28 the concept of a height [Chap. I

4.13. Notation. The models of X, together with all birational mor-

phisms between them, form an inverse system of schemes. This is a filtered

inverse system since, for two models X and X , the closure of the diagonal

Δ ⊂ X ×Spec X ⊂ X ×Spec X projects to both of them.

Thus, the arithmetic Picard groups Pic(X ) for all models X of X form a filtered

direct system. The injections Pic(X ) → Pic(X) fit together to yield an injection

ιX : lim

−→

Pic(X ) → Pic(X) .

Similarly, the usual Picard groups Pic(X ) form a filtered direct system, too. We get

a homomorphism

αX : lim

−→

ker(Pic(X ) → Pic(X)) ⊗ −→ Pic(X) .

4.14. Definition (Metric on

v−1(L

) ⊆ Pic(X)). Let X be a projective va-

riety over . On X, let (L , . ) and (L , . ) be two adelically metrized invertible

sheaves with the same underlying sheaf.

Then the distance between (L , . ) and (L , . ) is given by

δ((L , . ), (L , . )) :=

ν∈Val( )

δν( . ν, .

ν

)

for

δν( . ν, .

ν

) := sup

x∈X(

ν

)

log

.

ν

(x)

. ν(x)

.

4.15. Lemma. δ is a metric on the set

v−1(L

) of all metrizations of L .

Proof. We have to show that the sum is always finite.

For this, we note first that the metrics .

ν

and .

ν

are bounded by definition.

Therefore, each summand is finite.

We may thus ignore a finite set S of primes and assume that . and . are

given by triples (X , L , n) and (X , L , n ), respectively, in the sense of Defini-

tion 4.11.a.ii).

The isomorphism X

∼

=

−→ X

∼

=

−→ X may be extended to an open neigh-

bourhood of the generic fiber. Therefore, enlarging S if necessary, we have an

isomorphism X

∼

=

−→ X of schemes over Spec \S. Further, the triple (X , L , n)

may be replaced by (X , L

⊗n

, nn ) without any change of the induced metric.

Thus, without restriction, n = n .

To summarize, we are reduced to the case that . and . are given by (X , L , n)

and (X , L , n). We have an isomorphism

L |X

∼

=

−→ L

⊗n

∼

=

−→ L |X ,

which may be extended to an open neighbourhood of the generic fiber. There-

fore, in the definition of δ((L , . ), (L , . )), all the summands vanish, except

finitely many.

Positivity, symmetry, and the triangle inequality are clear.