions meromorphic in the preimage of an open neighborhood of zero in \overline{D}^{∗}(r), which are algebraic over F. It is an algebraic closure of F and, in particular, π_{1}(D^{∗}) acts on F^{a}. Let K^{a} be the corresponding algebraic closure of K, K̂^{a} the completion of K^{a}, and \overline{X}^{an} = (X^{an} ⊗_{F} K̂^{a})^{an}. As was mentioned in the introduction, the constructed homomorphisms of cohomology groups induce π_{1}(D^{∗})-equivariant homomorphisms H^{q}(\overline{X}^{an}, Z) → H^{q}(\overline{X}^{h}, Z), q ≥ 0.

Theorem 5.1. The above homomorphisms give rise to π_{1}(D^{∗})-equivariant isomorphisms

H^{q}(\overline{X}^{an}, Q) →^{∼} W_{0}H^{q}(\overline{X}^{h}, Q), q ≥ 0 .

Proof. Consider first the case when X = Y_{η}, where Y is a projective strictly semi-stable scheme over O_{C,0}. By Corollary 2.4, in the commutative diagram of Theorem 4.1 for such Y the maps from the lower row give rise to an isomorphism H^{q}(Y^{an}, Q) →^{∼} W_{0}H^{q}(Y_{s}^{h}, Q) and, by Steenbrink’s work [St], the homomorphisms H^{q}(Y_{s}^{h}, Z) →^{∼} H^{q}(Y^{h}, Z) → H^{q}(X^{h}, Z) → H^{q}(\overline{X}^{h}, Z) give rise to an isomorphism W_{0}H^{q}(Y_{s}^{h}, Q) →^{∼} W_{0}H^{q}(\overline{X}^{h}, Q), q ≥ 0. Since the residue field of K is algebraically closed, the canonical map \overline{X}^{an} → X^{an} is a homotopy equivalence (see [Ber5, §5]) and, in particular, H^{q}(X^{an}, Z) →^{∼} H^{q}(\overline{X}^{an}, Z). Thus, the required isomorphism follows from Theorem 4.1. Consider now the case when X is projective and smooth over F. One can find an integer n ≥ 1 such that, if F′ is the cyclic extension of F of degree n in F^{a}, then the scheme X′ = X ⊗_{F} F′ is of the previous type over F′. The extensions F^{a} ⊃ F′ ⊃ F correspond to morphisms \overline{D}^{∗}(r) → D^{∗}(r^{\frac{1}{n}}) ⟶^{z↦z^{n}} D^{∗}(r), and so there is a canonical isomorphism of complex analytic spaces \overline{X}^{′h} →^{∼} \overline{X}^{h}. The latter gives rise to isomorphisms H^{q}(\overline{X}^{h}, Z) →^{∼} H^{q}(\overline{X}^{′h}, Z) of cohomology groups provided with the limit mixed Hodge structures (see [GNPP, p. 126]). Similarly, one has canonical isomorphisms H^{q}(\overline{X}^{an}, Z) →^{∼} H^{q}(\overline{X}^{′an}, Z), and the required isomorphism follows from the previous case. Finally, if X is an arbitrary proper scheme over F, one gets the required isomorphism using the same reasoning as in the proof of Corollary 2.4, i.e., using a proper hypercovering X_{•} → X with projective and smooth X_{n}’s and the fact that the functor H ↦ W_{0}H on the category of rational mixed Hodge structures H with W_{i}H = 0 for i < 0 is exact. ⊓ ⊔

Corollary 5.2. In the above situation, the following is true:

(i) H^{q}(X^{an}, Q) →^{∼} (W_{0}H^{q}(\overline{X}^{h}, Q))^{T =1};

(ii)if X is projective and smooth, then

H^{q}(X^{an}, Q) →^{∼} ((T^{m} − 1)^{i}H^{q}(X_{t}^{h}, Q))^{T =1} .

Here T is the canonical generator of π_{1}(D^{∗}), and m is a positive integer for which the action of (T^{m} − 1)^{i+1} on H^{q}(X_{t}^{h}, Q) is zero (see the introduction).

Proof. It suffices to show that the canonical map

H^{q}(X^{an}, Q) → H^{q}(\overline{X}^{an}, Q)^{T =1}

is an isomorphism. For this we recall that, by [Ber5, Theorem 10.1], one has H^{q}((X ⊗_{F} K′)^{an}, Q) →^{∼} H^{q}(\overline{X}^{an}, Q) for some finite Galois extension K′ of K in K^{a}. Since the topological space X^{an} is the quotient of (X ⊗_{F} K′)^{an} by the action of the Galois group of K′ over K, the required fact follows from [Gro, Corollary 5.2.3]. ⊓ ⊔ Remark 5.3. As was mentioned at the end of the introduction, Theorem 5.1 is an analog of a similar description of the weight zero subspaces in the l-adic cohomology groups of algebraic varieties over a local field, which holds for arbitrary separated schemes of finite type (see [Ber6]). And so it is very likely that the isomorphism of Theorem 5.1 also takes place for arbitrary separated schemes of finite type over F. The latter would follow from the validity of Theorem 4.1 for that class of schemes (see Remark 4.4(iv)), and is easily extended to separated smooth schemes. (Recall that the theory of limit mixed Hodge structures on the cohomology groups H^{q}(\overline{X}^{h}, Q) for separated schemes X of finite type over F is developed in [EZ].)

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