A HEIGHT FUNCTION ON THE PICARD GROUP OF SINGULAR ARAKELOV VARIETIES

JÖRG JAHNEL Universität Göttingen, Mathematisches Institut, Bunsenstraße 3-5, 37073 Göttingen, Germany E-mail: ˜ jjahnel@wwwuser.gwdg.de

Université de Paris-Sud, Mathématique, Bâtiment 425, 91405 Orsay, France E-mail: ˜ jjahnel@geo.math.u-psud.fr

For line bundles on possibly singular arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of a model of an algebraic curve of genus g over a number field, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by the Θ-divisor. The behaviour of this equivalence under change of the base field is investigated. In arbitrary dimension a finiteness property is proven.

1 Introduction

1.1 In a previous paper^{17} there was given a construction for a height function on the Picard group of a regular arithmetic variety following the philosophy of Bost, Gillet and Soulé^{3} that heights should be objects in arithmetic geometry analogous to degrees in algebraic geometry. Let us shortly recall this construction and fix notation.

Let K be a number field, O_{K} its ring of integers and X /O_{K} an arithmetic variety, by which we mean a reduced scheme, projective and flat over O_{K}, whose generic fiber X is geometrically connected. In the main body of the paper we will additionally assume X to be regular in order to have the theory of Quillen metrics available. Denote the dimension of X by d.

Further we choose a Kähler metric ω on the complex manifold X(ℂ) associated to X being invariant under complex conjugation F_{∞}. This transforms X into a so-called Arakelov variety \overline{X}. Finally choose some metrized line bundle \overline{T} = (T , ∥.∥_{T}) ∈ P̂ic(X). 1.2 Definition. Let L ∈ Pic (X). We call a hermitian metric ∥.∥ on the associated line bundle L_{ℂ} on the complex manifold X(ℂ) distinguished if its Chern form c_{1}(L_{ℂ}, ∥.∥) is harmonic with respect to ω and

d̂eg (det Rπ_{∗}L, ∥.∥_{Q}) = 0.  (1)

Here π : X → Spec O_{K} is the structural morphism and ∥.∥_{Q} denotes the Quillen metric^{2} at the infinite places of K. 1.3 Lemma. Suppose the Euler characteristic χ(L_{K}) is different from zero.

a) Then there exists a distinguished metric on L.

b) If ∥.∥ and ∥.∥′ are distinguished metrics on L, then ĉ_{1}(L, ∥.∥) and ĉ_{1}(L, ∥.∥′) are numerically equivalent to each other. 1.4 Definition (The height). Let L ∈ Pic (X) be a line bundle and assume χ(L_{K}) ≠ 0. Then the height of L with respect to ω and \overline{T} is the arithmetic intersection number

h_{\overline{T} ,ω}(L) := d̂eg γ ((T , ∥.∥_{T}), …, (T , ∥.∥_{T}), (L, ∥.∥)) ,  (2)

where ∥.∥ is a distinguished metric on L. Here

γ : \underbrace{P̂ic(X) × P̂ic(X) × … × P̂ic(X)}_{d+1 times} → ĈH^{1}(Spec O_{K})_{ℚ}  (3)

is the multi-linear map defined in Lemma A.4. When X is regular, it is given by

(\overline{U_{1}}, …, \overline{U_{d+1}}) ↦ π_{∗}(ĉ_{1}(\overline{U_{1}}) · . . . · ĉ_{1}(\overline{U_{d+1}})) .  (4)

1.5 We note that the concept of an arithmetic variety considered here is compatible with finite field extensions L/K in that sense that the base change of an arithmetic variety with Spec O_{L} → Spec O_{K} is again an arithmetic variety, but over O_{L}. So one easily mimics the classical approach to heights for L-valued points on varieties defined over K. Definition (The normalized height). Let L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) be some line bundle and assume χ(L_{L}) ≠ 0. Then the normalized height of L with respect to ω and \overline{T} is given by

h_{\overline{T} ,ω}(L) := \frac{1}{[L : K]} h_{π_{1}^{∗}\overline{T} ,π_{1}^{∗}ω}(L),  (5)

where π_{1} : X ×_{Spec O_{K}} Spec O_{L} → X denotes the projection to the first factor. 1.6 Remark. The constructions above use the first Chern class of a hermitian line bundle and the intersection product on an arithmetic variety. These are the work of Gillet and Soulé^{7,8}. They make extensive use of algebraic K-Theory. On an arithmetic variety there is no moving lemma available and thus the classical method to define the intersection product of cycles fails. Instead Gillet and Soulé analyze the Brown-Gersten-Quillen spectral sequence

E_{1}^{pq} := ⨁_{x∈X^{(p)}} K_{−p−q}(k(x)) ⇒ K_{−p−q}(X).  (6)

They show by investigation of the Adams operations that, after tensoring with ℚ, it degenerates in the E_{2}-term, at least on the K_{0}-diagonal. Thus K_{0}(X)_{ℚ} ≅ ⨁_{i≥0} CH^{i}(X)_{ℚ} and this coincides with the splicing of K_{0}(X)_{ℚ} into eigenspaces under the Adams operations. Therefore the product structure on K_{0}(X) gives rise to a product, transforming ⨁_{i≥0} CH^{i}(X)_{ℚ} into a graded ring with unit.

In the appendix we will be going to discuss to some extent one point of view on the possible intersection products when X is singular. We will mainly follow the ideas of Fulton^{5} as well as Remark 2.3.1.ii of Bost, Gillet and Soulé^{3}.

2 Elementary Properties

2.1 Lemma (Field extensions). Let L′/L be an extension of number fields containing K and let p : X ×_{Spec O_{K}} Spec O_{L′} → X ×_{Spec O_{K}} Spec O_{L} be the natural morphism. Then, for line bundles L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) with χ(L_{L}) ≠ 0 one has

h_{\overline{T} ,ω}(L) = h_{\overline{T} ,ω}(p^{∗}L).  (7)

Proof. First we note that p^{∗}∥.∥_{dis} is a distinguished metric again. Indeed the canonical isomorphism det Rπ_{L}_{∗}p^{∗}L ⟶^{≅} p^{∗} det Rπ_{∗}L is isometric since the formation of the Quillen metric commutes with arbitrary base changes. Therefore

d̂eg (det Rπ_{L}_{∗}p^{∗}L, ∥.∥_{Q}) = [L : K] d̂eg (det Rπ_{∗}L, ∥.∥_{Q}) = 0.  (8)

Consequently, one has

h_{\overline{T} ,ω}(p^{∗}L) = \frac{1}{[L : K]} d̂eg γ (p^{∗}(L, ∥.∥), p^{∗}(T , ∥.∥), …)
= d̂eg γ ((L, ∥.∥), (T , ∥.∥), …)  (9)
= h_{\overline{T} ,ω}(L)

by Lemma A.4.ii). □ 2.2 Proposition (Special fibers). a) Let D be an effective Cartier divisor on X being non-trivial only in the special fiber over 𝔭 ∈ Specm O_{K}. Then, for L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}), where L/K is a finite field extension and χ(L_{L}) ≠ 0,

h_{\overline{T} ,ω} (L(D)) = h_{\overline{T} ,ω}(L) + log(♯O_{K}/𝔭) · [deg_{T} D − c_{1}(T_{K})^{d} · \frac{χ (L(D)/L)}{χ(L_{L})}] .

(10)

Note that L(D)/L ≅ L(D)|_{D} if on the right hand side we consider D as a Weil divisor.

b) In particular h_{\overline{T} ,ω}(L(D)) = h_{\overline{T} ,ω}(L) if D is a complete fiber. Proof. This is essentially contained in Proposition 2.4 of a previous paper^{16}. □ 2.3 Remarks. i) The finiteness and comparison results we show below assume that the line bundles under consideration have bounded degrees in the components of the special fibers. To the contrary, the proposition above shows the behaviour of the height under large perturbations in special fibers.

ii) In the forthcoming propositions we will study h_{..}(L) under changes of the initial data. The correction terms turn out to be of algebro-geometric and complex-analytic nature, i.e. they consist of degrees and Euler characteristics. The proofs were essentially given in a previous paper^{16}. 2.4 Proposition (Change of defining bundle). a) Let F be a Cartier divisor supported over 𝔭. Then for any L ∈ Pic(X ×_{Spec O_{K}} Spec O_{L}) with χ(L_{L}) ≠ 0

h_{\overline{T} (F),ω}(L)  (11)
= h_{\overline{T} ,ω}(L) + d log(♯O_{K}/𝔭) · deg_{T} L|\overline{F^{1}} + ∑_{k=2}^{d} \binom{d}{k} log(♯O_{K}/𝔭) · deg_{T} L|\overline{F^{k}},

where \overline{F^{k}} denotes the pull back to the geometric fiber \overline{X_{𝔭}} of a cycle representing F^{k} ∈ CH_{X_{𝔭}}^{k}(X)_{ℚ}. Note that the right summand disappears as F = [X_{𝔭}] or for an arithmetic surface.

b) On the line bundle T_{ℂ} on X(ℂ) let ∥.∥′ = e^{φ} · ∥.∥ be another hermitian metric. Then for any L ∈ Pic(X ×_{Spec O_{K}} Spec O_{L}) with χ(L_{L}) ≠ 0

h_{\overline{T} ′,ω}(L) = h_{\overline{T} ,ω}(L) + \frac{1}{2 [L : K]} ∑_{i+j=d−1} ∫_{X_{L}(ℂ)} φ c_{1}(\overline{T} )^{i}c_{1}(\overline{T}′)^{j}H (c_{1}(L_{ℂ})) ,

(12)

where \overline{T} and \overline{T}′ mean the pull-backs of these hermitian line bundles under the natural projection X_{L}(ℂ) → X(ℂ) and H denotes the harmonic projection. In particular, when φ is constant

h_{\overline{T} ′,ω}(L) = h_{\overline{T} ,ω}(L) + \frac{d}{2} φ [K : ℚ] deg_{T} L_{L}.  (13)
□

2.5 Proposition (Change of Kähler metric). Let ω, ω′ be Kähler metrics on X(ℂ). Then for every L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) with χ(L_{L}) ≠ 0 there exists a smooth function g_{L} on X(ℂ), depending only on the homological equivalence class of L_{ℂ}, such that

h_{\overline{T} ,ω′}(L) = h_{\overline{T} ,ω}(L) + \frac{1}{2} ∫_{X(ℂ)} g_{L} c_{1}(\overline{T}_{ℂ})^{d}.  (14)
□

2.6 Proposition (Birational morphisms). Let p : X′ ⟶ X be a morphism of arithmetic varieties inducing an isomorphism between the generic fibers. Then, for any line bundle L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) with χ(L_{L}) ≠ 0

h_{p^{∗}\overline{T} ,ω}(p^{∗}L) = h_{\overline{T} ,ω}(L)  (15)

−\frac{c_{1}(T_{K})^{d}}{[L : K] χ(L_{L})} ∑_{𝔭} log(♯O_{K}/𝔭) ∑_{j≥1}(−1)^{j}χ (\overline{X_{𝔭}}, (R^{j}p_{∗}O_{X′})|\overline{X_{𝔭}} ⊗ L|\overline{X_{𝔭}}) .

Note that, if R^{j}p_{∗}O_{X′} = 0 for all j ≥ 1, then the correction term vanishes. □ 2.7 As line bundles can be defined by divisors it is natural to expect a relation of the height of a line bundle with that of a corresponding divisor. In general, linearly equivalent divisors will have different heights, such that the relation can not be too simple. Proposition (Comparison with the height of divisor). Let P/O_{K} be a scheme of finite type whose generic fiber is proper. On X ×_{Spec O_{K}} P consider a line bundle U such that det Rπ_{2}_{∗}U|_{P_{K}} ≅ O_{P_{K}} and χ(U|_{X ×\overline{y}}) ≠ 0 for each \overline{y} ∈ P_{K}(\overline{K}). Assume that for some H ∈ Pic (P) and n ∈ ℕ the tensor power (U ⊗ π_{2}^{∗}H)^{⊗n} has a suitable section s, i.e. s|_{X×\{\overline{y}\}} ≠ 0 for each \overline{y} ∈ P_{K}(\overline{K}). Then for every number field L and any p ∈ P(O_{L})

h_{\overline{T} ,ω}(U|_{X ×p}) = \frac{1}{n[L : K]} h\overline{T} (div(s|_{X ×p})) − \frac{1}{[L : K]} h\overline{H}(p) · c_{1}(T_{L})^{d} + O(1),

(16)

where H is equipped with any hermitian metric, while h\overline{T} and h\overline{H} denote heights for cycles^{3}. O(1) is a bounded function on

⨆_{L/K finite} P(O_{L}),  (17)

but the bound depends on \overline{H} and s (and, of course, on the original data X , ω, \overline{T} , P, U). Proof. Equip the line bundle U_{ℂ} on ⨆_{σ:K↪ℂ} X (ℂ) × P(ℂ) with a hermitian metric ∥.∥ being invariant under F_{∞} and admitting a harmonic Chern form c_{1}(U_{ℂ}, ∥.∥) fiber-by-fiber. One easily sees that the function

⨆_{L/K finite}P(O_{L}) ⟶  ℝ
p ↦ \frac{1}{[L:K]} d̂eg (det Rπ_{2}_{∗}U|_{X ×p}, ∥.∥_{Q})

(18) is bounded. Hence there is a distinguished metric ∥.∥_{dis} = e^{C(p)}·∥.∥ on U|_{X ×p}, where C is a bounded function on ⨆_{L/K finite} P(O_{L}). Therefore

h_{\overline{T} ,ω}(U|_{X ×p}) = \frac{1}{[L : K]} d̂eg γ (\overline{T} , …, \overline{T} , (U|_{X ×p}, ∥.∥_{dis}))
= \frac{1}{[L : K]} d̂eg γ (\overline{T} , …, \overline{T} , (U|_{X ×p}, ∥.∥)) + O(1) (19)
= \frac{1}{[L : K]} d̂eg γ (\overline{T} , …, \overline{T} , ((U ⊗ π_{2}^{∗}H)|_{X ×p}, ∥.∥))

−\frac{1}{[L : K]} d̂eg γ (\overline{T} , …, \overline{T} , π_{2}^{∗}\overline{H}|_{X ×p}) + O(1).

Here the second summand is the same as

−\frac{1}{[L : K]} d̂eg γ (\overline{T} , …, \overline{T} , π^{∗}\overline{H}|_{p})

= −\frac{1}{[L : K]} d̂eg β (\overline{T} , …, \overline{T} , π^{∗}\overline{(0, \frac{2}{[L:ℚ]}h\overline{H}(p), …, \frac{2}{[L:ℚ]}h\overline{H}(p))}) (20)
= −\frac{1}{[L : K][L : ℚ]} ∫_{X_{L}(ℂ)} h\overline{H}(p) · ω∧d ; \overline{T}
= −\frac{1}{[L : K]} h\overline{H}(p) · c_{1}(T_{L})^{d},

where we used the properties of the various intersection products shown in the appendix. Note that the factor [L : ℚ] in the denominator disappears since X_{L}(ℂ) has [L : ℚ] mutually isomorphic connected components. For the first summand in the formula above we obtain

\frac{1}{n[L : K]} d̂eg β (\overline{T} , …, \overline{T} , (div(s|_{X ×p}), − log ∥s|_{X ×p}∥^{2}))

= \frac{1}{n[L : K]} d̂eg α (\overline{T} , …, \overline{T} , div(s|_{X ×p}))  (21)

−\frac{1}{2n[L : K]} ∫_{X_{L}(ℂ)} log ∥s|_{X ×p}∥^{2} · ω∧d ; \overline{T}

= \frac{1}{n[L : K]} h\overline{T} (div(s|_{X×p})) − \frac{1}{2n[L : K]} ∑_{σ:L↪ℂ} ∫_{X⊗_{K}ℂ (ℂ)}log ∥s|_{X ×σ(p)}∥^{2} · ω∧d ; \overline{T}.

Using the work of Stoll^{23} one shows easily that the integral

∫_{X⊗_{K}ℂ (ℂ)}log ∥s|_{X ×q}∥^{2} · ω∧d ; \overline{T}  (22)

is a continuous function on q ∈ P(ℂ), in particular the second part is bounded. □

3 A Finiteness Theorem

3.1 Proposition (Models of line bundles). Let π : X → O_{K} be an arithmetic variety and L/K an extension of number fields. Then any line bundle K ∈ Pic (X) that becomes trivial under base change by Spec O_{L} → Spec O_{K} is of the form K = π^{∗}I where I ∈ Pic (Spec O_{K}). Proof. It is sufficient to assume L/K to be Galois.

{X_{O_{L}} := X ×_{SpecO_{K}} Spec O_{L} → X }  (23)

is a covering for the fppf-site on X. We consider the Čech spectral sequence^{20}

E_{2}^{pq} := Ȟ^{p}(\{X_{O_{L}} → X \}, \underline{H}_{fppf}^{q}(𝔾_{m})) ⇒ H_{fppf}^{p+q}(X , 𝔾_{m})  (24)

associated to this covering. The beginning of its lower term exact sequence looks like this.

0 → Ȟ^{1}(\{X_{O_{L}} → X \}, 𝔾_{m}) → H_{fppf}^{1}(X , 𝔾_{m}) → Ȟ^{0}(\{X_{O_{L}} → X \}, \underline{H}_{fppf}^{1}(𝔾_{m}))
∥  ⋂||
Pic (X)  Pic (X_{O_{L}})

(25)

Therefore the line bundles under consideration are described by the Čech cohomology group Ȟ^{1}({X_{O_{L}} → X }, 𝔾_{m}).

We remark that our assumptions make sure that π_{∗}O_{X} = O_{K}, i.e. every fiber of X is geometrically connected. So we simply deal with the first cohomology group of the complex

0 ⟶ O_{L}^{∗} ⟶ (O_{L} ⊗_{O_{K}} O_{L})^{∗} ⟶ (O_{L} ⊗_{O_{K}} O_{L} ⊗_{O_{K}} O_{L})^{∗} ⟶ …, (26)

which means nothing but that π induces an isomorphism

Ȟ^{1}({Spec O_{L} → Spec O_{K}}, 𝔾_{m}) ⟶^{≅} Ȟ^{1}({X_{O_{L}} → X }, 𝔾_{m}).  (27)

On the other hand, repeating the spectral sequence argument above with X replaced by Spec O_{K} one sees that the group on the left hand side just describes the line bundles on Spec O_{K} becoming trivial under base change to Spec O_{L}. □ 3.2 As for effective cycles one can expect finiteness results only when working in a fixed equivalence class under the relation of algebraic equivalence. Definition. Let K be a field and X/K a proper variety. Let 𝔏 ∈ NS(X) be an element of the Néron-Severi group, i.e. an equivalence class of line bundles modulo algebraic equivalence. We will call 𝔏 appropriate, if the following two conditions hold.

(∗) For some finite field extension K′/K there is a K′-valued point x ∈ X(K′) = X_{K′}(K′) with the property below: On X_{K′} × Pic^{𝔏_{K′}} (X_{K′}) let P be the tautological line bundle with P|_{{x}×Pic^{𝔏_{K′}} (X_{K′})} ≅ O_{Pic}_{𝔏_{K′}}. Then (det Rπ_{2}_{∗}P)^{−1} is ample on Pic^{𝔏_{K′}} (X_{K′}).

(∗∗) If L ∈ 𝔏, then χ(L) > 0. 3.3 Theorem (Finiteness). A. Let (X /O_{K}, ω) be an Arakelov variety, 𝔏 ∈ NS(X_{K}) an appropriate class and assume that the line bundle T ∈ Pic (X), defining the height, is ample. Then for every H ∈ ℝ there are only finitely many L ∈ Pic(X) such that

i) L|_{X_{K}} ∈ 𝔏,

ii) for each 𝔭 ∈ Specm O_{K} the degree on every irreducible component \overline{X_{𝔭}^{i}} of the geometric fiber \overline{X_{𝔭}} is bounded,

|deg_{T} L|\overline{X_{𝔭}^{i}}| < H,  (28)

and iii) the height is bounded above,

h_{\overline{T} ,ω}(L) < H.  (29)

B. Assume in addition X /O_{K} to be normal and suppose that stably, i.e. after any finite field extension L/K, for any special fiber [X_{O_{L},𝔮}] ∈ Div(X ×_{Spec O_{K}} Spec O_{L}) the coefficients in its decomposition into irreducible Weil divisors are relatively prime. Then for every H ∈ ℝ there are only finitely many L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) such that

i) L|_{X_{L}} ∈ 𝔏_{L},

ii) for each 𝔮 ∈ Specm O_{L} the degree on every irreducible component \overline{X_{𝔮}^{i}} of the geometric fiber \overline{X_{𝔮}} is bounded,

|deg_{T} L|\overline{X_{𝔮}^{i}}| < H,  (30)

iii) if 𝔭 ∈ Specm O_{K} splits in O_{L}, say 𝔭O_{L} = 𝔮_{1}^{α_{1}} · . . . · 𝔮_{l}^{α_{l}}, then L|\overline{X_{𝔮_{1}}} , …, L|\overline{X_{𝔮_{l}}} become mutually numerically equivalent under the canonical isomorphisms \overline{X_{𝔮_{1}}} ≅ … ≅ \overline{X_{𝔮_{l}}} and

iv) the normalized height is bounded above,

h_{\overline{T} ,ω}(L) < H,  (31)

where

v) L/K is an arbitrary finite field extension with [L : K] < H, if no distinction is made between L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) and its pull back to X ×_{Spec O_{K}} Spec O_{L′} when L′ is a finite field extension of L.

3.4 Remarks. i) The convention in B on the pull back to a field extension is in obvious analogy to the situation of cycles. Note that it implies automatically that no distinction is made between L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) and L ⊗ π_{L}^{∗}I, where I ∈ Pic (Spec O_{L}) is an element of the class group. That identification is necessary for finiteness for trivial reasons. Indeed, for K = ℚ and X = Spec ℤ there are infinitely many quadratic fields with non-trivial class group.

ii) The assumption on the special fibers of the stabilization of X is fulfilled, in particular, if every special fiber of X is geometrically reduced. Unfortunately, as Example 3.5 will show, we can not get rid of an assumption like this. In fact, the minimal assumption being necessary in this place is that stably there exist no torsion line bundles being defined by a Cartier divisor supported in a special fiber except those coming from the class group. Lemma 3.9 below shows this property, being the one actually used in the proof, under our assumption.

iii) Condition B.iii) seems to be a relatively restrictive one. But note that this restriction is only concerned with the behaviour of L in the non-smooth fibers X_{𝔭}, where 𝔭 splits in O_{L}. At least if X is regular and the irreducible components of its fibers are geometrically irreducible, then, after a finite field extension, every line bundle can be brought into the form required by adding a suitably chosen divisor being supported in special fibers. See Proposition 3.10 for this.

iv) Pic^{𝔏_{K′}} (X_{K′}) denotes the scheme representing the open subfunctor of the Picard functor assigning to each K′-scheme T the the set of all line bundles on X_{K′} × T, which are rigidified at x and belong to 𝔏_{K′}, the class of the base changes of the line bundles of 𝔏, fiber-by-fiber. In order to work with the tautological line bundle one has to extend the ground field such that there is a K′-valued point on X. In general there exists only a Picard scheme representing the fppf-sheafification of the naive Picard functor and the relation with line bundles becomes less direct.

v) Condition (∗) above is fairly independent of choices. It does not depend on the choice of the K′-valued point x. Indeed, the tautological line bundle P′ defined by another K′-valued point x′ differs from P by some π_{2}^{∗}Z, where Z is algebraically equivalent to zero on Pic^{𝔏_{K′}} (X_{K′}). Consequently, (det Rπ_{2}_{∗}P′)^{−1} is algebraically equivalent to (det Rπ_{2}_{∗}P)^{−1} and therefore it is ample, too, by the Nakai-Moishezon criterion. Furthermore, (∗) does obviously still hold when K′ is replaced by a finite field extension.

vi) Some cases where (∗) is actually fulfilled are listed in Theorem 3.6 below.

3.5 Example. Let K = ℚ and consider the quadric

X = Proj ℤ[X, Y, Z]/(X^{2} + Y^{2} − 2Z^{2}).  (32)

One shows without difficulty that this is a regular arithmetic variety. Its fiber over 𝔭 = (2) is equal to Proj 𝔽_{2}[X, Y, Z]/(X + Y)^{2}. Therefore O([X_{(2)}]_{red}) is a non-trivial torsion bundle. In particular, it is numerically equivalent to zero in any fiber.

Let \overline{x} := (1:1:1) ∈ X (ℤ). By Théorème 2.3.1 of Raynaud^{22} the rigidified Picard functor

Pic_{X /ℤ} : (Sch/ℤ) ⟶ (Sets)  (33)
T ↦ \{(L, i) | L ∈ Pic (T ×_{Spec ℤ} X), i : L|_{T ×\overline{x}} ⟶^{≅} O_{T ×\overline{x}}\}

is representable as an algebraic space. Further, its subfunctor Pic_{X /ℤ}^{τ} collecting the line bundles, which are numerically equivalent to zero in each geometric fiber of T ×_{Spec ℤ} X → T, is representable by an open group subspace Pic_{X /ℤ}^{τ} being of finite type over Spec ℤ as shown in SGA6, Exp. XIII, Thm. 4.7.

It is easy to see that Pic_{X /ℤ}^{τ} can not be separated. In fact there are two different maps i_{1}, i_{2} : Spec ℤ → Pic_{X /ℤ}^{τ} that are both extensions of i : Spec ℚ → Pic_{X /ℤ}^{τ} corresponding to the trivial line bundle on X. This effect has serious consequences already on quadratic extensions of ℚ split over (2). Namely, if (2)O_{L} = 𝔭_{1}𝔭_{2}, then in order to extend i_{L} : Spec L → Pic_{X /ℤ}^{τ} to Spec O_{L} we have two possibilities for the image of 𝔭_{1} and two possibilities for the image of 𝔭_{2}, thus producing two new line bundles in the sense of our statement above. For any other choice of L there will be different ones and we end up with infinitely many of them. Note that in order to produce this pathology we needed extensions of the ground field.

At this point it becomes clear, what condition B.iii) is good for. Pic_{X /O_{K}} is not separated in general, even when Pic_{X /O_{K}}^{τ} is. A typical case is a regular arithmetic variety X having reducible fibers. We could therefore produce infinitely many line bundles not being of much interest by the same procedure. 3.6 Theorem. Let K be a field and X/K be a regular, proper and geometrically connected scheme. Further, let 𝔏 ∈ NS(X) be an equivalence class of line bundles modulo algebraic equivalence. Then 𝔏 is appropriate, if

a) X is a curve of genus g and 𝔏 consists of line bundles of degree at least g,

b) X is torsor over an abelian variety and 𝔏 is an ample equivalence class,

c) (X is arbitrary), 𝔏′ ∈ NS(X) is any equivalence class and 𝔄 is an ample equivalence class, for 𝔏 = 𝔏′ + n𝔄 when n ≫ 0. Proof. This follows directly from Theorem 1.7 in a previous paper^{16} and Remark ii) above. □

3.7 Proof of Theorem 3.3. B. 1st step. We may assume without restriction that X(K)≠∅. This is just Lemma 3.1.

2nd step. The Picard functor. Denote by \overline{x} an O_{K}-valued point of X and consider the rigidified Picard functor as in Example 3.5. We claim that the algebraic space Pic_{X /O_{K}}^{τ} is separated here. In fact, by the valuative criterion we have to consider the commutative diagram

Spec F → Pic_{X /O_{K}}^{τ}
↓ ↗↗ ↓
Spec S → Spec O_{K},

(34)

where S is a discrete valuation ring and F is fraction field. By the modular interpretation we have two line bundles L_{1}, L_{2} ∈ Pic (X ×_{Spec O_{K}} Spec S) becoming isomorphic under restriction to X ×_{Spec O_{K}} Spec F. Lemma 3.9 shows that under our assumptions the two must be isomorphic. The claim is proven. By Théorème 3.3.1 of Raynaud^{22} it follows that Pic_{X /O_{K}}^{τ} is a scheme already. We define Pic_{X /O_{K}}^{σ} to be the closure of Pic_{X /O_{K},K}^{τ} in Pic_{X /O_{K}}^{τ}.

A little bit extending the collection of line bundles under consideration we look at the subfunctor Pic_{X /O_{K}}^{δ} of Pic_{X /O_{K}}, which collects the line bundles satisfying ii) in each characteristic p geometric point of T and, instead of i), being of the same numerical type as the bundles in 𝔏_{C} for each Cvalued point of T, when C is a field in characteristic zero. This subfunctor is given by numerical conditions only, therefore it is represented by an open subspace Pic_{X /O_{K}}^{δ} of Pic_{X /O_{K}}. We will not be interested in line bundles on special fibers that can not be generalized. Therefore we consider the closure Pic_{X /O_{K}}^{γ} of Pic_{X /O_{K},K}^{δ} in Pic_{X /O_{K}}^{δ}. Proposition 3.8 shows now that we are dealing with finitely many numerical types only, so, after a finite field extension, Pic_{X /O_{K},O_{L_{0}}}^{γ} is the union of finitely many cosets of Pic_{X /O_{K},O_{L_{0}}}^{σ}. In particular, it is of finite type over Spec O_{K}. Condition B.iii) translates into the requirement, that we consider O_{L}-valued points lying in one of the cosets already.

We note that by FGA, Exp. 232, Théorème 3.1 and Exp. 236, Théorème 2.1 the scheme Pic_{X /O_{K}}^{γ} is projective outside the fibers of finitely many primes 𝔭_{1}, …, 𝔭_{l} ∈ Specm O_{K}. By enlarging the exceptional set if necessary we may assume smoothness, i.e. Pic_{X /O_{K}}^{γ} ×_{Spec O_{K}\{𝔭_{1}, ... ,𝔭_{l}}} Spec O_{K} is a disjoint union of finitely many abelian schemes.

3rd step. For every n ∈ ℕ there exist C(n) ∈ ℕ and a morphism p′ : P′ → Pic_{X /O_{K}}^{δ} satisfying the conditions below.

i) P′ is a separated scheme of finite type over O_{K}.

ii) p′ is quasi-finite.

iii) p′ is finite outside the fibers over the primes 𝔭_{1}, …, 𝔭_{l}.

iv) For every number field L and any i ∈ Pic_{X /O_{K}}^{δ} (O_{L}) lying after lift to O_{LL_{0}} in one of the cosets under Pic_{X /O_{K},O_{L_{0}}}^{σ} there exist a field extension L′/L with [L′ : L] ≤ C(n) and i′ ∈ P′(O_{L′}) lifting i.

v) For every F ∈ Pic (Pic_{X /O_{K},\overline{ℚ}}^{δ}) the pull-back p′∗ ; \overline{ℚ}F is divisible by n. We may choose finitely many sections i_{α} : Spec O_{L_{0}} → Pic_{X/O_{K}}^{γ} to obtain

a surjection P′_{1} := ∐_{α∈A} Pic_{X/O_{K},O_{L_{0}}}^{σ} → Pic_{X/O_{K},O_{L_{0}}}^{γ} → Pic_{X/O_{K}}^{γ}, where the morphisms Pic_{X/O_{K},O_{L_{0}}}^{σ} → Pic_{X/O_{K},O_{L_{0}}}^{γ} are just the multiplications with the O_{L_{0}}-valued points i_{α}. This satisfies i), ii) and iii).

For v) there is a cohomological argument. The obstruction against F ∈ Pic ( .) to be divisible by n is its Chern class c_{1}(F) ∈ H_{ét}^{2}( . , μ_{n}). Since each connected component of Pic_{X/O_{K},\overline{ℚ}}^{δ} is an abelian variety, its second cohomology group is generated by the first one. So we only have to annihilate (some of) the classes in H_{ét}^{1}(P′_{1,\overline{ℚ}}, μ_{n}) and this is done by an étale cover. But this one will be given by finitely many data and therefore be nothing but the base change of some cover P̃ → P′_{1,L} defined over a finite field extension L/K already. Consider the composition P̃ → P′_{1} ×_{Spec O_{K}} Spec O_{L} → P′_{1} being a finite morphism in the generic fibers and define an extension P′ of P̃ being finite over P′_{1} as the normalization of P′_{1} in K(P̃). By construction, P fulfills v), while i), ii) and iii) remain valid by the finiteness over P′_{1}.

Finally, let us show iv). It is sufficient to prove this for P′_{1}. In fact, P′ is finite over P′_{1} of some degree C. By SGA1, Exp. I, Théorème 10.11 over any L-valued point of P′_{1} there is an L′-valued point with [L′ : L] ≤ C. Extend it to an O_{L′}-valued point. But for lifts to P′_{1} the only field extension we need is the composite with L_{0}. Indeed, any i : Spec O_{L} → Pic_{X /O_{K}}^{δ} corresponds to some line bundle L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}). But the sections i_{α} chosen represent all the possible numerical types in the special fibers. Therefore, as soon as L ⊇ L_{0}, we may find α ∈ A such that i − i_{α} is a section in Pic_{X/O_{K}}^{σ}. This is just the assertion.

4th step. The tautological line bundle. Let P be the rigidified tautological line bundle on X ×_{Spec O_{K}} Pic_{X/O_{K}}^{γ}. A direct application of Riemann-Roch shows that for any \overline{ℚ}-valued point \overline{x} ↪ Pic_{X/O_{K}}^{γ} the Euler characteristic χ := χ(P|_{X ×\overline{x}}) is the same. We put n := χ and consider the morphism P′ →^{p′} Pic_{X/O_{K}}^{γ} constructed in Step 3. We claim: There is a projective morphism q : P → P′ satisfying the following conditions.

i) q is an étale cover outside the fibers over 𝔭_{1}, …, 𝔭_{l}.

ii) There exists C ∈ ℕ such that for any number field L and any i ∈ P′(O_{L}) there exist a field extension L′/L with [L′ : L] ≤ C and a lift i′ ∈ P (O_{L′}) of i. iii) There exists E ∈ Pic(P) such that the line bundle

det Rπ_{2}_{∗} ((id × p)^{∗}P ⊗ π_{2}^{∗}E)  (35)

is trivial on the generic fiber P_{K}. Here p denotes the composition p = p′q : P → Pic_{X/O_{K}}^{γ}.

By the projection formula

det Rπ_{2}_{∗} ((id × p)^{∗}P ⊗ π_{2}^{∗}E) = p^{∗} det Rπ_{2}_{∗}P ⊗ E^{⊗χ}.  (36)

So we are looking for a morphism, the pull-back under which makes the line bundle det Rπ_{2}_{∗}P divisible by χ. p′ yields that on the geometric generic fiber by the last step. Therefore it is clear that there exists a number field L/K such that the pull-back of det Rπ_{2}_{∗}P to P_{1} := P′ ×_{Spec O_{K}} Spec O_{L} will be divisible by χ on the generic fiber. It remains to extend the quotient D from the generic fiber P_{1,K} to a model. For this take a Weil divisor D with D = O(D) and consider its Zariski closure \overline{D} in P_{1}. In general it will not define a line bundle there, but the blow-up P := Bl\overline{D}(P_{1}) fulfills properties i), ii) and iii). Note that the restriction of E to P_{K} is an ample line bundle by assumption (∗) and (∗∗).

5th step. Comparison with height for cycles. Put U := (id × p)^{∗}P ⊗ π_{2}^{∗}E. For the x ∈ X(K) used in the rigidification the line bundle U|_{{x}×P_{K}} ≅ E_{K} is ample. Thus, U_{K} is ample fiber-by-fiber since all the U|_{{.}×P} are mutually numerically equivalent (up to extension of ground field). By EGA III, Théorème 4.7.1 it is relatively ample. But then EGA II, Proposition 4.6.13.ii shows that there exists some H ∈ Pic(X) such that U ⊗ π_{1}^{∗}H induces an ample line bundle on X × P_{K}. This implies that there is some n ∈ ℕ such that (U ⊗ π_{1}^{∗}H)^{⊗n}|_{X×P_{K}} admits a suitable section s, i.e. s|_{X×\{\overline{y}\}} ≠ 0 for every geometric point \overline{y} ↪ P_{K}, as worked out in a previous paper^{16}. Multiplying with elements of O_{K} if necessary, we may assume that s can be extended to X ×_{Spec O_{K}} P. By Lemma 2.7 we know for any number field L and any y ∈ P (O_{L})

h_{\overline{T} ,ω} (U|_{X ×y}) = \frac{1}{n[L : K]} h\overline{T} (div(s|_{X ×y})) + O(1)  (37)
= \frac{1}{n[L : K]} h\overline{T} (π_{L}_{∗} div(s|_{X ×y})) + O(1),

where π_{L} : X ×_{Spec O_{K}} P ×_{Spec O_{K}} Spec O_{L} → X ×_{Spec O_{K}} P is the natural projection. But the cycles div(s|_{X ×y}) are pairwise different, even when restricted to the geometric generic fiber, since they represent different line bundles there. One easily sees that each cycle on X ×_{Spec O_{K}} P has only finitely many possibilities to be a push-forward under one of the π_{L}. So the proof is finished by the finiteness result for the height for effective cycles^{3}.

For A we have Pic_{X/O_{K}}^{γ} and Pic_{X/O_{K}}^{σ} as well, but we neither know Pic_{X/O_{K}}^{σ} to be separated nor to be a scheme. So we have to deal carefully with algebraic spaces. We claim: For every n ∈ ℕ there exist C(n) ∈ ℕ and a morphism p′ : P′ → Pic_{X /O_{K}}^{γ} satisfying the conditions below.

i) P′ is a separated scheme of finite type over O_{K}.

ii) Each component of P′ is finite over its image in Pic_{X/O_{K}}^{γ}. iii) p′ is finite outside the fibers over the primes 𝔭_{1}, …, 𝔭_{l}. iv) For any i ∈ Pic_{X /O_{K}}^{γ} (O_{K}) there are a field extension K′/K with

[K′ : K] ≤ C(n) and i′ ∈ P′(O_{K′}) lifting i.

v) For every F ∈ Pic (Pic_{X /O_{K},\overline{ℚ}}^{γ}) the pull-back p′∗ ; \overline{ℚ}F is divisible by n. By the definition of an algebraic space there are a scheme U and an étale

covering U → Pic_{X /O_{K}}^{γ}. As Pic_{X /O_{K}}^{γ} is the closure of its generic fiber, we may, not losing the property of being étale, decompose U into its components and consider ∐_{α∈A} U_{α} → Pic_{X /O_{K}}^{γ}, where A is a finite index set and U_{α} are irreducible affine schemes. Without restriction assume A = A_{0} ∪ … ∪ A_{l}, where for α ∈ A_{i} with i ∈ {1, …, l} the component U_{α} meets the generic fiber and among the fibers over 𝔭_{1}, …, 𝔭_{l} exactly that over 𝔭_{i}, while it does not meet the fibers over 𝔭_{1}, …, 𝔭_{l} for α ∈ A_{0}. Define \overline{U_{α}} to be the normalization of U_{α,red} in the normal closure K(U_{α,red})^{n} of its function field, considered as a finite extension of K(Pic_{X/O_{K}}^{γ}). Corresponding to each σ ∈ G = Gal(K(U_{α,red})^{n}/K(Pic_{X/O_{K}}^{γ})) there is a birational map j_{σ} : \overline{U_{α}} − → \overline{U_{α}} making the obvious diagram of maps to Pic_{X/O_{K}}^{γ} commutative. We consider these copies of \overline{U_{α}} as different schemes connected by a birational map j_{id,σ} : \overline{U_{α}^{id}} − → \overline{U_{α}^{σ}}, put j_{τ,σ} := j_{σ}j_{τ}^{−1} : \overline{U_{α}^{τ}} − → \overline{U_{α}^{σ}} and claim that all these can be glued together to give a scheme \overline{U_{α}^{n}}. In fact there is a maximal open set \overline{U_{α}^{τ,σ}}, where j_{τ,σ} is a morphism. Since \overline{U_{α}} is normal and quasi-finite over Pic_{X/O_{K}}^{γ}, j_{τ,σ} is quasi-finite and therefore an open embedding by Zariski’s Main Theorem. The same argument is true for the inverse map and the cocycle relations are trivial.

\overline{U_{α}^{n}} clearly fulfills i) and is quasi-finite over Pic_{X/O_{K}}^{γ}. In order to show it is finite over its image we need that \overline{p}_{α}^{T} : \overline{U_{α}^{n}} ×_{Pic}_{X/O_{K}}^{γ} T → T is finite as soon as it is surjective for a scheme T being étale over Pic_{X/O_{K}}^{γ}. As by EGA II, Corollaire 5.4.3.ii) properness descends under proper surjective base changes we may assume T to be normal (but no more étale) and K(T) ⊇ K(\overline{U_{α}^{n}}) = K(U_{α,red})^{n}. But in that case p_{α}^{T} splits completely into a disjoint union of birational maps, i.e. open embeddings. As the full Galois group G acts on \overline{U_{α}^{n}} ×_{Pic}_{X/O_{K}}^{γ} T surjectivity implies that all of them must be isomorphisms.

Now, for every l-tuple (α_{1}, …, α_{l}) ∈ A_{1} × … × A_{l} let K_{(α_{1}, ... ,α_{l})} := K(\overline{U_{α_{1}}^{n}}) · . . . · K(\overline{U_{α_{l}}^{n}}) be the composite of the corresponding function fields. We put U_{(α_{1}, ... ,α_{l})}^{i} for i ∈ {1, …, l} to be the normalization of \overline{U_{α_{i}}^{n}} in the field extension K_{(α_{1}, ... ,α_{l})} and U_{(α_{1}, ... ,α_{l})}^{0} to be the normalization of O := Pic_{X/O_{K}}^{γ} ×_{Spec O_{K}} (Spec O_{K} \{𝔭_{1}, …, 𝔭_{l}}) in K_{(α_{1}, ... ,α_{l})}. The U_{(α_{1}, ... ,α_{l})}^{i} can be glued together giving an integral scheme U_{(α_{1}, ... ,α_{l})} being equipped with a map to Pic_{X/O_{K}}^{γ}, which is finite onto its image. Indeed, by our construction there is something to be glued only outside the the exceptional fibers. But there \overline{U_{α_{i}}^{n}} is just an open subscheme of the normalization of O in K(U_{α_{i},red})^{n}. Consequently, outside the exceptional fibers each U_{(α_{1}, ... ,α_{l})}^{i} is nothing but an open subscheme of the normalization of O in K_{(α_{1}, ... ,α_{l})}. So there are natural identifications and the cocycle relations are trivial. We put

P′′ := ∐_{(α_{1}, ... ,α_{l})∈A_{1}× ... ×A_{l}} U_{(α_{1}, ... ,α_{l})}.  (38)

By construction P′′ admits properties i), ii) and iii). For v) there is exactly the same argument as in step B.3 giving the scheme P′ desired. Again to show iv) it is sufficient to consider P′′. By the definition of an algebraic space a map i : Spec O_{K} → Pic_{X /O_{K}}^{γ} gives rise to a morphism of schemes ∐_{α∈A} \overline{V_{α}} → ∐_{α∈A} \overline{U_{α}^{n}}. By our construction, \overline{V_{α}} is finite over its image in Spec O_{K}. As we started with a covering there is necessarily one combination (α_{1}, …, α_{l}) ∈ A_{1} × … A_{l} giving a base change map V_{(α_{1}, ... ,α_{l})} → U_{(α_{1}, ... ,α_{l})}, where V_{(α_{1}, ... ,α_{l})} is surjective, hence necessarily finite, over Spec O_{K}. That implies the assertion.

The proof is finished along the lines of steps B.4 and B.5. □ 3.8 Proposition. Let R be a discrete valuation ring with maximal ideal 𝔪 and fraction field K. Further, let X be an integral scheme, projective and flat over R whose generic fiber is geometrically connected. Equip it with some ample line bundle T . Assume L ∈ Pic(X) satisfies the following conditions.

i) Its restriction L_{K} ∈ Pic (X_{K}) to the generic fiber is trivial.

ii) On every component \overline{X_{𝔪}^{i}} of the geometric fiber \overline{X_{𝔪}}

deg_{T} L|\overline{X_{𝔪}^{i}} = 0.  (39)

Then L|_{X_{𝔪}} is numerically equivalent to zero. Proof. We may assume dim X ≥ 2. Let p : X′ → X be the normalization. Note, since p is finite, p^{∗}T is again an ample line bundle. Thus p^{∗}L also satisfies conditions i) and ii). In fact this is trivial for i), while for ii) let \overline{X_{𝔪}^{′i}} be a component of the geometric fiber \overline{X′_{𝔪}}. Then p_{∗} [\overline{X_{𝔪}^{′i}}] = n [\overline{X_{𝔪}^{j}}] ∈ CH^{0}(\overline{X_{𝔪}}) for some component \overline{X_{𝔪}^{j}} and a non-negative integer n. Therefore the assumption deg_{T} L|\overline{X_{𝔪}^{j}} = deg c_{1}(L) · c_{1}(T )^{dim X −2}|\overline{X_{𝔪}^{j}} = 0 implies deg_{p^{∗}T} p^{∗}L|\overline{X_{𝔪}^{′i}} = deg c_{1}(p^{∗}L) · c_{1}(p^{∗}T )^{dim X −2}|\overline{X_{𝔪}^{′i}} = 0. By Lemma 3.9 below p^{∗}L is torsion, at least after a finite unramified extension of R. But then L|_{X_{𝔪}} is numerically equivalent to zero as for this by SGA6, Exp. XIII, Théorème 4.6 one only has to verify that certain intersection numbers of line bundles are 0 and this can be checked after pull-back under a finite morphism (cp. Lemma A.4). □ 3.9 Lemma. Let X be as in Proposition 3.8 and assume further it is normal and dim X ≥ 2.

a) Let D_{𝔪} be the vector space of Weil ℚ-divisors supported over 𝔪. Then there is a symmetric bilinear form ⟨ . , . ⟩ : D_{𝔪} × D_{𝔪} → ℚ with the properties below.

i) ⟨D, [X_{𝔪}]⟩ = 0 for every D ∈ D_{𝔪}.

ii) If D is effective with coefficient 0 at the component [X_{𝔪}^{i}], then ⟨D, [X_{𝔪}^{i}]⟩ ≥ 0.

ii’) If, in addition, D is Cartier and |D| ∩ |[X_{𝔪}^{i}]| ≠ ∅, then ⟨D, [X_{𝔪}^{i}]⟩ > 0. iii) If D is a Cartier divisor, then ⟨D, D′⟩ = deg D·c_{1}(T )^{dim X −2}|_{D′}.

b) On the vector space C_{𝔪} of Cartier ℚ-divisors supported over 𝔪, the form ⟨ . , . ⟩ is negative semi-definite, whereas only rational multiples of the fiber [X_{𝔪}] have square zero. In particular, any line bundle L = O(D), where D is supported over 𝔪, whose degree vanishes at every component of X_{𝔪}, is torsion. One even has L ≅ O_{X} if the coefficients of the divisor [X_{𝔪}] are relatively prime. Proof. a) Let X_{𝔪}^{i} and X_{𝔪}^{j} be two different components of X_{𝔪}. We let

X_{𝔪}^{i} · X_{𝔪}^{j} := ∑_{x∈X^{(2)}} l (O_{X ,x}/(J_{X_{𝔪}^{i}} + J_{X_{𝔪}^{j}})O_{X ,x}) · x  (40)

be their naive intersection product and put

⟨[X_{𝔪}^{i}], [X_{𝔪}^{j}]⟩ := deg c_{1}(T^{dim X −2})|_{X_{𝔪}^{i}·X_{𝔪}^{j}}.  (41)

This can easily be extended to a symmetric bilinear form satisfying i). But then iii) is clear, since there are no higher Tor’s occurring when intersecting with a Cartier divisor. ii) is trivial from the construction. For ii’) note that by dimension theory dim |D| ∩ |[X_{𝔪}^{i}]| ≥ dim X − 2 as soon as they meet. For two arbitrary Weil divisors this would be wrong in general.

b) Let [X_{𝔪}] = ∑_{i} C_{i}[X_{𝔪}^{i}]. Then

0 = ⟨[X_{𝔪}^{i}], [X_{𝔪}]⟩ = C_{i}⟨[X_{𝔪}^{i}], [X_{𝔪}^{i}]⟩ + ∑_{k≠i} C_{k}⟨[X_{𝔪}^{i}], [X_{𝔪}^{k}]⟩,  (42)

hence ⟨C_{i}[X_{𝔪}^{i}], C_{i}[X_{𝔪}^{i}]⟩ = − ∑_{k≠i}⟨C_{i}[X_{𝔪}^{i}], C_{k}[X_{𝔪}^{k}]⟩. With rational coefficients b_{i} one obtains

⟨ ∑_{i} b_{i}C_{i}[X_{𝔪}^{i}], ∑_{i} b_{i}C_{i}[X_{𝔪}^{i}]⟩ = ∑_{i} b_{i}^{2}⟨C_{i}[X_{𝔪}^{i}], C_{i}[X_{𝔪}^{i}]⟩  (43)

+ ∑_{j≠k} b_{j}b_{k}⟨C_{j}[X_{𝔪}^{j}], C_{k}[X_{𝔪}^{k}]⟩

= −\frac{1}{2} ∑_{j≠k}(b_{j} − b_{k})^{2}⟨C_{j}[X_{𝔪}^{j}], C_{k}[X_{𝔪}^{k}]⟩.

Therefore ⟨ . , . ⟩ is negative semi-definite. Now assume D = ∑_{i} b_{i}C_{i}[X_{𝔪}^{i}] to be ℚ-Cartier with square zero, while not all the b_{i} are equal. Without restriction we may suppose some of the b_{i} are zero, b_{i_{1}} = … = b_{i_{l}} = 0, and the others are not, b_{j_{1}} ≠ 0, …, b_{j_{m}} ≠ 0. But this implies ⟨C_{i_{n}}[X_{𝔪}^{i_{n}}], C_{j_{o}}[X_{𝔪}^{j_{o}}]⟩ = 0 for any n and o, therefore ⟨[X_{𝔪}^{i_{n}}], D⟩ = 0 for any n being a contradiction to a.ii’) since X_{𝔪} is connected. □ 3.10 Proposition. Let L/K be an extension of number fields, X /O_{K} a regular arithmetic variety, the irreducible components of all whose fibers are geometrically irreducible, and L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}). Then there exist some finite field extension L′/L and H ∈ Pic (X ×_{Spec O_{K}} Spec O_{L′}) being non-trivial only in non-smooth fibers such that

L′ := L_{O_{L′}} ⊗ H ∈ Pic (X ×_{Spec O_{K}} Spec O_{L′})  (44)

admits the property that for each 𝔭 ∈ Specm O_{K} splitting in O_{L}, say 𝔭O_{L} = 𝔮_{1}^{α_{1}} · . . . · 𝔮_{l}^{α_{l}}, the restrictions L′|_{X\overline{𝔮_{i}}} are mutually numerically equivalent. Proof. The assumption makes sure that on X ×_{Spec O_{K}} Spec O_{L} every vertical Weil divisor admits a non-zero multiple which is Cartier. Therefore, on the vector spaces D_{𝔮_{i}}/[X_{𝔮_{i}}] of Weil ℚ-divisors supported over 𝔮_{i}, there is a negative definite intersection pairing. We would be finished, if we were allowed to use a vertical ℚ-divisor for H. Lemma 3.11 below completes the proof. □ 3.11 Lemma. Let n be a positive integer, S be a Dedekind ring and X /S a proper scheme whose generic fiber is smooth. Denote its open subscheme consisting of all the smooth fibers by X^{(s)}. Consider L ∈ Pic (X) such that L|_{X^{(s)}} ≅ O_{X^{(s)}}. Then there exists a finite flat morphism p : Spec T → Spec S of spectra of Dedekind rings such that p_{X}^{∗} L ∈ Pic (X ×_{Spec S} Spec T) is divisible by n in that sense that there exists Q ∈ Pic (X ×_{Spec R} Spec T) with Q|_{X^{(s)}} ≅ O_{X^{(s)}} and Q^{⊗n} ≅ p_{X}^{∗} L. Proof. The question is local in the base. We may even assume S to be a strictly henselian discrete valuation ring. We note that, if the residue characteristic is l > 0, then pull-back under Frobenius of any line bundle is divisible by l. So assume from now on n to be prime to the residue characteristic.

Let (S, s) → (T, t) be a totally ramified homomorphism of degree n between strictly henselian discrete valuation rings and denote the associated morphism of affine schmes by p.. The only obstruction against L ∈ Pic (X) to be divisible by n in the sense above is

c_{1}(L) ∈ H_{ét,X_{s}}^{2}(X , μ_{n}) = H_{ét,s}^{0}(Spec S, R^{2}π_{∗}μ_{n}).  (45)

On the other hand

H_{ét,(X_{T})_{t}}^{2}(X ×_{Spec S} Spec T, μ_{n}) = H_{ét,t}^{0}(Spec T, R^{2}π_{T}_{∗}p_{X}^{∗} μ_{n})  (46)
= H_{ét,s}^{0}(Spec S, R^{2}π_{∗}(p_{X}_{∗}p_{X}^{∗} μ_{n}))

as p_{X} : X ×_{Spec S} Spec T → X is acyclic being a finite morphism. We have to study the canonical map between these two spaces. i.e. the map between the stalks of the direct image sheaves occurring. By proper base change this is a map H_{ét}^{2}(X\overline{s}, μ_{n}) → H_{ét}^{2}(X\overline{s}, p_{X\overline{s}}_{∗}p_{X\overline{s}}^{∗}μ_{n}). But one easily sees p_{X\overline{s}}_{∗}p_{X\overline{s}}^{∗}μ_{n} ≅ μ_{n}. We claim the map is multiplication by deg p_{X\overline{s}}, hence zero. It comes from the pull-back p_{X\overline{s}}^{∗} : H_{ét}^{2}(X\overline{s}, μ_{n}) → H_{ét}^{2}(X\overline{s} ×_{Spec S} Spec T, p_{X\overline{s}}^{∗}μ_{n}). But there is an isomorphism of topoi (p_{X\overline{s}}_{∗}, p_{X\overline{s}}^{∗}) : (X\overline{s} ×_{Spec S} Spec T)_{ét} → (X\overline{s})_{ét}. Finally, note that p_{X\overline{s}}_{∗}p_{X\overline{s}}^{∗}s = deg p_{X\overline{s}} · s for any class in étale cohomology. □

4 The Case of an Arithmetic Surface

4.1 Conventions. An arithmetic surface is an arithmetic variety of dimension 2. Note that this is exactly the case d = 1 in the notation introduced in the beginning of the paper. An Arakelov surface is an Arakelov variety of dimension 2, i.e. an arithmetic surface X equipped with a Kähler metric ω being invariant under complex conjugation. 4.2 Proposition. Let X /O_{K} be an arithmetic surface. Then there exists C ∈ ℝ with the property below: Let L′/L/K be finite field extensions and

p : X̃ → X ×_{Spec O_{K}} Spec O_{L′} → X ×_{SpecO_{K}} Spec O_{L}  (47)

the composition of a proper birational morphism with a base extension. Then for each \overline{T} ∈ P̂ic(X), each Kähler metric ω on X(ℂ) being invariant under F_{∞} and every L ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) with χ(L_{L}) ≠ 0

|h_{p^{∗}\overline{T} ,p^{∗}ω}(p^{∗}L) − h_{\overline{T} ,ω}(L)| ≤ C · deg(T_{K}).  (48)

Proof. We put C := c_{1}(T_{K})^{d} · ∑_{𝔭} log(♯O_{K}/𝔭) C(𝔭), where the C(𝔭) come from Lemma 4.3 below. In the case that L = K Proposition 2.6 gives the assertion directly, since R^{j}p_{∗}O_{X′} = 0 for j ≥ 2 and the Euler characteristic breaks down to the length of the space of global sections. For arbitrary L note that X ×_{Spec O_{K}} Spec O_{L} → X is acyclic for coherent sheaves. □ 4.3 Lemma. Let X /O_{K} be an arithmetic surface. Then, there exists some function C : Specm O_{K} → ℝ such that

i) Only finitely many values of C are different from zero.

ii) For any composition p : X̃ → X ×_{Spec O_{K}} Spec O_{L} → X of a birational morphism and a base extension one has

l (R^{1}p_{∗}OX̃|_{X_{𝔭}}) ≤ C(𝔭) · [L : K].  (49)

Proof. First note, if X_{1} →^{p_{1}} X_{2} →^{p_{2}} X_{3} is any composition of proper birational maps of surfaces, then the lower term sequence associated to the Leray spectral sequence for O_{X_{1}} reads 0 → R^{1}p_{2}_{∗}O_{X_{2}} → R^{1}(p_{2}p_{1})_{∗}O_{X_{1}} → p_{2}_{∗}R^{1}p_{1}_{∗}O_{X_{1}} → 0. So in order to show an estimate for p_{2} it is sufficient to prove it for p_{2}p_{1}.

For arithmetic surfaces there exists a semi-stable reduction, i.e. there is some field extension L_{0} and a birational desingularization p′ : X′ → X ×_{Spec O_{K}} Spec O_{L_{0}} → X such that X′ is semi-stable^{19}. We put C(𝔭) := l (R^{1}p′_{∗}O_{X′}|_{X_{𝔭}})/[L_{0} : K]. Note that R^{1}p′_{∗}O_{X′} is a coherent sheaf supported in finitely many points.

To prove the assertion for X̃ → X ×_{Spec O_{K}} Spec O_{L} → X we may assume by flat base change that L ⊇ L_{0}. Further, it is sufficient to consider X′ ×_{(X ×_{Spec O_{K}} Spec O_{L_{0}} )} X̃ instead of X̃ and we may even regard its desingularization X̃′. So everything we need is that X̃′ → X′ ×_{Spec O_{L_{0}}} Spec O_{L} is acyclic for the structure sheaf, i.e. that X′ ×_{Spec O_{L_{0}}} Spec O_{L} has only rational singularities.

But X′ ×_{Spec O_{L_{0}}} Spec O_{L} is regular in codimension 1 as X′ is smooth in codimension 1 and Cohen-Macaulay as it is flat of relative dimension zero over X′. So it is normal. In formal coordinates its singularities are given by Spf Ô_{L_{0},𝔭}[[X, Y]]/(XY − 𝔭^{n}) for some positive integer n. Therefore they are toroidal embeddings in the sense of Section IV.3 of Kempf, Knudsen, Mumford and Saint-Donat^{19}, in particular they are rational. □ 4.4 Theorem. Let (X /O_{K}, ω) be an Arakelov surface whose generic fiber is of genus g. Assume x ∈ X (K) is a K-valued point such that O(x) can be extended to some T ∈ Pic (X), which we suppose to be equipped with a hermitian metric. Transfer the Θ-divisor to Pic^{g}(X_{K}) via the map Pic^{g−1}(X_{K}) → Pic^{g}(X_{K}) defined by x. Then there exists a function C = C_{X ,ω,\overline{T}} : ℝ → ℝ with the property below: For each finite field extension L/K and each L ∈ Pic(X ×_{Spec O_{K}} Spec O_{L}) of degree g on X_{L} satisfying

| deg L|\overline{X_{𝔭}^{i}}| < H  (50)

for every component \overline{X_{𝔭}^{i}} of the geometric fibers \overline{X_{𝔭}}, the inequality

|h_{\overline{T} ,ω}(L) − \frac{1}{[L : K]} h_{Θ}([L_{K}])| < C(H)  (51)

is true, where h_{Θ} is the height on Pic^{g}(X_{K}) defined by Θ. Proof. By the existence of semi-stable reduction and Proposition 4.2 we may assume X to be semi-stable. In that situation the statement of Lemma 3.9 is known for a long time. But then, using Proposition 3.10 and Proposition 2.2 one easily sees that it is sufficient to assume that over any 𝔭 ∈ Specm O_{K} that splits in O_{L}, the restrictions of L to the corresponding geometric fibers are mutually numerically equivalent. By Proposition 3.8 we deal with finitely numerical types only. We may consider them separately. Exactly the same argument as in 3.7, step B.ii), yields a separated scheme Pic_{X /O_{K}}^{σ} being of finite type over O_{K}. We are forced to work with a coset P ↪ Pic_{X /O_{K}} of it, whose generic fiber is Pic_{X/K}^{g}.

Consider the line bundle

O((π_{1}×π_{21})^{∗}Δ + … + (π_{1}×π_{2g})^{∗}Δ) ∈ Pic (X × X^{g}),  (52)

where Δ denotes the diagonal in X × X. It defines a morphism

c : X^{g} → Pic_{X/K}^{g}.  (53)

Let P′ denote the normalization of P in K(X^{g}). It is equipped with a proper morphism p : P′ → P such that P′ ×_{Spec O_{K}} Spec K ≅ X^{g} and p goes over into c under this isomorphism. By functoriality there is a line bundle V ∈ Pic (X × P′) such that V|_{X×X^{g}} ≅ O((π_{1}×π_{21})^{∗}Δ + … + (π_{1}×π_{2g})^{∗}Δ). We claim that D := det Rπ_{2}_{∗}V|_{X^{g}} ≅ c^{∗}O(−Θ) ⊗ ⨂_{i=1}^{g} π_{i}^{∗}O(x). In fact, there is an exact sequence

0 → V′|_{X×X^{g}} → V|_{X×X^{g}} → V|_{{x}×X^{g}} → 0,  (54)

where V′ := V ⊗ π_{1}^{∗}T^{−1} ∈ Pic (X × P′). Consequently,

det Rπ_{2}_{∗}V|_{X^{g}} ≅ det Rπ_{2}_{∗}(V|_{X×X^{g}})  (55)
≅ det Rπ_{2}_{∗}(V′|_{X×X^{g}}) ⊗ O(π_{1}^{∗}(x) + … + π_{g}^{∗}(x)).

But by Faltings^{4} or Moret-Bailly^{21} one has det Rπ_{2}_{∗}(V′|_{X×X^{g}}) ≅ c^{∗}O(−Θ).

We are in the situation of Proposition 2.7 with n = 1, U := V ⊗ π_{2}^{∗}D^{−1} and H := D. Therefore

h_{\overline{T} ,ω} (V|_{X ×\overline{(p_{1}, ... ,p_{g})}})  (56)
= \frac{1}{[L : K]} h\overline{T} (\overline{(p_{1})} + … + \overline{(p_{g})}) − \frac{1}{[L : K]} h_{D} (\overline{(p_{1}, …, p_{g})}) + O(1).

The statement follows easily from Lemma 4.5 below. □ 4.5 Lemma. Let g be a positive integer and X /O_{K} be an arithmetic variety equipped with \overline{T} ∈ P̂ic(X). Let P/O_{K} be a separated scheme of finite type whose generic fiber is X^{g} and let \overline{G} ∈ P̂ic(P) be such that G|_{X^{g}} ≅ ⨂_{i=1}^{g} π_{i}^{∗}T |_{X^{g}}. Then, for any number field L/K and any p_{1}, …, p_{g} ∈ X(L)

h\overline{T} ((\overline{p_{1}}) + … + (\overline{p_{g}})) = h\overline{G} (\overline{(p_{1}, …, p_{g})}) + [L : K] · O(1) (57)

as soon as (p_{1}, …, p_{g}) ∈ X^{g}(L) can be extended to some \overline{(p_{1}, …, p_{g})} ∈ P(O_{L}). Here O(1) is a bounded function on ⨆_{L/K finite} X^{g}(L), but the bound depends on \overline{T} and \overline{G}. Proof. Consider X^{g} instead of P first. By Lemma A.2 one has

h\overline{T} ((\overline{p_{1}}) + … + (\overline{p_{g}})) = ∑_{i=1}^{g} h_{π_{i}^{∗}\overline{T}} (\overline{(p_{1}, …, p_{g})}) = h_{⨂_{i=1}^{g}π_{i}^{∗}\overline{T}} (\overline{(p_{1}, …, p_{g})}).

(58)

But there is a scheme P′ equipped with birational morphisms X^{g} ←^{q} P′ →^{p} P, where p is proper. Indeed let P′ be the closure of the diagonal in X^{g}×_{Spec O_{K}} P. Then h\overline{G} (\overline{(p_{1}, …, p_{g})}) = h_{p^{∗}\overline{G}} (\overline{(p_{1}, …, p_{g})}) and

h_{⨂_{i=1}^{g}π_{i}^{∗}\overline{T}} (\overline{(p_{1}, …, p_{g})}) = h_{q^{∗}⨂_{i=1}^{g}π_{i}^{∗}\overline{T}} (\overline{(p_{1}, …, p_{g})}).  (59)

The difference of the two terms on the right hand side is bounded as there are changes only in the special fibers (and the metric). □ 4.6 Remark (Asymptotic behaviour). If E, M ∈ Pic (X ×_{Spec O_{K}} Spec O_{L}) are equipped with distinguished metrics, then there is a formula for h_{\overline{T} ,ω}(E ⊗ M^{n}) in terms of arithmetic intersection numbers^{17}. In particular, if deg(E_{L}) = g, deg(M_{L}) = 0 and deg(T_{L}) = 1, then the dominating term for n → ∞ is

−\frac{1}{2} d̂eg γ (\overline{M}, \overline{M}) · n^{2}.  (60)

By Theorem 4.4 we recover the formulas h_{NT,Θ}([M_{K}]) = −\frac{1}{2} d̂eg γ (\overline{M}, \overline{M}) and

h_{NT,Θ+Θ^{−}}([M_{K}]) = − d̂eg γ (\overline{M}, \overline{M})  (61)

for the Néron-Tate, which are due to Faltings and Hriljac.

Appendix

A Some Facts Concerning Intersection Theory on Singular Arithmetic Varieties

A.1 Remark. It is a well-known phenomenon that there is no good intersection product available for arbitrary cycles on singular varieties. Nevertheless it is possible to intersect cycles with Chern classes of vector bundles. The same situation occurs in arithmetic intersection theory. It seems that the right context to describe possible intersection products is arithmetic K-Theory^{9} more than arithmetic Chow theory. A.2 Lemma. A. For any projective arithmetic variety X /O_{K}, whose generic fiber X is regular, any natural number r and any partition (r_{1}, …, r_{s}) of r there is a unique multi-linear map

α_{r_{1}, ... ,r_{s}} : \underbrace{K̂_{0}(X) × … × K̂_{0}(X)}_{s times} × Z_{r}(X) ⟶ ĈH^{1}(Spec O_{K})_{ℚ}, (62)

called restriction product, satisfying the following conditions.

i) If X is regular, then α_{r_{1}, ... ,r_{s}} coincides with the product

(ĉh_{r_{1}}(.) · . . . · ĉh_{r_{s}}(.) | . )  (63)

defined by Bost, Gillet and Soulé^{3}.

ii) For any morphism f : X → X′ there is a projection formula

α_{r_{1}, ... ,r_{s}} (v_{1}, …, v_{s}, f_{∗}(Z)) = α_{r_{1}, ... ,r_{s}} (f^{∗}(v_{1}), …, f^{∗}(v_{s}), Z) . (64)

B. If (r_{1}, …, r_{s}) = (1, …, 1), the specialized product

α : \underbrace{P̂ic(X) × … × P̂ic(X)}_{r times} × Z_{r}(X) ⟶ ĈH^{1}(Spec O_{K})_{ℚ}  (65)

exists without the assumption on X to be regular and fulfills i) and ii) below. i) If X is regular, then α coincides with the product (ĉ_{1}(.) · . . . · ĉ_{1}(.) | . ) . ii) For any morphism f : X → X′ there is a projection formula

α (\overline{U_{1}}, …, \overline{U_{r}}, f_{∗}(Z)) = α (f^{∗}(\overline{U_{1}}), …, f^{∗}(\overline{U_{r}}), Z) .  (66)

Proof. We will show A only. If \overline{V_{1}}, …, \overline{V_{s}} are hermitian vector bundles on X, then there are a morphism ι : X → P, where P is projective and smooth over Spec O_{K}, and vector bundles V_{1,P} , …, V_{r,P} on P such that ι^{∗}(V_{j,P}) = V_{j}. One may even choose ι such that the hermitian metric on \overline{V}_{j} is a pullback of one on V_{j,P} and any differential form on X(ℂ) extends to P (ℂ).

ι^{∗}(v_{j,P}) = v_{j}  (67)

Then for an r-dimensional cycle Z in X define

α_{r_{1}, ... ,r_{s}} (v_{1}, …, v_{s}, Z) := (ĉh_{r_{1}}(v_{1,P}) · . . . · ĉh_{r_{s}}(v_{s,P}) | ι_{∗}(Z)) , (68)

where (.|.) denotes the restriction product ĈH^{r}(P)×Z_{r}(P)→ĈH^{1}(Spec O_{K})_{ℚ}. In their remark after Proposition 3.2.1 Bost, Gillet and Soulé^{3} have shown independence of the ι chosen. Uniqueness of α_{r_{1}, ... ,r_{s}} is clear. □ A.3 Lemma. a) For any projective arithmetic variety π : X → O_{K}, whose generic fiber X is regular, any natural number r and any partition (r_{1}, …, r_{s}) of r, there is a unique multi-linear map

β_{r_{1}, ... ,r_{s}} : \underbrace{K̂_{0}(X) × … × K̂_{0}(X)}_{s times} × ĈH_{r}(X) ⟶ ĈH^{1}(Spec O_{K})_{ℚ}, (69)

called intersection product, satisfying the following condition.

If y ∈ ĈH_{r}(X) is represented by a cycle (Y, g_{Y}), where g_{Y} is a differential form with logarithmic singularities along |Y |, then

β_{r_{1}, ... ,r_{s}} (v_{1}, …, v_{s}, y) := α_{r_{1}, ... ,r_{s}} (v_{1}, …, v_{s}, Y)  (70)

+\overline{(0, ( ∫_{X (ℂ)} g_{Y} ω_{ĉh_{r_{1}} (v_{1})} · . . . · ω_{ĉh_{rs} (v_{s})})_{σ:K↪ℂ})}.

b) In particular, if (r_{1}, …, r_{s}) = (1, …, 1), there is a specialized product

β : \underbrace{P̂ic(X) × … × P̂ic(X)}_{r times} × ĈH_{r}(X) ⟶ ĈH^{1}(Spec O_{K})_{ℚ}  (71)

c) If X is regular, then β_{r_{1}, ... ,r_{s}} coincides with π_{∗}[ĉh_{r_{1}}(.) · . . . · ĉh_{r_{s}}(.) · . ].

d) For any morphism f : X → X′ of projective arithmetic varieties having regular generic fibers such that f_{K} is smooth, there is the projection formula

β_{r_{1}, ... ,r_{s}} (f^{∗}(v_{1}), …, f^{∗}(v_{s}), y) = β_{r_{1}, ... ,r_{s}} (v_{1}, …, v_{s}, f_{∗}(y)) .  (72)

e) If f : X → X′ is a proper and flat morphism such that f_{K} is étale, then

β_{r_{1}, ... ,r_{s}} (f^{∗}(v_{1}), …, f^{∗}(v_{s}), f^{∗}(z)) = deg f · β_{r_{1}, ... ,r_{s}} (v_{1}, …, v_{s}, z) . (73)

Proof. a) Take the property given as a definition. The integral over X (ℂ) converges because of the logarithmic singularities of g_{Y}. Independence of the cycle chosen immediately carries over from the regular case. Indeed, assume (Y, g_{Y}) is an arithmetic r-cycle, rationally equivalent to zero. If ι : X → P is a closed embedding into a regular arithmetic variety, then (ι_{∗}Y, ι_{∗}g_{Y}) is also an arithmetic r-cycle, rationally equivalent to zero.

b) and c) are trivial now and d) follows from the projection formula for α_{r_{1}, ... ,r_{s}} and the same argument for the integrals as in the regular case.

e) If f : X → X′ is flat inducing an étale covering on generic fibers, then there is a pull-back homomorphism f^{∗}: ĈH_{d}(X′) → ĈH_{d}(X) satisfying f_{∗}f^{∗}z = deg f · z. □ A.4 Lemma. A. a) For any projective arithmetic variety π : X → O_{K}, whose generic fiber X is regular and equidimensional of dimension d, and any partition (d_{1}, …, d_{s}) of d + 1 there is a unique multi-linear map

γ_{d_{1}, ... ,d_{s}} : \underbrace{K̂_{0}(X) × K̂_{0}(X) × … × K̂_{0}(X)}_{s times} ⟶ ĈH^{1}(Spec O_{K})_{ℚ}  (74)

satisfying the following conditions.

i) If X is regular, then

γ_{d_{1}, ... ,d_{s}} (v_{1}, …, v_{s}) = π_{∗}[ĉh_{d_{1}}(v_{1}) · . . . · ĉh_{d_{s}}(v_{s})] .  (75)

ii) If f : X′ → X is an alteration, i.e. a surjective and generically finite morphism, then

γ_{d_{1}, ... ,d_{s}} (f^{∗}(v_{1}), …, f^{∗}(v_{s})) = deg f · γ_{d_{1}, ... ,d_{s}} (v_{1}, …, v_{s}) .  (76)

b) γ_{.} ( .) is compatible with the natural action of the symmetric group.

c) One has

γ_{d_{1}, ... ,d_{s}} (v_{1}, …, v_{s}) = α_{d_{1}, ... ,d_{s}} (v_{1}, …, v_{s}, [X]) ,  (77)

where [X] denotes the fundamental class of X.

d) If d_{s} = 1 and X is regular, then

γ_{d_{1}, ... ,d_{s−1}, 1} (v_{1}, …, v_{s−1}, v_{s}) = β_{d_{1}, ... ,d_{s−1}} (v_{1}, …, v_{s−1}, ĉ_{1}(v_{s})) . (78)

B. If (d_{1}, …, d_{s}) = (1, …, 1), then the specialized product

γ : \underbrace{P̂ic(X) × P̂ic(X) × … × P̂ic(X)}_{d+1 times} ⟶ ĈH^{1}(Spec O_{K})_{ℚ}  (79)

exists without assuming X to be regular. It has the properties below. i) If X is regular, then

γ (\overline{U_{1}}, …, \overline{U_{d+1}}) = π_{∗} [ĉ_{1}(\overline{U_{1}}) · . . . · ĉ_{1}(\overline{U_{d+1}})] .  (80)

ii) If f : X′ → X is an alteration, i.e. a surjective and generically finite morphism, then

γ (f^{∗}(\overline{U_{1}}), …, f^{∗}(\overline{U_{d+1}})) = deg f · γ (\overline{U_{1}}, …, \overline{U_{d+1}}) .  (81)

iii) γ is symmetric.

iv) One has

γ (\overline{U_{1}}, …, \overline{U_{d+1}}) = α (\overline{U_{1}}, …, \overline{U_{d+1}}, [X]) ,  (82)

where [X] denotes the fundamental class of X.

v) If the generic fiber X of X is regular, then

γ(\overline{U_{1}}, …, \overline{U_{d}}, \overline{U_{d+1}}) = β(\overline{U_{1}}, …, \overline{U_{d}}, ĉ_{1}(\overline{U_{d+1}})).  (83)

Proof. Again we show A only. By de Jong^{18} for every X there is a resolution of singularities by alterations, so let p : X′ → X be surjective and generically finite with X′ regular and put

γ_{d_{1}, ... ,d_{s}} (v_{1}, …, v_{s}) := \frac{1}{deg p} (πp)_{∗} [ĉh_{d_{1}}(p^{∗}v_{1}) · . . . · ĉh_{d_{s}}(p^{∗}v_{s})] . (84)

As ii) is true for X regular, this definition is independent of the choice of p. Property i) is trivial, ii) and b) immediately carry over from the regular case.

c) Lemma A.2.A.ii) implies

α_{d_{1}, ... ,d_{s}} (v_{1}, …, v_{s}, [X]) = \frac{1}{deg p} α_{d_{1}, ... ,d_{s}} (v_{1}, …, v_{s}, p_{∗}[X′])  (85)
= \frac{1}{deg p} (ĉh_{d_{1}}(p^{∗}v_{1}) · . . . · ĉh_{d_{s}}(p^{∗}v_{s})| X′)

being equal to our formula for γ_{d_{1}, ... ,d_{s}} (v_{1}, …, v_{s}) above.

d) Assume without loss of generality that v_{s} is given by a hermitian line bundle \overline{U} and there is some section 0 ≠ s ∈ Γ(X , U_{d+1}). Then, by definition,

β_{d_{1}, ... ,d_{s−1},1} (v_{1}, …, v_{s−1}, ĉ_{1}(\overline{U}))  (86)
= α_{d_{1}, ... ,d_{s−1},1} (v_{1}, …, v_{s−1}, div(s))

+\overline{(0, ( ∫_{X (ℂ)}− log ∥s∥^{2} ω_{ĉh_{d_{1}} (v_{1})} · . . . · ω_{ĉh_{d_{s−1}} (v_{s−1})})_{σ:K↪ℂ})}.

On the other hand, we have p_{∗}(div(p^{∗}s)) = deg p · div(s). This is clear in the case when X is normal, as then p is flat over some X \ Y with codim Y ≥ 2. If p is the normalization of X, then it follows from Example 1.2.3 in Fulton’s book^{6}. Consequently,

α_{d_{1}, ... ,d_{s−1},1} (v_{1}, …, v_{s−1}, div(s))  (87)
= \frac{1}{deg p} α_{d_{1}, ... ,d_{s−1},1} (p^{∗}(v_{1}), …, p^{∗}(v_{s−1}), div(p^{∗}s)) .

Finally, it is obvious that

∫_{X (ℂ)}− log ∥s∥^{2} ω_{ĉh_{d_{1}} (v_{1})} · . . . · ω_{ĉh_{d_{s−1}} (v_{s−1})}  (88)

= \frac{1}{deg p} ∫_{X′(ℂ)}− log ∥p^{∗}s∥^{2} ω_{ĉh_{d_{1}} (p^{∗}v_{1})} · . . . · ω_{ĉh_{d_{s−1}} (p^{∗}v_{s−1})}.

□

Acknowledgments

While doing this work the author was supported in part by a European Union Training and Mobility of Researchers Grant and by a Research Scholarship of the Deutsche Forschungsgemeinschaft (DFG). Further he wishes to express his gratitude to the International Centre for Theoretical Physics at Trieste for its hospitality. Finally the referee should be mentioned who read the first version of the manuscript very carefully and found several inaccuracies.

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