j. differential geometry 60 (2002) 345-354

COHOMOLOGY THEORY IN BIRATIONAL GEOMETRY

CHIN-LUNG WANG

Abstract

This is a continuation of [9], where it was shown that K-equivalent complex projective manifolds have the same Betti numbers by using the theory of p-adic integrals and Deligne’s solution to the Weil conjecture. The aim of this note is to show that with a little more book-keeping work, namely by applying Faltings’ p-adic Hodge Theory, our p-adic method also leads to the equivalence of Hodge numbers — a result which was previously known via motivic integration.

1. Introduction

Mori’s minimal model theory has proven to be important in various geometric problems and is also important in our philosophical view point of birational geometry. In order to understand the relation between birational but not isomorphic minimal models in dimensions bigger than two, the notion of K-equivalence was developed to serve as a formal analogue, but with the advantage of independence from the existence problem of minimal models. This applies to the most interesting case of birational Calabi-Yau manifolds which was studied extensively in the last decade.

In dimension three, any birational map between minimal models can be decomposed as a composite of flops [7]. This gives very precise information needed in analyzing birational minimal threefolds. However, the only known proof of this result relies on detailed classification of terminal singularities, hence is out of reach in higher dimensions. Because of this, new dimension-free approaches are needed for the study of K-equivalent manifolds.

Supported by NSC 89-2115-M-002-012.

Received April, 2002.

In [9], an integration formalism was formulated to compare numerical invariants of K-equivalent manifolds. In particular, the p-adic integral has been used to prove the equivalence of Betti numbers and the motivic integral was used to prove the equivalence of Hodge numbers (cf. [9], §5.5, [4], [1], [2]). More recently, by formally viewing the intersection theory as an integration theory, the author has shown that the complex elliptic genera are the most general Chern numbers invariant under K-equivalence [10].

The aim of this note is to present a proof of the equivalence of Hodge numbers along the original line using p-adic integral and the Weil conjecture. This was announced with a sketched proof in [11]. The new input needed here is the so-called p-adic Hodge Theory developed by Fontaine and Messing [6] and completed by Faltings [5]. It turns out that one may apply the existing theorems quite straightforwardly except a few minor technicalities. One is related to the Cěbotarev density theorem that the zeta functions determine only the semisimplifications of the p-adic étale cohomology as Galois representation. The other is the reduction procedure from finitely generated fields over ℚ to number fields. Fortunately, these problems can all be handled by quite standard tricks and the proof goes through.

The author is grateful to C.-L. Chai and J.-K. Yu for useful discussions on p-adic Hodge Theory. Thanks are also to I.-H. Tsai for his criticism on the proof of [9], Theorem 1.4 and to J.-D. Yu for his careful reading of an early draft.

2. K-partial ordering in a birational class

For a birational map f : X −−→ X′ between two ℚ-Gorenstein (complex projective) varieties, we say that X ≤_{K} X′ (resp. X <_{K} X′) if there is a birational correspondence (ϕ, ϕ′) : Y → X × X′ extending f with Y smooth, such that ϕ^{∗}K_{X} ≤_{ℚ} ϕ′^{∗}K_{X′} (resp. <_{ℚ}) as divisors. These relations are easily seen to be independent of the choice of Y. Notice that X ≤_{K} X′ and X ≥_{K} X′ imply X =_{K} X′, that is ϕ^{∗}K_{X} =_{ℚ} ϕ′^{∗}K_{X′}. In this case, we say that X and X′ are K-equivalent.

In this K-partial ordering, divisorial contractions and flips will decrease its K-level while flops induce K-equivalence. It is easy to see that K-equivalent terminal varieties are isomorphic in codimension one. In fact, more is true in general:

Theorem 2.1. Let f : X −−→ X′ be a birational map between two varieties with canonical singularities. Suppose that the exceptional locus Z ⊂ X is proper and that K_{X} is nef along Z, then X ≤_{K} X′. Moreover, If X′ is terminal then codim_{X}Z ≥ 2.

(This is Theorem 1.4 in [9]; a better presentation of its proof is included in the appendix.) In particular, birational minimal models are K-equivalent.

Conjecture 2.2. For complex projective manifolds X and X′ with X ≤_{K} X′, the canonical morphism T : H^{k}(X, ℚ) → H^{k}(X′, ℚ) induced from the graph closure \overline{Γ}_{f} ⊂ X ×X′ is a monomorphism which preserves the rational Hodge structures. More generally, T induces a monomorphism of motives, e.g., in the sense of Galois representations or in the sense of Chow motives.

While it is of fundamental importance to study the cycle \overline{Γ}_{f} ⊂ X×X′ directly, we instead restrict ourselves to a numerical version in this note. Namely, we prove in §4 the main result of this note:

Theorem 2.3. Let X and X′ be two K-equivalent complex projective manifolds. Then h^{p,q}(X) = h^{p,q}(X′) for all p, q. More precisely, if X =_{K} X′ is defined over a finitely generated field F over ℚ, then for any prime ℓ,

H_{et}^{j}(X_{F} , ℚ_{ℓ})^{ss} ≅ H_{et}^{j}(X′_{F} , ℚ_{ℓ})^{ss}

as Gal (F /F) representations.

3. p-adic integration and étale cohomology

We start by recalling the construction in [9]. We will assume that X and X′ are smooth projective and K-equivalent, though the construction also works for log-terminal varieties. Take an integral model of the K-equivalence diagram over Spec S with S a finitely generated ℤ-algebra:

\xymatrix{ & Y \ar[dl]_{ϕ} \ar[dd] \ar[dr]^{ϕ′} \\ X \ar[dr] &  & X′ \ar[dl] \\  & Spec S}

For almost all maximal ideals P in S, in fact a Zariski open dense set in the maximal spectrum of S, we have good reductions of X, X′, Y, ϕ and ϕ′. In such cases, let R = Ŝ_{P} be the completion of S at P with residue field k_{P} := R/P ≅ 𝔽_{q}, q = p^{r} for some r. For ease of notation, we use the same symbol to denote the corresponding object over Spec R. Let {U_{i}} be a Zariski open cover of X such that K_{X}|_{U_{i}} is trivial for each i with generator Ω_{i} a regular n-form on U_{i}, where n = dim X. Then for a compact open subset A ⊂ U_{i}(R) ⊂ X(R), we define its p-adic measure by

μ_{X}(A) ≡ ∫_{A} |Ω_{i}|_{p}.

This is independent of the choice of the generator Ω_{i}. The p-adic measure of X(R) and X′(R) are the same by the change of variable formula and X =_{K} X′. By a direct extension of Weil’s formula [12], we see in [9] that (let X be the special fiber over Spec 𝔽_{q})

μ_{X}(X(R)) = \frac{|X(𝔽_{q})|}{q^{n}}.

By applying this to finite extensions of 𝔽_{q}, we conclude that X and X′ have the same local zeta functions Z(X, t) = Z(X′, t) with

Z(X, t) := exp ( ∑_{k≥1} |X(𝔽_{q^{k}})|\frac{t^{k}}{k} ).

Knowing this for one P already allows us to apply Grothendieck-Deligne’s solution to the celebrated Weil conjecture [3] to conclude that K-equivalent manifolds have the same Betti numbers. Indeed, let

P_{j}(t) = det (1 − t Fr_{q} | H_{et}^{j}(X_{\overline{𝔽}_{q}}, ℚ_{ℓ}))

be the characteristic polynomial of the Frobenius map Fr_{q} acting on the ℓ-adic étale cohomologies of X_{\overline{𝔽}_{q}} for any fixed ℓ ≠ p, then Grothendieck’s Lefschetz trace formula implies that

Z(X, t) = \frac{P_{1}(t) … P_{2n−1}(t)}{P_{0}(t)P_{2}(t) … P_{2n}(t)}.

Moreover, Deligne showed that P_{j}(t) ∈ ℤ[t] which is independent of the choices of ℓ and all roots of P_{j}(t) have absolute value q^{−j/2} (Riemann Hypothesis). This clearly implies that X and X′ have the same ℓ-adic Betti numbers, hence by comparison theorem X and X′ have the same ordinary Betti numbers.

4. Global ℓ-adic representations

In this section we prove Theorem 2.3 using two technical devices discussed in §5 and §6. The basic observation here is that in fact more is true by putting together the information provided by all such P ’s. Let us assume that X =_{K} X′ are defined over S such that the quotient field F of S is a number field. The general case can be reduced to the number field case by the standard trick in §6. Under this assumption we then know that X and X′ have good reductions and have the same local zeta functions for all P ∈ Spec S\A with A a finite set.

Consider the following two semisimplifications of ℓ-adic cohomologies

H_{et}^{j}(X_{F} , ℚ_{ℓ})^{ss} and H_{et}^{j}(X′_{F} , ℚ_{ℓ})^{ss}

as (global) ℓ-adic representations of Gal(F /F), denoted by ρ and ρ′. In the language of [8], these are integral representations, meaning that the associate local representations are unramified and have integral characteristic polynomial for the Frobenius generator of Gal(k_{P} /k_{P}) for all but finitely many P. This is indeed the case by Deligne’s result since for P with good reduction, the local ℓ-adic representation is exactly H_{et}^{j}(X_{k_{P}} , ℚ_{ℓ})^{ss} with characteristic polynomial P_{j}(t) as before.

By the Cěbotarev density theorem ([8], Ch1, §2) that ⋃_{P ∉A} Gal(k_{P} /k_{P})

is dense in Gal (F /F)/(ker ρ∩ker ρ′) and the fact that rational semisimple representations are characterized by their trace functions (characters), this implies that

H_{et}^{j}(X_{F} , ℚ_{ℓ})^{ss} ≅ H_{et}^{j}(X′_{F} , ℚ_{ℓ})^{ss}

as Gal(F /F) representations.

Remark 4.1. From the point of view of motive theory, the above argument is simply the one showing that L function is equivalent to the semisimplification of the corresponding Galois representations. Also there is a semisimplicity conjecture stating that the cohomological ℓ-adic representations are always semisimple.

Now we select a prime P ∈ Spec S with char k_{P} = ℓ. Let K be the completion of F at P, then by base change theorem we also get

H_{et}^{j}(X_{K}, ℚ_{ℓ})^{ss} ≅ H_{et}^{j}(X′_{K}, ℚ_{ℓ})^{ss}

as Gal(K/K) representations (usually highly ramified!) — here we do not even need to require X or X′ to have good reductions at P. By Faltings’ Hodge-Tate decomposition theorem in the next section, this then implies the equivalence of ℚ_{ℓ} (and hence ℚ) Hodge numbers.

5. p-adic Hodge theory

In this section we recall the p-adic Hodge Theory that we are going to apply. Following the usual convention, we will switch the prime number ℓ to p.

Let X a smooth projective manifold over a p-adic field K. Let G = Gal(K/K) and ℂ_{p} be the completion of K. Then there exists a natural G-equivariant isomorphism [5], the so-called Hodge-Tate decomposition:

⨁_{i} (ℂ_{p} ⊗_{K} H^{m−i}(X_{K}, Ω^{i})(−i)) ≅ ℂ_{p} ⊗_{ℚ_{p}} H_{et}^{m}(X_{K}, ℚ_{p}),

where G acts on H^{m−i}(X_{K}, Ω^{i}) trivially and on the right hand side diagonally. Here (i) means the Tate twist by i-th power of cyclotomic character (\underleftarrow{lim} μ_{p^{n}})^{⊗i}. Since ℂ_{p}^{G} = K and ℂ_{p}(i)^{G} = 0 for i ≠ 0, it is clear that

h^{i,m−i} = dim_{K} (ℂ_{p} ⊗_{ℚ_{p}} H_{et}^{m}(X_{K}, ℚ_{p})(i))^{G}.

Now the key observation is that the semisimplification is already enough to determine the Hodge numbers.

Proposition 5.1. In the notation as above, then

h^{i,m−i} = dim_{K} (ℂ_{p} ⊗_{ℚ_{p}} H_{et}^{m}(X_{K}, ℚ_{p})^{ss}(i))^{G}.

Proof. Let H_{et}^{m}(X_{K}, ℚ_{p}) =: V = V_{0} ⊃ V_{1} ⊃ ⋯ ⊃ V_{k} be a filtration of G-submodules such that V_{j}/V_{j+1}’s are simple G-modules. Then by definition V^{ss} = ⨁_{j} V_{j}/V_{j+1}. Since ℂ_{p} is a flat G-module and the functor of taking G-invariants A ↦ A^{G} is left exact, simple induction shows that

dim_{K}(ℂ_{p} ⊗ V)^{G} ≤ dim_{K}(ℂ_{p} ⊗ V^{ss})^{G}.

The same inequality applies to V (i) as well for any i ∈ ℤ, hence

∑_{i} h^{i,m−i} = ∑_{i} dim_{K}(ℂ_{p} ⊗ V (i))^{G}

≤ ∑_{i} dim_{K}(ℂ_{p} ⊗ V^{ss}(i))^{G} ≤ dim_{ℚ_{p}} V^{ss},

where the last inequality is a general fact about G-modules. Now since both ends are equal to dim_{ℚ_{p}} V, all the inequalities are equalities. In particular,

h^{i,m−i} = dim_{K} (ℂ_{p} ⊗ V (i))^{G} = dim_{K} (ℂ_{p} ⊗ V^{ss}(i))^{G}.

q.e.d.

Remark 5.2 (Suggested by C.-L. Chai). Proposition 5.1 follows immediately from the fact that Hodge-Tate representations form a Tannakian category, in particular it is stable under extensions.

6. Deformation to the number field case

The aim of this section is to explain that given a finite number of complex projective varieties and a finite number of morphisms between them, denoted by 𝒳, one may always deform the system 𝒳 a little bit to make it defined over a number field F. Moreover, when the issue of smoothness is imposed, one may find a ℤ-algebra S integral over ℤ with F as its quotient field such that all the given objects are defined over S and the imposed smoothness condition is preserved under reductions over all but a finite number of primes.

As we will only apply it to the K-equivalence diagram here, we restrict ourself to this case for simplicity. We start with a model 𝒳 → Spec S with S a finitely generated ℤ-algebra. Namely, (ϕ, ϕ′) : Y → X × X′ → Spec S with relation (E is a relative normal crossing divisor over Spec S)

K_{Y/S} = ϕ^{∗}K_{X/S} + E = ϕ′^{∗}K_{X′/S} + E.

Consider a large enough number field F such that there exists an F - valued point η : Spec F → Spec S in the regular values of 𝒳 → Spec S. Then, by considering the resulting fiber diagram over η, we get a Kequivalence diagram 𝒳_{η} over F by taking base change of the above relation to the fiber over η. By selecting an integral model of 𝒳_{η} again then we are done.

Notice that these algebraic (number field) points η are dense in Spec S and the union of Galois groups Gal(F /F) among all η is dense in Gal(F_{S}/F_{S}) for F_{S} the quotient field of S. This allows us to deduce that

H_{et}^{j}(X_{F_{S}}, ℚ_{ℓ})^{ss} ≅ H_{et}^{j}(X′_{F_{S}}, ℚ_{ℓ})^{ss}

as Gal(F_{S}/F_{S}) representations from the number field case.

Remark 6.1. For people not familiar with this procedure, the following example provides a trivial illustration. Let e and π be any two algebraically independent numbers. Take say two plane conics X and X′ defined by 2x^{2} +ey^{2} +z^{2} = 0 and πx^{2} +y^{2} +z^{2} = 0 with birational map (in fact an isomorphism) given by (x, y, z) ↦ (\sqrt{\frac{π}{2}} x, \frac{1}{\sqrt{e}}y, z). This sys-

tem is defined over the ring S = ℤ [e, π, \frac{1}{\sqrt{e}}, \sqrt{\frac{π}{2}} ] ≅ ℤ[u, v, w, s]/(uw^{2} − 1, v − 2s^{2}). Take F = ℚ, an F -rational point of Spec S could be taken to be η = (u, v, w, s) = (1, 2, 1, 1). Over this point η, both X and X′ are deformed into 2x^{2} + y^{2} + z^{2} = 0 and the birational map is deformed into the identity map.

7. Appendix: Proof of Theorem 2.1

Let us recall the proof briefly. Let (ϕ, ϕ′) : Y → X × X′ be a resolution of f so that the union of the exceptional set of ϕ and ϕ′ is a normal crossing divisor of Y. Let K_{Y} =_{ℚ} ϕ^{∗}K_{X} + E =_{ℚ} ϕ′^{∗}K_{X′} + E′. So

ϕ′^{∗}K_{X′} =_{ℚ} ϕ^{∗}K_{X} + F, with F := E − E′.

It suffices to show that F ≥ 0. Let F = ∑_{j=0}^{n−1} F_{j} with dim ϕ′(Supp F_{j}) = j. We will show that F_{j} ≥ 0 for j = n − 1, n − 2, …, 1, 0 inductively. As E′ is ϕ′-exceptional, F_{n−1} ≥ 0 is clear. Suppose that we have already shown that F_{j} ≥ 0 for j ≥ k + 1.

Consider the surface S_{k} := H^{n−2−k}.ϕ′^{∗}L^{k} on Y where H is very ample on Y and L is very ample on X′. We get a relations of divisors on S_{k}:

ϕ′^{∗}K_{X′}|_{S_{k}} =_{ℚ} ϕ^{∗}K_{X}|_{S_{k}} + a − b,

where H^{n−2−k}.ϕ′^{∗}L^{k}.F = a − b with both a and b effective. Notice that b can only come from F_{k} since ∑_{j≥k+1} F_{j} ≥ 0 and L^{k} ∩ ϕ′(F_{j}) = ∅ for j < k. Now we look at

b.ϕ′^{∗}K_{X′} =_{ℚ} b.ϕ^{∗}K_{X} + b.a − b^{2}.

The left hand side is always zero since ϕ′(b) ⊂ L^{k} ∩ ϕ′(F_{k}) is zero dimensional. Moreover, since ϕ′^{∗}K_{X′} =_{ℚ} ϕ^{∗}K_{X} on ϕ^{−1}(X\Z), we must have that ϕ(Supp F) ⊂ Z. In particular, b.ϕ^{∗}K_{X} ≥ 0. It is also clear that b.a ≥ 0. However, if b ≠ 0 then it is a nontrivial combination of ϕ′ exceptional curves in S_{k}. By the Hodge index theorem for surfaces we then have that b^{2} < 0, a contradiction. So b = 0 and F_{k} ≥ 0. The codimension statement is easy and we omit its proof.

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354 chin-lung wang

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National Tsing-Hua University, Hsinchu, Taiwan