On the distribution of small points on abelian and toric varieties

Jörg Jahnel

Abstract

We give a unified proof for a theorem of S. Zhang and a theorem of Yu. Bilu. First, if A is a polarized abelian variety over ℚ and X ⊂ A a closed subvariety containing a generic sequence of points with Néron-Tate height tending to zero then X is torsion. Secondly, if X ⊂ P_{ℚ}^{n} contains a generic sequence of points whose naive heights tend to zero then X is preperiodic under coordinate-wise formation of d-th powers for every d ⩾ 2.

1. Introduction

Classical results. In 1972, J.-P. Serre proved the following remarkable result.

Theorem 1.1 (Serre). Let K be an algebraic number field and E be an elliptic curve over K without complex multiplication. Then, for almost every prime number l, the Galois group Gal(\overline{K}/K) acts transitively on the l-torsion points of E.

Notation 1.2. For x ∈ E(\overline{K}), we denote by δ_{x} the Dirac measure associated to its Galois orbit. I.e., if x ∈ E(F) for some number field F ⊇ K then δ_{x}(φ) := \frac{1}{♯ Gal(F/K)} ∑_{σ: F ↪ℂ} φ(σ(x)) for each φ ∈ C^{0}(E(ℂ)).

1.3. In this language, Serre’s theorem immediately implies the following.

Corollary. Let K be an algebraic number field and E be an elliptic curve over K without complex multiplication. Further, let (x_{i})_{i∈ℕ} be a sequence of torsion points in E not containing any constant subsequence.

Then, the associated sequence (δ_{x_{i}})_{i∈ℕ} of measures on E(ℂ) converges in the weak sense to the Haar measure normalized to volume one.

1.4. In 1997, L. Szpiro, E. Ullmo and S. Zhang [SUZ] showed a generalization from torsion points (which are of Néron-Tate height zero) to points of a small positive height. Their result is as follows.

Theorem 1.5 (Szpiro, Ullmo, Zhang). Let A be an abelian variety over a number field K and (x_{i})_{i∈ℕ} be a sequence of closed points in A. Assume that (x_{i})_{i∈ℕ} converges to the generic point of A in the sense of the Zariski topology. Suppose further that lim_{i→∞} h_{NT} (x_{i}) = 0.

Then, the associated sequence (δ_{x_{i}})_{i∈ℕ} of measures on A(ℂ) converges in the weak sense to the Haar measure of volume one.

1.6. Somewhat surprisingly, there is a completely parallel theorem for the naive height on projective space due to Yu. Bilu [Bi97].

2000 Mathematics Subject Classification 14G40 (primary), 11G35, 37P30 (secondary) Keywords: Equidistribution, small points, canonical height

Jörg Jahnel

Theorem 1.7 (Bilu). Let (x_{i})_{i∈ℕ} be a sequence of closed points in P_{ℚ}^{n} . Assume that (x_{i})_{i∈ℕ} converges to the generic point in the sense of the Zariski topology. Assume further that lim_{i→∞} h_{naive}(x_{i}) = 0.

Then, the associated sequence (δ_{x_{i}})_{i∈ℕ} of measures on P^{n}(ℂ) converges weakly to the Haar measure of volume one on (S^{1})^{n} ⊂ ℂ^{n} ⊂ P^{n}(ℂ).

Remark 1.8. For n = 1, Yu. Bilu proved this result already in 1988 [Bi88].

Canonical heights.

Definition 1.9. Let P be a projective variety over ℚ equipped with an ample invertible sheaf ℒ ∈ Pic(P). Further, we assume there is given a self-map f: P → P such that there is an isomorphism Φ: ℒ^{⊗d} ⟶^{≅} f^{∗}ℒ .

Then, the canonical height function h_{f,ℒ} on P is given by

h_{f,ℒ} (x) := lim_{n→∞} \frac{1}{d^{n}} h_{ℒ} (f^{(n)}(x))

for x ∈ P a closed point. Here, f^{(n)} means the n-fold iteration of f. By h_{ℒ} , we denote the height defined by the ample invertible sheaf ℒ.

Remark 1.10. The limit process defining h_{f,ℒ} is a generalization of the classical one for the Néron-Tate height [CS, Chapter VI, §4]. We will show in Example 1.15.ii) that the naive height is a canonical height, too.

Remark 1.11. The height h_{ℒ} is defined only up to a bounded summand. Nevertheless, the canonical height function h_{f,ℒ} is independent of the choice of a particular representative. Indeed, if |h_{ℒ}^{(1)}(x) − h_{ℒ}^{(2)}(x)| ⩽ C for every x ∈ P then

|\frac{1}{d^{n}} h_{ℒ}^{(1)}(f^{(n)}(x)) − \frac{1}{d^{n}} h_{ℒ}^{(2)}(f^{(n)}(x))| ⩽ \frac{C}{d^{n}} .

An equidistribution result. The goal of the present article is to prove a common generalization of Theorems 1.5 and 1.7. We will give an Arakelovian approach for a situation covering the case of the Néron-Tate height as well as that of the naive height.

Situation 1.12. Let P be a regular projective variety over ℚ containing a group scheme G ⊆ P as an open dense subset. Further, let a morphism f: P → P be given and assume that

i) the unit e is a repelling fixed point. I.e., all eigenvalues of the tangent map T_{e}f: T_{e}G → T_{e}G are of absolute value strictly bigger than 1. ii) f|_{G} is a group homomorphism f|_{G}: G → G. iii) There is some compact, Zariski dense subgroup K ⊆ G(ℂ) which is both backward and forward invariant under f_{ℂ} .

Finally, let ℒ ∈ Pic(P) be ample and Φ: ℒ^{⊗d} ⟶^{≅} f^{∗}ℒ be an isomorphism. Then, there is the canonical height h_{f,ℒ} on P.

Theorem 1.13 (Equidistribution on P (ℂ)). Let P, f, ℒ, and Φ be as in 1.12. Further, let (x_{i})_{i∈ℕ} be a sequence of closed points in P which is convergent to the generic point of P in the sense of the Zariski topology. Finally, suppose that h_{f,ℒ} (x_{i}) → 0.

On the distribution of small points on abelian and toric varieties

Then, the associated sequence (δ_{x_{i}})_{i∈ℕ} of measures on P (ℂ) converges in the weak sense to the measure τ on P (ℂ) being the zero measure on P (ℂ)\K and the Haar measure of volume one on K.

Remark 1.14. For a general self-map f, iterating will lead to a fractal. Unfortunately, not much is known about the corresponding fractal heights.

Examples 1.15. The condition that e is a repelling fixed point is fulfilled, in particular, when G is commutative and f: g ↦ g^{l} is the homomorphism raising to the l-th power for l ⩾ 2. This includes the two standard cases. Namely,

i) let P = G = A be an abelian variety and ℒ ∈ Pic(A) a symmetric, ample invertible sheaf. One has

f = [l]: A → A

and puts K = A(ℂ). Then, independently of the choice of l, h_{f,ℒ} is the Néron-Tate height corresponding to ℒ. ii) Let P = P^{n}, G = 𝔾_{m}^{n+1}/𝔾_{m} ≅ 𝔾_{m}^{n} and ℒ = 𝒪(1). We have

f = [l]: (x_{0}, …, x_{n}) ↦ (x_{0}^{l}, …, x_{n}^{l})

and put K := U(1)^{n}. Here, h_{f,ℒ} is the naive height. iii) One may combine abelian varieties and projective spaces and consider a split semi-abelian variety A × P^{n}. Here, G = A × 𝔾_{m}^{n}, K = A × U(1), and ℒ = 𝒮 ⊠ 𝒪(p) for 𝒮 a symmetric, ample invertible sheaf on A. One may put f := [l] × [l^{2}]. A. Chambert-Loir’s equidistribution result [CL] for quasi-split semi-abelian varieties is easily deduced from Theorem 1.13. iv) Consider a free abelian group N of finite rank and a finite rational polyhedral decomposition {σ_{α}}_{α} of N ⊗_{ℤ} ℝ. These data define a proper toric variety X. The multiplication map N → N, n ↦ ln induces a self-morphism f = [l]: X → X. Further, consider a function g: N ⊗_{ℤ} ℝ → ℝ which is strictly convex in the sense of [KKMS, Chapter I, Theorem 13]. Then, ℒ := ℱ_{g} is a T -invariant ample invertible sheaf on X. One has [l]^{∗}ℱ_{g} ≅ ℱ_{lg} = ℱ_{g}^{⊗l}. In this situation, P = X, G = T ≅ 𝔾_{m}^{n} is the Zariski dense torus in X, and K ⊂ T is its maximal compact subgroup. We have a canonical height on the toric variety X such that equidistribution is fulfilled for sequences which are generic and small.

Remark 1.16. The latter example clearly contains the case of the naive height on projective space. Note that it also contains the blow-up of P^{2} in one point and the canonical height defined by the ample divisor nH − mE for n, m ∈ ℕ, n > m.

Remark 1.17. For abelian varieties, our proof will coincide with that of Szpiro, Ullmo, and Zhang. In the case of the projective space, it seems to us that our proof is different from Bilu’s.

2. The dynamics of f

Arakelovian interpretation of the canonical height. Let P, f, ℒ, and Φ be as in Definition 1.9. In order to understand the canonical height better, we have to use the language of Arakelov geometry [GS90].

First, recall that in Definition 1.9 we may replace h_{ℒ} by any other height function which differs from h_{ℒ} only by a bounded summand. For example, we may work with h′_{ℒ} (x) := \frac{1}{m} h_{l^{2}}(i_{ℒ^{⊗m}} (x)). Further, there is a projective model 𝒫 of P over Spec ℤ to which ℒ^{⊗m} extends. Indeed, ℒ^{⊗m} defines a closed embedding i_{ℒ^{⊗m}} : P → P^{N} and one may put 𝒫 := \overline{i_{ℒ^{⊗m}} (P)} .

Then, \underline{ℒ} := 𝒪(1)|_{𝒫} is an extension of ℒ^{⊗m} to 𝒫. We equip \underline{ℒ} with the restriction of the Fubini-Study metric. This guarantees that, for every closed point x ∈ P,

h_{ℒ ,0} (x) := h_{ℒ} (x) = \frac{1}{m} h_{(\underline{ℒ} ,∥ .∥)}(x) .

Here, h_{(\underline{ℒ} ,∥ .∥)} is the absolute height defined [BGS] by the smooth Hermitian line bundle (\underline{ℒ} , ∥ .∥).

The recursion is given by h_{ℒ ,i+1} (x) := \frac{1}{d} h_{ℒ ,i} (f(x)). Unfortunately, the isomorphism Φ: ℒ^{⊗d} ⟶^{≅} f^{∗}ℒ does not extend to 𝒫. If it would then we could put ∥ .∥_{0} := ∥ .∥ and, recursively,

∥ .∥_{i} := Φ^{−1}f^{∗}∥ .∥_{i−1}^{1/d} .

This would then lead to h_{ℒ ,i} (x) = \frac{1}{m} h_{(\underline{ℒ} ,∥ .∥_{i})}(x).

2.1. To make the idea above precise, one has to work in the more flexible context of adelic Picard groups. This is a theory due to S. Zhang [Zh95a]. In an appendix, we will recall it shortly and fix notation.

There is the adelic metric ∥ .∥_{∼}^{(0)} on ℒ^{⊗m} induced by \underline{ℒ} ,

(ℒ^{⊗m}, ∥ .∥_{∼}^{(0)}) := i_{𝒫} (\underline{ℒ} , ∥ .∥_{0}) .

Since \underline{ℒ} is very ample and ∥ .∥_{0} is a restriction of the Fubini-Study metric, we see that (ℒ^{⊗m}, ∥ .∥_{∼}^{(0)}) ∈ C_{P}^{⩾0}. Further, we define, recursively,

∥ .∥_{∼}^{(i)} := (Φ^{⊗m})^{−1}f^{∗}[∥ .∥_{∼}^{(i−1)}]^{\frac{1}{d}} .

We have (ℒ^{⊗m}, ∥ .∥_{∼}^{(i)}) ∈ C_{P}^{⩾0} for every i ∈ ℕ. We also observe that the isomorphism Φ does extend to an open neighbourhood of the generic fiber of 𝒫. Therefore, the sequence (∥ .∥_{∼}^{(i)})_{i∈ℕ} is actually constant at all but finitely many valuations. Furthermore,

δ((ℒ^{⊗m}, ∥ .∥_{∼}^{(i)}), (ℒ^{⊗m}, ∥ .∥_{∼}^{(i+1)})) ⩽ \frac{1}{d^{i}} ·δ((ℒ^{⊗m}, ∥ .∥_{∼}^{(0)}), (ℒ^{⊗m}, ∥ .∥_{∼}^{(1)})) .

This means, for every ν ∈ Val(ℚ), the sequence (∥ .∥_{ν}^{(i)})_{i∈ℕ} of metrics is uniformly convergent. The limit ∥ .∥_{∼} clearly fulfills all the requirements of Definition A.4. I.e., ∥ .∥_{∼} is an adelic metric on ℒ^{⊗m}. Even more,

\overline{ℒ} := (ℒ^{⊗m}, ∥ .∥_{∼}) ∈ \overline{C_{P}^{⩾0}} .

Finally, we have h_{f,ℒ} = \frac{1}{m} h\overline{ℒ} = \frac{1}{m} h_{(ℒ^{⊗m},∥ .∥_{∼})} .

Lemma 2.2. f is automatically finite of degree d^{dim P}.

Proof. P is a projective variety over ℚ and f: P → P is a morphism such that ℒ^{⊗d} ≅ f^{∗}ℒ for some ample ℒ ∈ Pic(P). In particular, f^{∗}ℒ is ample which implies f is quasi-finite. As f is a projective morphism, this is sufficient for finiteness. Further, we have deg_{ℒ} P ≠ 0 and deg_{f^{∗}ℒ} P = deg_{ℒ^{⊗d}} P = d^{dim P} deg_{ℒ} P.

Lemma 2.3. i) p_{Z}^{P}((ℒ^{⊗m}, ∥ .∥_{∼}), …, (ℒ^{⊗m}, ∥ .∥_{∼})) = 0, ii) e_{1}(h\overline{ℒ} ) = … = e_{dim P +1}(h\overline{ℒ} ) = 0.

Proof. i) For every i ∈ ℕ, write

P_{i} := p_{Z}^{P}((ℒ^{⊗m}, ∥ .∥_{∼}^{(i)}), …, (ℒ^{⊗m}, ∥ .∥_{∼}^{(i)})) .

Then, since f is of degree d^{dim P},

d^{dim P} P_{i−1} := p_{Z}^{P}((f^{∗}ℒ^{⊗m}, f^{∗}∥ .∥_{∼}^{(i−1)}), …, (f^{∗}ℒ^{⊗m}, f^{∗}∥ .∥_{∼}^{(i−1)})) = 0 .

Applying the isomorphism Φ^{⊗m} and dividing each of the (dim P + 1) operators by d yields

\frac{P_{i−1}}{d} = p_{Z}^{P}((ℒ^{⊗m}, ∥ .∥_{∼}^{(i)}), …, (ℒ^{⊗m}, ∥ .∥_{∼}^{(i)})) = P_{i} .

(P_{i})_{i∈ℕ} is, therefore, a zero sequence.

ii) There is a constant C such that h_{ℒ ,0} (x) > C for every x ∈ P. Consequently, h_{ℒ ,i} (x) > C/d^{i} and h\overline{ℒ} (x) ⩾ 0 for every x ∈ P. This means e_{dim P +1}(h\overline{ℒ} ) ⩾ 0 . Theorem A.14 implies the claim.

Remark 2.4. As the construction of ∥ .∥_{∼} depends on the iterated pull-backs under f, one has to understand the dynamics of the system (f^{i})_{i⩾1}. For our purposes, it will be sufficient to pay attention to the infinite place. I.e., to the dynamics of that system on P (ℂ).

Analyzing the dynamics of f. Let us analyze the special situation described in 1.12.

i) G ⊆ P is an open dense subset being a group scheme. The self-map f has a restriction m := f|_{G}: G → G which is a homomorphism of group schemes. Therefore, ker m is a group scheme of finite order d^{dim P}. As f is surjective, im m is a priori Zariski dense. Being a homomorphism, m is surjective, too. There is a compact subgroup K ⊆ G(ℂ) which is both forward and backward invariant under m_{ℂ} . In particular, all the finite groups ker(m_{ℂ} ∘ … ∘ m_{ℂ} ) are contained in K. ii) All eigenvalues of the tangent map T_{e}m have absolute value strictly bigger than 1. Therefore, on G(ℂ), there is a left invariant Riemannian metric μ such that q := ∥(T_{e}m_{ℂ} )^{−1}∥_{max} < 1. Indeed, based on the Jordan normal form, one easily finds a Hermitian scalar product on T_{e}G(ℂ) satisfying the analogous inequality. We take its real part. By transport of structure, we find a left invariant Riemannian metric μ on G(ℂ). We will denote by δ the metric on G(ℂ) given by the lengths of the μ-shortest paths.

Convention 2.5. In order to simplify notation, we will usually write f and m instead of f_{ℂ} and m_{ℂ} when there is no danger of confusion.

Lemma 2.6. a) m induces a diffeomorphism \overline{m}: G(ℂ)/K → G(ℂ)/K.

b) \overline{m} is expanding for the Riemannian metric \overline{μ} on G(ℂ)/K induced by μ. More precisely,

\overline{δ}(\overline{m}_{ℂ} (z_{1}), \overline{m}_{ℂ} (z_{2})) ⩾ \frac{1}{q} \overline{δ}(z_{1}, z_{2})

for any z_{1}, z_{2} ⊆ G(ℂ)/K.

Proof. a) \overline{m} is well-defined since K is forward invariant under m. Backward invariance of K implies that \overline{m} is even an injection. Further, as m: G(ℂ) → G(ℂ) is a surjection, the same is then true for \overline{m}. Finally, the tangent map T_{e}\overline{m}: T_{e}G(ℂ)/K → T_{e}G(ℂ)/K is expanding, too. In particular, T_{e}\overline{m} is invertible. This yields that \overline{m} is a local diffeomorphism.

b) The preimage \overline{m}^{−1}(w) of a path w in G(ℂ)/K is a path, too. All tangent maps of m are expanding by a factor ⩾ \frac{1}{q}. Therefore, ℓ(\overline{m}^{−1}(w)) ⩽ q · ℓ(w). This implies the claim.

Corollary 2.7. Let \overline{G(ℂ)} = G(ℂ) ∪ \{∞\} be the one-point compactification of G(ℂ).

Then, for each x ∈ G(ℂ)\K, the sequence (x_{i})_{i∈ℕ} given by x_{0} = x and x_{i+1} = m(x_{i}) converges to ∞. In other words, K is the Julia set of the dynamical system (m^{i})_{i⩾1} on G(ℂ).

Proof. We have to show that, for each compact set A ⊆ G(ℂ), there are only finitely many i ∈ ℕ such that x_{i} ∈ A. Replacing A by KA, if necessary, we may assume that A is left K-invariant. This means, we are given a compact set \overline{A} ⊆ G(ℂ)/K and have to verify that \overline{x}_{i} ∈ \overline{A} for only finitely many values of i. As \overline{m}: G(ℂ)/K → G(ℂ)/K is expanding and \overline{x}_{0} ≠ [K], we have \overline{δ}(\overline{x}_{i}, [K]) → ∞. However, \overline{A} is a compact set. In particular, \overline{A} is bounded.

Corollary 2.8. The union over all the groups ker(m ∘ … ∘ m) is dense in K.

Proof. Otherwise, the topological closure of K′ := ⋃_{i∈ℕ} ker(\overbrace{m ∘ … ∘ m}^{i times}) would be a compact Lie group properly contained in K. It is obvious that K′ is forward and backward invariant under f_{ℂ} . Therefore, the map induced by f on the compact homogeneous space K/K′ would be injective and expanding.

Notation 2.9. For N ∈ ℝ, we will write U_{N}(K) := {x ∈ G(ℂ) | δ(x, K) ⩽ N}.

Proposition 2.10. Let μ_{i} be the measure induced by the smooth differential form

c_{1}(ℒ , ∥ .∥_{∞}^{(i)}) ∧ … ∧ c_{1}(ℒ , ∥ .∥_{∞}^{(i)})

of type (dim P, dim P) on P (ℂ). Then, the sequence (μ_{i})_{i∈ℕ} converges weakly to the measure μ which is the zero measure on P (ℂ)\K and the Haar measure of volume deg_{ℒ} P on K.

Proof. First step. Generalities.

Each μ_{i} is of volume deg_{ℒ} P. Further, the sequence (μ_{i})_{i∈ℕ} obeys the recursive low μ_{i+1} = \frac{1}{d^{dim P}} m^{∗}μ_{i}.

Second step. μ_{i}|_{P (ℂ)\K} → 0.

Let g ∈ C_{0}(P (ℂ) \ K) be a continuous function with compact support. We consider the sequence (g_{i})_{i∈ℕ} given by g_{0} := g and g_{i+1} := \frac{1}{d^{dim P}} f_{∗}g_{i}. As usual, the push-forward of a function is given by summation over the fibers of the morphism f. By definition, μ_{i}(g) = μ_{0}(g_{i}). Thus, let us show μ_{0}(g_{i}) → 0 for i → ∞.

For this, we note that |g| is bounded by some constant C. This implies |g_{i}| ⩽ C for every i. Further, there is some ε > 0 such that supp(g) ⊆ P (ℂ)\U_{ε}(K). Consequently,

|μ_{i}(g)| = |μ_{0}(g_{i})| ⩽ C ∫_{P (ℂ)} χ_{(P (ℂ)\U_{ε/q^{i}})} dμ_{0} .

Here, χ_{A} denotes the characteristic function of a measurable set A. The integrand converges monotonically to χ_{P (ℂ)\G(ℂ)} which is of integral zero. The theorem of Beppo Levi yields μ_{i}(g) → 0.

Third step. The assertion in general.

Let g ∈ C_{0}(G(ℂ)) be a continuous function with compact support. As above, we put g_{0} := g and g_{i+1} := \frac{1}{d^{dim P}} m_{∗}g_{i}. Thus, we have μ_{i}(g) = μ_{0}(g_{i}). The sequence (g_{i})_{i∈ℕ} is uniformly bounded. Hence, it suffices to show that it converges pointwise to the constant

I := \frac{1}{vol K} ∫_{K} g dρ .

We claim that the convergence is actually uniform in every compact set A ⊆ G(ℂ).

For this, we may assume without loss of generality that A = U_{N}(K) for some N ∈ ℝ. Further, note that

g_{i}(x) = \frac{1}{♯ ker m^{i}} ∑_{k∈ker m^{i}}g(ky)

for any y such that f(y) = x. Here, we clearly have (m^{i})^{−1}(A) ⊆ U_{Nq^{i}}(K) and Nq^{i} → 0 for i → ∞.

In addition, g is uniformly continuous on A and m is expanding. Thus, for each ε > 0, there is some i ∈ ℕ such that, for two arbitrary points x, x′ ∈ A, one can find y, y′ such that f(y) = x, f(y′) = x′, and δ(y, y′) < ε. Consequently, (max g_{i}|_{A} − min g_{i}|_{A}) tends to zero for i → ∞. Finally, for every i ∈ ℕ, I is the mean value of g_{i} on K.

Some further observations. It is not true in general that the curvature current, given by c_{1}(ℒ_{ℂ} , ∥ .∥_{∞}) := \frac{1}{2πi}∂\overline{∂} log ∥s∥^{2} for a non-zero rational section of ℒ_{ℂ} , vanishes on P (ℂ)\K. Nevertheless, one has at least the following.

Lemma 2.11. The curvature current c_{1}(ℒ_{ℂ} , ∥ .∥_{∞}) has the properties below.

a) The restriction c_{1}(ℒ_{ℂ} , ∥ .∥_{∞})|_{G(ℂ)} is left and right K-invariant.

b) It satisfies the equation f^{∗}c_{1}(ℒ_{ℂ} , ∥ .∥_{∞}) = d · c_{1}(ℒ_{ℂ} , ∥ .∥_{∞}).

Proof. a) By construction, c_{1}(ℒ_{ℂ} , ∥ .∥_{∞,i}) is invariant under ker(m_{ℂ}^{i} ). Thus, c_{1}(ℒ_{ℂ} , ∥ .∥_{∞}), being the weak limit of that sequence of currents, is invariant under ⋃_{i∈ℕ} ker(m_{ℂ}^{i} ). Invariance under any k ∈ K follows as k_{i} → k implies that the test forms k_{i}ω converge to kω in the Schwartz space.

b) is clear.

3. Perturbing almost semiample metrics

3.1. One of the most natural ways to produce a new adelic metric from an old one is to replace ∥ .∥_{∞} by ∥ .∥_{∞} ·exp(−g) for some continuous function g ∈ C(P (ℂ)). For the new metric, we want to use the theorem of successive minima A.14 as a fundamental tool. It is, therefore, necessary to understand for which g this metric is almost semiample, i.e., the limit of a uniformly convergent sequence of metrics with positive curvature.

Notation 3.2. In this section, we will continue to use the notation of 1.12. Further, we will denote by S the set of all continuous functions g ∈ P (ℂ) such that ∥ .∥_{∞} · exp(−g) is almost semiample.

Lemma 3.3. a) One has C ∈ S for every constant C.

b) Let g_{1}, g_{2} ∈ S and 0 < a < 1. Then, ag_{1} + (1 − a)g_{2} ∈ S.

c) If g ∈ S then \frac{1}{d}f^{∗}g ∈ S.

d) Let g ∈ S be a function such that supp g ⊆ G(ℂ). Then, for each k ∈ K, one has g · k ∈ S for

(g · k)(x) := \{ \begin{matrix}g(kx) & if x ∈ G(ℂ) , \\ 0 & otherwise . \end{matrix}

e) S is closed under uniform convergence.

Proof. e) is trivial. a) ∥ .∥_{∞} is almost semiample. Thus, there is a sequence (∥ .∥_{∞}^{i})_{i∈ℕ} of smooth Hermitian metrics with strictly positive curvature on ℒ_{ℂ} which is uniformly convergent to ∥ .∥_{∞}. The metrics ∥ .∥_{∞,i} · exp(−C) have the same curvatures and converge uniformly to ∥ .∥_{∞} · exp(−C).

b) Let (∥ .∥′_{i})_{i∈ℕ} and (∥ .∥′′_{i})_{i∈ℕ} be sequences uniformly convergent versus ∥ .∥_{∞} · exp(−g_{1}) and ∥ .∥_{∞} · exp(−g_{2}), respectively. Put ∥ .∥_{i} := ∥ .∥_{i}^{′ a} · ∥ .∥_{i}^{′′ (1−a)}. These metrics have positive curvature and, uniformly,

∥ .∥_{i} → ∥ .∥_{∞} · exp(− ag_{1} − (1 − a)g_{2}) .

c) Put ∥ .∥′_{∞} := ∥ .∥_{∞} · exp(−g). Then,

Φ^{−1}f^{∗}∥ .∥_{∞}^{′ \frac{1}{d}} = Φ^{−1}f^{∗}∥ .∥∞^{\frac{1}{d}} · exp(− \frac{1}{d}f^{∗}g) = ∥ .∥_{∞}· exp(− \frac{1}{d}f^{∗}g) .

Therefore, if (∥ .∥′_{∞,i})_{i∈ℕ} is a sequence of metrics with positive curvature uniformly convergent to ∥ .∥′_{∞} then (Φ^{−1}f^{∗}∥ .∥_{∞,i}^{′ \frac{1}{d}})_{i∈ℕ} converges uniformly to ∥ .∥_{∞} · exp(−\frac{1}{d}f^{∗}g).

d) We denote by e_{k}: G(ℂ) → G(ℂ) the multiplication by k from the left. By e), we may assume that k ∈ ker(m^{j}) for some j ∈ ℕ. To show ∥ .∥_{∞} · exp(−g · k) is almost semiample, it suffices to consider the Hermitian metric ∥ .∥_{∞}^{d^{j}} · exp(−d^{j}g · k) on ℒ_{ℂ}^{⊗d^{j}} . Let us consider its restriction to G(ℂ), first. Iterated application of Φ induces an isomorphism

e_{k}^{∗}ℒ_{ℂ}^{⊗d^{j}}|_{G(ℂ)} ≅ e_{k}^{∗}(f^{j})^{∗}ℒ_{ℂ} |_{G(ℂ)} ≅ (f^{j})^{∗}ℒ_{ℂ} |_{G(ℂ)} ≅ ℒ_{ℂ}^{⊗d^{j}}|_{G(ℂ)} .  (1)

Here, the isomorphism in the middle is canonical as f^{j} ∘e_{k} = f^{j}.

By construction 2.1, ∥ .∥_{∞} is the uniform limit of a sequence (∥ .∥_{∞}^{(i)})_{i∈ℕ} such that, for every i ⩾ j,

e_{k}^{∗}[∥ .∥_{∞}^{(i)}]^{d^{j}} = [∥ .∥_{∞}^{(i)}]^{d^{j}}

under the identification (1). By consequence, e_{k}^{∗}∥ .∥_{∞}^{d^{j}} = ∥ .∥_{∞}^{d^{j}}, too, which shows

e_{k}^{∗}[∥ .∥_{∞}^{d^{j}} · exp(−d^{j}g)] = ∥ .∥_{∞}^{d^{j}} · exp(−d^{j}g · k) .

Clearly, ∥ .∥_{∞}^{d^{j}} · exp(−d^{j}g · k) is almost semiample on G(ℂ). In fact, almost semiampleness is preserved under holomorphic pull-back.

On the other hand, there is some neighbourhood U of P (ℂ) \ G(ℂ) such that g|_{U} = 0. This implies −d^{j}g · k = 0 and, therefore, ∥ .∥_{∞}^{d^{j}} · exp(−d^{j}g · k) = ∥ .∥_{∞}^{d^{j}} on U. In particular, ∥ .∥_{∞}^{d^{j}} · exp(−d^{j}g · k) is almost semiample on U. The assertion follows from Sublemma 3.4, below.

Sublemma 3.4. Let M be a projective complex manifold, 𝒢 ∈ Pic(M) be ample, and {U_{1}, …, U_{n}} an open covering. Further, let ∥ .∥_{1}, …, ∥ .∥_{n} be continuous Hermitian metrics on 𝒢 |_{U_{1}}, …, 𝒢 |_{U_{n}} such that, for some δ > 0, the sets D_{1,δ}, …, D_{n,δ}, given by

D_{i,δ} := {x ∈ U_{i} | ∥ .∥_{i,(x)} ⩽ (1 + δ) · min(∥ .∥_{1,(x)}, …, ∥ .∥_{n,(x)})} ⊆ U_{i}

are compact. Assume, each ∥ .∥_{i} is almost semiample. Then, the continuous Hermitian metric ∥ .∥ := min_{i} ∥ .∥_{i} is almost semiample.

Proof. We may assume that 𝒢 is very ample. For every i, let (∥ .∥_{ij})_{j∈ℕ} be a sequence of smooth and positively curved metrics on 𝒢 |_{U_{i}} which converges uniformly to ∥ .∥_{i}. Put

∥ .∥_{j}^{−} := min(∥ .∥_{1j}, …, ∥ .∥_{nj}) .

For j ≫ 0, ∥ .∥_{j}^{−} is a continuous Hermitian metric on the whole of 𝒢. The sequence (∥ .∥_{j}^{−})_{j} converges uniformly to ∥ .∥.

We choose an embedding i: M ↪ P^{N} such that i^{∗}𝒪(1) ≅ 𝒢. Further, we extend the smooth metrics ∥ .∥_{ij} to smooth metrics ∥ .∥′_{ij} on 𝒪(1) which are defined on open subsets W_{i} ⊇ U_{i} of P^{N}.

Under the tautological action PGL_{n}(ℂ) × P^{N} → P^{N}, each γ ∈ PGL_{n}(ℂ) defines an automorphism e_{γ}: P^{N} → P^{N} such that there is a natural identification e_{γ}^{∗}𝒪(1) ≅ 𝒪(1). We find an open neighbourhood O of e ∈ PGL_{n}(ℂ) such that, for every γ ∈ O and every i ∈ {1, …, n}, the pull-back e_{γ}^{∗} ∥ .∥′_{ij} is well-defined on D_{i,δ} and e_{γ}^{∗}∥ .∥′_{ij} |_{U_{i}∩ e_{γ}^{−1}(W_{i})} has strictly positive curvature in every point of D_{i,δ}.

We claim that, for each γ ∈ O, the perturbation

∥ .∥_{j}^{−,γ} := e_{γ}^{∗}( min(∥ .∥′_{1j}, …, ∥ .∥′_{nj}))|_{M}

of ∥ .∥_{j}^{−} has a positive curvature current on each holomorphic curve inside M.

Indeed, this is a local statement. Let x ∈ M. Fix a holomorphic section 0 ≠ s ∈ Γ(U, 𝒢) defined in some neighbourhood U of x. Then,

c_{1}(𝒢 , ∥ .∥_{j}^{−,γ})|_{U} = −dd^{c} log (∥s∥_{j}^{−,γ})^{2}
= dd^{c}(max(− log e_{γ}^{∗}∥s∥_{1j}^{′ 2}, …, − log e_{γ}^{∗}∥s∥_{nj}^{′ 2})).

Observe that the maximum of a finite system of plurisubharmonic functions is plurisubharmonic, again.

Now, we use Sobolev’s averaging procedure. For every non-negative smooth function φ ≠ 0 on PGL_{n}(ℂ) such that supp φ ⊆ O, we get the approximation

∥ .∥_{j}^{−,φ} := ∫_{PGL_{n}(ℂ)}φ(γ)·∥ .∥_{j}^{−,γ} dρ_{γ}/ ; ∫_{PGL_{n}(ℂ)}φ(γ) dρ_{γ}

of ∥ .∥_{j}^{−}. Here, ρ is the left Haar measure on PGL_{n}(ℂ).

Finally, consider a sequence (φ_{k})_{k∈ℕ} of functions as above which converges weakly to the delta distribution δ_{e}. Every ∥ .∥_{j}^{−,φ_{k}} is smooth and positively curved. The sequence (∥ .∥_{k}^{−,φ_{k}})_{k∈ℕ} converges uniformly to ∥ .∥.

Corollary 3.5. If g_{1}, g_{2} ∈ S then g := max{g_{1}, g_{2}} ∈ S.

Corollary 3.6. Let l ∈ ℕ, 0 < a < 1, and 0 ≠ s ∈ Γ(P (ℂ), ℒ_{ℂ}^{⊗l}). Then,

g := max{a log ∥s∥^{l}, 0} ∈ S .

Proof. Consider the function \overline{g} on U := \{x ∈ P (ℂ) | log ∥s∥(x) >−1\} given by

\overline{g}(x) := a log ∥s∥^{l}(x) .

We have to show that the Hermitian metric

∥ .∥′_{∞} := ∥ .∥_{∞}^{l} · exp(−\overline{g})

on ℒ_{ℂ}^{⊗l}|_{U} is the limit of a uniformly convergent sequence of smooth metrics with positive curvature. We put ∥ .∥′_{∞,i} := ∥ .∥_{∞,i}^{l}/ (∥s∥_{∞,i}^{l})^{a}. It is clear that, uniformly, ∥ .∥′_{∞,i} → ∥ .∥′_{∞}. Further,

c_{1}(ℒ_{ℂ} |_{U}, ∥ .∥′_{∞,i}) = dd^{c}(− log ∥s∥_{∞,i}^{′2})
= dd^{c}(− log(∥s∥_{∞,i}^{l})^{2(1−a)})
= (1 − a)l · c_{1}(ℒ_{ℂ} |_{U}, ∥ .∥_{∞,i}) > 0 .

Lemma 3.7. Let g ∈ S such that supp g ⊆ G(ℂ). Consider the K-invariant function \underline{g} given by

\underline{g}(x) := \{ \begin{matrix}max_{k∈K} g(kx) & if x ∈ G(ℂ) , \\ g(x) & otherwise . \end{matrix}

Then, \underline{g} ∈ S, too.

Proof. \underline{g} is the uniform limit of sequence (g_{i})_{i∈ℕ} , defined by g_{i}(x) := max_{k∈ker(m^{i})} g(kx).

Proposition 3.8. Let ε > 0 and U ⊆ G(ℂ) be an open set containing K. Then, there is some non-negative K-invariant function 0 ≠ g ∈ ℝ_{+}·S such that

i) g(e) = 1,

ii) max_{x∈P (ℂ)} g(x) ⩽ 1 + ε,

iii) supp g ⊆ U.

Proof. Put D := P \ G. For some j ≫ 0, the coherent sheaf ℒ^{⊗j} ⊗ℐ_{D} has a section which does not vanish in e ∈ G. I.e., there is a section s ∈ Γ(P, ℒ^{⊗j}) vanishing in D but not in e. Using Corollary 3.6, we see

g̃_{C} := max \{\frac{1}{2j} log ∥Cs∥^{j}, 0\} = max \{\frac{1}{2j} log ∥s∥^{j} + \frac{log C}{2j} , 0\} ∈ S

for every C > 0. It is clear that supp g̃_{C} ⊆ G(ℂ) is a compact set.

We put A := max_{x∈K} \frac{1}{2j} log ∥s∥^{j}(x) and B := max_{x∈P (ℂ)} \frac{1}{2j} log ∥s∥^{j}(x). Then, we choose C such that log C + A > 0 and \frac{log C+B}{log C+A} ⩽ 1 + ε. Further, let g_{0} be the K-invariant function associated to g̃_{C}. I.e.,

g_{0} := \{ \begin{matrix}max_{k∈K} g̃_{C}(kx) & if x ∈ G(ℂ) , \\ g(x) & otherwise . \end{matrix}

By construction, g_{0}(e) > 0 and max_{x∈P (ℂ)} g_{0}(x)/g_{0}(e) ⩽ 1 + ε. Further, supp g_{0} ⊆ K ·supp g̃_{C} is compact. Lemma 3.7 shows that g_{0} ∈ S.

Finally, we define a sequence (g_{i})_{i∈ℕ} of functions on P (ℂ) by putting, recursively, g_{i+1} := \frac{1}{d}f^{∗}g_{i}. Then, one clearly has g_{i}(e) = g_{0}(e) > 0 and

max_{x∈P (ℂ)} g_{i}(x)/g_{i}(e) = max_{x∈P (ℂ)} g_{0}(x)/g_{0}(e) ⩽ 1 + ε

for every i ∈ ℕ. Lemma 3.3.c) implies that g_{i} ∈ S. As supp g_{0} is compact, there is some N ∈ ℝ such that supp g_{0} ⊆ U_{N}(K). This yields, by Lemma 2.6.b), that supp g_{i} ⊆ U_{Nq^{i}}(K). Therefore, the function g, given by g(x) := g_{i}(x)/g_{i}(e) for some i ≫ 0, has all the properties desired.

4. Equidistribution

Definition 4.1. Let X be a projective variety over ℚ. Then, a sequence (x_{i})_{i∈ℕ} of closed points on X is called generic if no infinite subsequence is contained in a proper closed subvariety of X.

Remark 4.2. In other words, (x_{i})_{i∈ℕ} is generic if it converges to the generic point with respect to the Zariski topology.

Definition 4.3. Let X be a projective variety over ℚ and \overline{ℒ} ∈ P̃ic(X). Suppose that e_{dim X+1}(h\overline{ℒ} ) = 0. Then, a sequence (x_{i})_{i∈ℕ} of closed points on X is called small if h\overline{ℒ} (x_{i}) → 0.

Lemma 4.4. Let X be a projective variety over ℚ and \overline{ℒ} ∈ C_{X}^{⩾0}. Assume e_{dim X+1}(\overline{ℒ} ) = 0.

Then, p_{Z}^{X}(\overline{ℒ} , …, \overline{ℒ} ) = 0 is equivalent to the existence of a sequence of closed points on X which is generic and small.

Proof. “⟹” There are only countably many closed subvarieties X_{1}, X_{2}, … ⊂ X. Further, Theorem A.14 implies e_{1}(h\overline{ℒ} ) = 0. Therefore, we may choose a sequence (x_{i})_{i∈ℕ} of closed points on X such that x_{i} ∈ X\X_{1}\X_{2}\…\X_{i} and h\overline{ℒ} (x_{i}) < \frac{1}{i}. The sequence (x_{i})_{i∈ℕ} is generic and small.

“⟸” Let (x_{i})_{i∈ℕ} be a sequence of closed points on X which is generic and small. Let Y ⊂ X be a closed subset of codimension one. Then, only finitely many of the x_{i} are contained in Y. Hence, inf_{x∈X\Y} h\overline{ℒ} (x) = 0. As this is true for every Y, we see that e_{1}(h\overline{ℒ} ) = 0. Theorem A.14 yields p_{Z}^{X}(\overline{ℒ} , …, \overline{ℒ} ) = 0.

Lemma 4.5. Let P, f, \overline{ℒ} , and Φ be as in 1.12. Denote by S the set of all continuous functions g ∈ C(P (ℂ)) such that ∥ .∥_{∞}· exp(−g) is almost semiample.

Let φ ∈ C(P (ℂ)). Then, for every ε > 0, there exist a function φ_{1} ∈ C(P (ℂ)) supported in P (ℂ)\K and a function φ_{2} ∈ ℝ_{+}·S such that

∥φ − φ_{1} − φ_{2}∥_{max} < ε .

Proof. The set T := {h ∈ C(K) | h = φ|_{K} for some φ ∈ ℝ_{+}·S} fulfills all the assumptions of Sublemma 4.6, except for closedness under uniform convergence. Indeed, Lemma 3.3.a) and b) imply that assumptions i) and ii) are satisfied. Corollary 3.6 is general enough to guarantee assumption iii). Lemma 3.3.d) yields assumption iv). Finally, Corollary 3.5 implies that assumption v) is fulfilled.

Therefore, there is some φ_{2} ∈ ℝ_{+}·S such that |φ(x) − φ_{2}(x)| < \frac{ε}{2} for every x ∈ K. A decomposition of the unit adapted to a suitable open covering {P (ℂ)\K, U} of P (ℂ) yields some φ_{1} ∈ C(P (ℂ)) such that supp φ_{1} ⊆ P (ℂ)\K and

|(φ(x) − φ_{2}(x)) − φ_{1}(x)| < ε

for every x ∈ P (ℂ).

Sublemma 4.6. Let L be a compact topological group and T ⊆ C(L) a set of continuous realvalued functions fulfilling the following conditions.

i) T contains all the constant functions. ii) T is closed under addition and multiplication by positive constants. iii) For each x ∈ L such that x ≠ e, there is some g ∈ T such that g(e) > g(x). iv) For each x ∈ L and g ∈ T, the shift g · x is in T, too.

v) If g ∈ T then g_{+} ∈ T for g_{+}(x) := max {g(x), 0}. vi) T is closed under uniform convergence.

Then, T = C(L).

Proof. We fix a Haar measure ρ on L. Then, conditions ii,) iv) and vi) together imply the following.

(†) Let h ∈ T and g be a non-negative, measurable, and bounded function. Then, for the convolution, we have g∗h ∈ T.

Therefore, it suffices to show that, for each open set U ⊆ L containing e, there is some nonnegative function g_{U} ∈ T such that g_{U} ≠ 0 and supp g_{U} ⊆ U. For this, by i), ii), and v), we only need a function g ∈ T taking its maximum entirely in U.

Assume, for some open U_{0}, there would be no such function. Then, for every g ∈ T taking its maximum in e, there is a non-empty set

A_{g} := {y ∈ L\U_{0} | g(y) = g(e)} ⊆ L\U_{0}

where the maximum is taken, too. If ⋂_{g∈T} A_{g} = ∅ then, by compactness, already a finite intersection A_{g_{1}} ∩ … ∩ A_{g_{n}} would be empty. Then, g_{1} + … + g_{n} ∈ S had its maximum in U_{0}, only.

Consequently, A_{U_{0}} := ⋂_{g∈T} A_{g} is a non-empty set. Put A := {e} ∪ ⋃_{U∋e,U open} A_{U}. By construction, this set has the property below.

(‡) Every function g ∈ T taking its maximum in e is necessarily constant on A. Further, A ⊆ L is the largest set containing e with this property.

The property (‡) implies that A is closed. Further, for y ∈ A and g ∈ T, the shift g·y^{−1} has its maxima in yA. Therefore, A = yA for every y ∈ A. This means that A ⊆ L is a subgroup. Fix a Haar measure ρ_{A} on A and denote by i: A → L the natural inclusion. Property (‡) together with i), ii) and v) implies that there is a sequence of functions of T such that the associated distributions converge weakly to i_{∗}ρ_{A}. Finally, property (†), together with vi), guarantees that every A-left invariant continuous function is an element of T.

Now, we use condition iii). Choose some x_{0} ∈ A different from e. There exists some g ∈ T such that g(e) > g(x_{0}). Shifting by an element of A, if necessary, we may assume that g(e) = max_{x∈A} g(x).

Define another continuous function M on L by M(x) := − max_{a∈A} g(ax). Obviously, M is Ainvariant. Thus, M ∈ T. Consequently, g̃:= g + M ∈ T, too. However, g̃(x) ⩽ 0 for every x ∈ L, g̃(e) = 0, and g̃(x_{0}) < 0 together form a contradiction to property (‡).

Proposition 4.7. Let P, f, \overline{ℒ} , and Φ be as in 1.12. Further, let (x_{i})_{i∈ℕ} be a sequence of closed points on P which is generic and small. Assume, some non-negative φ ∈ C(P (ℂ)) fulfills either supp φ ⊆ P (ℂ)\K or φ ∈ ℝ_{+}·S. Then,

lim inf_{i→∞} ∫_{P (ℂ)} φ dδ_{x_{i}} ⩾ ∫_{P (ℂ)} φ dτ .

Here, τ is the zero measure on P (ℂ) \ K and the Haar measure of volume one on K.

Proof. For φ such that supp φ ⊆ P (ℂ)\K, the right hand side is zero. Thus, in this case, the assertion is clear.

Let φ ∈ ℝ_{+}·S. Then, for any positive λ ∈ ℝ, let ∥ .∥_{∼}^{λ} be the adelic metric on ℒ^{⊗m} given by ∥ .∥_{p}^{λ} := ∥ .∥_{p} for the non-archimedean valuations and by ∥ .∥_{∞}^{λ} := ∥ .∥_{∞} · exp(−λφ) for the archimedean valuation. For λ → 0, the adelic metric ∥ .∥^{λ} fulfills the assumptions of Theorem A.14. Further, one clearly has

h_{(ℒ^{⊗m},∥ .∥^{λ})} (x_{i}) = h\overline{ℒ} (x_{i}) + λ ∫_{X(ℂ)}φ dδ_{x_{i}} .

As (x_{i})_{i∈ℕ} is generic and h\overline{ℒ} (x_{i}) → 0, we obtain, according to the very definition of e_{1},

λ·lim inf_{i→∞} ∫_{X(ℂ)} φ dδ_{x_{i}} ⩾ e_{1}(h_{(ℒ^{⊗m},∥ .∥^{λ})} ) .  (4)

On the other hand,

\frac{1}{(dim X + 1) c_{1}(ℒ_{ℚ}^{⊗m})^{dim X}} p_{Z}^{X}((ℒ^{⊗m}, ∥ .∥_{∼}^{λ}), …, (ℒ^{⊗m}, ∥ .∥_{∼}^{λ}))

= \frac{1}{(dim X + 1) m^{dim X} deg_{ℒ} X} p_{Z}^{X}(\overline{ℒ} , …, \overline{ℒ} )

+ \frac{1}{m^{dim X} deg_{ℒ} X} ·

p_{Z}^{X}((𝒪_{X}, ∥ .∥_{∞}· exp(−λφ)), (ℒ^{⊗m}, ∥ .∥_{∼}), …, (ℒ^{⊗m}, ∥ .∥_{∼}))

+ O(λ^{2}) .

By virtue of Lemma 4.4, we have p_{Z}^{X}(\overline{ℒ} ,…, \overline{ℒ} ) = 0. Further,

\frac{1}{m^{dim X}deg_{ℒ} X} ·p_{Z}^{X}((𝒪_{X}, ∥ .∥_{∞}· exp(−λφ)), (ℒ^{⊗m}, ∥ .∥_{∼}),…, (ℒ^{⊗m}, ∥ .∥_{∼}))

= λ lim_{i→∞} I_{i}(φ)

for

I_{i}(φ) := \frac{1}{m^{dim X}} p_{Z}^{X}((𝒪_{X}, ∥ .∥_{∞}· exp(−φ)), \underbrace{(ℒ^{⊗m}, ∥ .∥_{∼}^{(i)}),…, (ℒ^{⊗m}, ∥ .∥_{∼}^{(i)})}_{dim X times}) .

At this level, we can make the arithmetic intersection product explicit and find

I_{i}(φ) = ∫_{P (ℂ)}φc_{1}(ℒ , ∥ .∥_{∞}^{(i)}) ∧…∧ c_{1}(ℒ , ∥ .∥_{∞}^{(i)}) .

Theorem A.14 therefore yields in view of Proposition 2.10

e_{1}(h_{(ℒ^{⊗m},∥ .∥^{λ})} ) ⩾ λ· ∫_{P (ℂ)} φ dτ .  (5)

The equations (4) and (5) together imply the assertion.

Theorem 4.8 (Equidistribution on P (ℂ)). Let P, f, \overline{ℒ} , and Φ be as in 1.12. Then, for each sequence (x_{i})_{i∈ℕ} of closed points on P which is generic and small, the associated sequence of measures (δ_{x_{i}})_{i∈ℕ} converges weakly to the measure τ which is the zero measure on P (ℂ)\K and the Haar measure of volume one on K.

Proof. First step. δ_{x_{i}}|_{P (ℂ)\K} → 0.

By Proposition 3.8, for every ε_{1}, ε_{2} > 0, there is some non-negative g_{ε_{1},ε_{2}} ∈ S such that supp g_{ε_{1},ε_{2}} ⊆ U_{ε_{1}}(K),

max_{x∈P (ℂ)} g_{ε_{1},ε_{2}} ⩽ 1 + ε_{2} ,

and

∫_{P (ℂ)}g_{ε_{1},ε_{2}} dτ = 1 .

Then, by Proposition 4.7,

lim inf_{i→∞} ∫_{P (ℂ)} g_{ε_{1},ε_{2}} dδ_{x_{i}} ⩾ 1 .

This yields

lim sup_{i→∞} ∫_{P (ℂ)} (1 − g_{ε_{1},ε_{2}}) dδ_{x_{i}} ⩽ 0 .

But

1 − g_{ε_{1},ε_{2}}(x) ⩾ \{ \begin{matrix}1 & if x ∈ P (ℂ)\U_{ε_{1}}(K) , \\ −ε_{2} & if x ∈ U_{ε_{1}}(K) . \end{matrix}

Consequently,

lim sup_{i→∞} [δ_{x_{i}}(P (ℂ)\U_{ε_{1}}(K)) − ε_{2}δ_{x_{i}}(U_{ε_{1}}(K))] ⩽ 0 .

This shows

lim sup_{i→∞} δ_{x_{i}}(P (ℂ)\U_{ε_{1}}(K)) ⩽ ε_{2} .

As the latter is true for every ε_{2} > 0, we have

lim_{i→∞} δ_{x_{i}}(P (ℂ)\U_{ε_{1}}(K)) = 0 .

Further, this formula is true for each ε_{1} > 0. Hence, (δ_{x_{i}}|_{P (ℂ)\K} )_{i∈ℕ} converges weakly to the zero measure.

Second step. The assertion in general.

Let φ ∈ C(P (ℂ)) be an arbitrary continuous function. As we can interchange the roles of φ and (−φ), it suffices to prove

lim inf_{i→∞} ∫_{P (ℂ)} φ dδ_{x_{i}} ⩾ ∫_{P (ℂ)} φ dτ.

By Lemma 4.5, we have continuous functions φ_{1} and φ_{2} on P (ℂ) such that supp φ_{1} ⊆ P (ℂ)\K and φ_{2} ∈ ℝ_{+}·S. The result of the first step shows

lim_{i→∞} ∫_{P (ℂ)} φ_{1} dδ_{x_{i}} = 0 .

Further, by Proposition 4.7,

lim inf_{i→∞} ∫_{P (ℂ)} φ_{2} dδ_{x_{i}} ⩾ ∫_{P (ℂ)} φ_{2} dτ .

Consequently,

lim inf_{i→∞} ∫_{P (ℂ)} φ dδ_{x_{i}} ⩾ ∫_{P (ℂ)} φ dτ − ε .

The assertion follows since ε > 0 is arbitrary.

Corollary 4.9. Let X ⊂ P be a closed subvariety and (x_{i})_{i∈ℕ} be a sequence of closed points on X which is generic and small.

Then, the sequence (δ_{x_{i}}|_{X(ℂ)\K} )_{i∈ℕ} converges in the weak sense to the zero measure.

Proof. As there are only finitely many i such that x_{i} ∉ G, let us assume that the whole sequence (x_{i})_{i∈ℕ} is contained in G. Choose some K-invariant metric on G(ℂ) and assume that, for some ε > 0, we would have

lim inf_{i→∞} δ_{x_{i}}(X(ℂ)\U_{ε}(K)) = δ > 0 .

We will construct another sequence (y_{i})_{i∈ℕ} which is generic and small on the whole of P. For this, note first that we have h\overline{ℒ} (x) = \frac{1}{deg f} h\overline{ℒ} (f(x)). Consequently, h\overline{ℒ} is invariant under shift by any torsion point t ∈ K_{tor} = ⋃_{j∈ℕ} ker(m^{j}). Further, all the sequences (t·x_{i})_{i∈ℕ} fulfill

lim inf_{i→∞} δ_{t·x_{i}}(P (ℂ)\U_{ε}(K)) = δ .

Finally, K ⊆ G(ℂ) is Zariski dense. Therefore, for each i, the union ⋃_{t∈K_{tor}}{t·x_{i}} is a Zariski dense subset of P.

Recall, at this point, the fact there are only countably many proper closed subvarieties P_{0}, P_{1}, P_{2}, … ⊂ P. We may choose a sequence (y_{i})_{i∈ℕ} such that, for every i ∈ ℕ, one has

y_{i} ∈ ⋃_{t∈K_{tor},i⩾0} {t·x_{i}} and y_{i} ∈ P \P_{0}\…\P_{i}. Then,

δ_{y_{i}}(P (ℂ)\U_{ε}(K)) > δ/2

for i ≫ 0 and h\overline{ℒ} (y_{i}) → 0. This is a contradiction to Theorem 4.8.

Remark 4.10. This corollary shows, in particular, the following. If X ∩ K = ∅ then

p_{Z}^{X}(\overline{ℒ} , …, \overline{ℒ} ) > 0 .

In other words, there are no small and generic sequences on X.

Appendix A. The adelic Picard group

In this appendix, we will recall S. Zhang’s adelic Picard group [Zh95a] and fix notation.

The local case. Preparations. Let K be an algebraically closed valuation field. The cases we have in mind are K = \overline{ℚ}_{p} for a prime number p and K = \overline{ℚ}_{∞} = ℂ but, of course, there are many other examples, all of which are non-archimedean.

Definition A.1. Let X be a projective scheme over K. Then, a metric on an invertible sheaf ℒ ∈ Pic(X) is a system of K-norms on the K-vector spaces ℒ (x) for all x ∈ X(K).

Example A.2. Let K be non-archimedean and 𝒪_{K} the ring of integers in K. Then, a model of X, i.e. a flat projective scheme π: 𝒳 → 𝒪_{K} such that 𝒳_{K} ≅ X, together with an invertible sheaf ℒ̃∈ Pic(𝒳 ) with ℒ̃|_{X} ≅ ℒ^{⊗n} for some n > 0 induces a metric ∥.∥ on ℒ as follows:

The point x ∈ X(K) has a unique extension \overline{x}: Spec 𝒪_{K} → 𝒳 . Then \overline{x}^{∗}ℒ̃is a projective 𝒪_{K}-module of rank 1. Each l ∈ ℒ (x) induces l^{⊗n} ∈ ℒ^{⊗n}(x) and a rational section of \overline{x}^{∗}ℒ̃. Put

∥l∥_{x} := [inf \{|a| | a ∈ K, l ∈ a · \overline{x}^{∗}ℒ̃\}]^{\frac{1}{n}} .

This metric is called the metric on ℒ being induced by the model (𝒳 ,ℒ̃).

Definition A.3. Let X be a smooth, projective scheme over K.

a) Assume K = ℂ. Then, a metric ∥.∥ on ℒ ∈ Pic(X) is called continuous respectively smooth if, for each x ∈ X(ℂ), there exist some open neighborhood U_{x} and a holomorphic section s ∈ Γ(U_{x}, ℒ |_{U_{x}}) such that ∥s∥ is continuous, respectively smooth.

b) Assume K is non-archimedean. Then, a metric ∥.∥ on ℒ ∈ Pic(X) is called continuous if ∥.∥ = f · ∥.∥′ for f a continuous function on X(K) and ∥.∥′ a metric being induced by a model.

The global case. The adelic Picard group. Let now X be a projective variety over ℚ and ℒ ∈ Pic(X) be a invertible sheaf.

Definition A.4. An adelic metric ∥.∥ on ℒ is a system {∥.∥_{p}}_{p∈{2,3,5,...;∞}} of continuous and bounded metrics on ℒ_{\overline{ℚ} _{p}} ∈ Pic(X_{\overline{ℚ} _{p}}) such that

i) for each p ∈ \{2, 3, 5, …; ∞\} the metric ∥.∥_{p} is Gal(\overline{ℚ}_{p}/ℚ_{p})-invariant,

ii) for some n ∈ ℕ, there exist a projective model 𝒳 of X over Spec ℤ[\frac{1}{n}] and an invertible sheaf ℒ̃∈ Pic(𝒳 ) with ℒ̃|_{X} ≅ ℒ , which induces ∥.∥_{p} for almost all p.

ℒ being equipped with an adelic metric is called an adelicly metrized invertible sheaf. The group of all adelicly metrized invertible sheaves on X will be denoted by Pic^{ad}(X).

Remark A.5. Let 𝒳 → Spec ℤ be a model of X. Then, there are the following two natural homomorphisms given by the induced metrics.

i_{𝒳} : P̂ic(𝒳 ) → Pic^{ad}(X)
a_{𝒳} : ker(Pic(𝒳 ) → Pic(X)) ⊗_{ℤ} ℚ → Pic^{ad}(X)

Here, P̂ic(𝒳 ) denotes the arithmetic Picard group of Gillet and Soulé [GS90]. Further, one has the forgetful homomorphism v: Pic^{ad}(X) → Pic(X). Note, if ζ ∈ im a_{𝒳} then there exists some n ∈ ℕ such that nζ ∈ im i_{𝒳} .

Definition A.6. a) The group

Pic^{in}(X) := ⟨im i_{𝒳} , im a_{𝒳} ⟩_{𝒳 model of X} ⊆ Pic^{ad}(X)

is called the group of the invertible sheaves on X with induced metrics.

b) ζ ∈ Pic^{in}(X) is called semipositive, ζ ∈ C_{X}^{⩾0}, if there are a natural number n, a model 𝒳 of X, and \overline{ℒ} ∈ P̂ic(𝒳 ) such that nζ = i_{𝒳} (\overline{ℒ} ) where

i) \overline{ℒ} has a non-negative curvature form c_{1}(\overline{ℒ} ), ii) for every prime number p, the underlying bundle ℒ is of non-negative degree on each curve in 𝒳_{𝔽_{p}} .

Definition A.7 (Metric on v^{−1}(ℒ) ⊆ Pic^{ad}(X)). Let (ℒ , ∥.∥) and (ℒ , ∥.∥′) be two adelicly metrized invertible sheaves with the same underlying bundle. Then, the distance between (ℒ , ∥.∥) and (ℒ , ∥.∥′) is given by

δ ((ℒ , ∥.∥), (ℒ , ∥.∥′)) := ∑_{p∈\{2,3,5,...;∞\}} sup_{x∈X(\overline{ℚ} _{p})} |log \frac{∥.∥′_{p}}{∥.∥_{p}} | .

Lemma A.8. Let 𝒳 be a flat, projective scheme over ℤ of dimension d + 1. Denote its generic fiber by X.

Further, let (ℒ′, ∥.∥′), (ℒ′′, ∥.∥′′), \overline{ℒ} _{1}, …, \overline{ℒ} _{d} ∈ P̂ic(𝒳 ) where ℒ′ and ℒ′′ are extensions of one and the same invertible sheaf ℒ ∈ Pic(X). If i_{𝒳} (\overline{ℒ} _{1}), …, i_{𝒳} (\overline{ℒ} _{d}) are semipositive then

\begin{matrix}|[ĉ_{1}(ℒ′, ∥.∥′)−ĉ_{1}(ℒ′′, ∥.∥′′)] · ĉ_{1}(\overline{ℒ} _{1}) · . . . · ĉ_{1}(\overline{ℒ} _{d})| ⩽ \\ ⩽ δ(i_{𝒳} (ℒ′, ∥.∥′), i_{𝒳} (ℒ′′, ∥.∥′′)) · c_{1}(ℒ_{1}|_{X}) · . . . · c_{1}(ℒ_{d}|_{X}) . \end{matrix}

Proof. This result is contained in [Zh95a, Theorem 1.4]. The main ingredient is the fact that on a variety over a field the intersection product of semipositive divisors is always non-negative.

The adelic Picard group.

Definition A.9. Let \overline{C_{X}^{⩾0}} be the closure of the semipositive cone in Pic^{ad}(X).

i) Then, the subgroup P̃ic(X) ⊆ Pic^{ad}(X) generated by \overline{C_{X}^{⩾0}} is called the adelic Picard group of X. ii) If (ℒ , ∥.∥) ∈ P̃ic(X) then (ℒ , ∥.∥) is said to be an integrably metrized invertible sheaf. The metric ∥.∥ is called an integrable metric on ℒ.

Theorem A.10 (Zhang). Let X be a scheme which is smooth, projective, and equidimensional over ℚ.

i) Then, there is exactly one continuous map, the adelic intersection product

p_{Z}^{X}: \underbrace{P̃ic(X) × … × P̃ic(X)}_{dim X+1 times} → ℝ,

such that, for every model 𝒳 of X, the diagram

\xymatrix{P̃ic(X) ×…× P̃ic(X) \ar[r]^{p_{Z}^{X}} & ℝ \\ P̂ic(𝒳 ) ×…× P̂ic(𝒳 ) \ar[u]^{i_{𝒳}^{dim X+1}} \ar[ur]_{p_{GS}^{𝒳}}}

commutes. Here, p_{GS}^{𝒳} is the arithmetic intersection product of Gillet and Soulé. ii) p_{Z}^{X} is multi-linear and symmetric.

Example A.11. For L a number field, P̃ic(Spec L) = P̂ic(Spec 𝒪_{L}). I.e., there is an exact sequence 0 → Cl_{L} → P̃ic(Spec L) → ℝ → 0.

Example A.12. Let V be a semistable projective curve over ℚ. Then, there is an exact sequence 0 → 𝔸 → P̃ic(V) → Pic(V) → 0 where

𝔸 = ⨁_{ν∈Val(ℚ)}C_{0}(V (\overline{ℚ}_{ν})) .

Here, C_{0}(V (\overline{ℚ}_{∞})) = C(V (ℂ))^{F_{∞}} is the space of all continuous F_{∞}-invariant functions on V (ℂ).

For p a prime number, C_{0}(V (\overline{ℚ}_{p})) is given as follows. We choose a minimal semistable model 𝒱 of V over Spec ℚ_{p}. Let D_{1},…, D_{r} be the irreducible divisors in the special fiber. Further, for each k ∈ ℕ, we define a countable disjoint covering \{U_{i}^{(k)}\}_{i∈ℕ} of V (\overline{ℚ}_{p}) by the requirement that x, y ∈ V (\overline{ℚ}_{p}) belong to the same set if they coincide modulo p^{k}. Then, C_{0}(V (\overline{ℚ}_{p})) consists of all Gal(\overline{ℚ}_{p}/ℚ_{p})-invariant functions V (\overline{ℚ}_{p}) → ℝ of the type

∑_{i=1}^{r} α_{i}∥1∥_{(𝒱 ,𝒪(D_{i}))} + ∑_{l=1}^{∞} ∑_{ik} β_{ikl} χ_{U_{i}^{(k)}} .

Here, ∥ .∥_{(𝒱 ,𝒪(D_{i}))} denotes the metric, induced by the model (𝒱 , 𝒪(D_{i})), χ_{U_{i}^{(k)}} is the characteristic function of U_{i}^{(k)}, and α_{i} as well as β_{ikl} are real numbers. We require that the inner sum is finite for each l ∈ ℕ and that the outer series is uniformly convergent.

A.13. Finally, let us recall the theorem of successive minima. We suppose that X is a scheme which is smooth, projective, and equidimensional over ℚ. Further, \overline{ℒ} ∈ P̃ic(X).

i) Then, the height of an L-valued point x ∈ X(L) for L a number field is given by

h\overline{ℒ} (x) := \frac{1}{[L : ℚ]} deg \overline{ℒ} |_{X}.

For i = 1, 2, …, dim X + 1, put

e_{i}(\overline{ℒ} ) := sup_{cod Y = i} inf_{x∈X\Y} h\overline{ℒ} (x) .

ii) The height of X itself with respect to \overline{ℒ} is given by the formula

h\overline{ℒ} (X) := \frac{c_{1}(\overline{ℒ} )^{dim X+1}}{(dim X + 1) c_{1}(ℒ_{ℚ} )^{dim X}} .

Theorem A.14 (Zhang). Let X be a scheme which is smooth, projective, and equidimensional over ℚ and \overline{ℒ} ∈ P̃ic(X). Assume \overline{ℒ} ∈ \overline{C_{X}^{⩾0}}. Then,

e_{1}(\overline{ℒ} ) ⩾ h\overline{ℒ} (X) ⩾ \frac{e_{1}(\overline{ℒ} ) + … + e_{dim X+1}(\overline{ℒ} )}{dim X + 1}.

Proof. This is [Zh95a, Theorem 1.10]. It is obtained by an easy limit argument from an analogous statement for arithmetic varieties [Zh95b, (5.2)]. Note that the arithmetic Riemann-Roch theorem [GS92] is the main ingredient here.

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Jörg Jahnel ˜ jjahnel@mathematik.uni-siegen.de Fachbereich 6 Mathematik, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany