Sec. 4] the adelic picard group 27

4.8. Definition. Let K be an algebraically closed valuation field. Assume

K to be non-Archimedean.

Then a metric . on L ∈ Pic(X) is called continuous, respectively bounded,

if . = f · . for . a metric induced by some model and f a function on X(K)

that is continuous or bounded, respectively.

4.9. Remark. If K = , then we adopt the concepts of bounded, con-

tinuous, and smooth metrics in their the usual meaning from complex geometry.

Note that smooth metrics are continuous and that continuous metrics are automat-

ically bounded in the case K = .

ii. The global case. Adelically metrized invertible sheaves.

4.10. Definition. Let X be a projective variety over and m ∈ .

Then, by a model of X over Spec [

1

m

], we mean a scheme X that is projective

and flat over Spec [

1

m

] such that the generic fiber of X is isomorphic to X.

4.11. Definition. Let X be a projective variety over and L ∈ Pic(X)

be an invertible sheaf.

a) Then an adelic metric on L is a system

. = { .

ν

}ν∈Val(

)

of continuous and bounded metrics on L

ν

∈ Pic(X

ν

) such that

i) for each ν ∈ Val( ), the metric .

ν

is Gal(

ν

/

ν

)-invariant,

ii) for some m ∈ , there exist a model X of X over Spec [

1

m

], an invertible

sheaf L ∈ Pic(X ), and a natural number n such that

L |X

∼

=

L

⊗n

and, for all prime numbers p m, the metric .

νp

is induced by (Xp, L |Xp , n).

b) An invertible sheaf equipped with an adelic metric is called an adelically metrized

invertible sheaf.

c) All adelically metrized invertible sheaves on X form an abelian group, which will

be called the adelic Picard group of X and denoted by Pic(X).

4.12. Notation. Let X be a model of X over Spec . Then taking the

induced metric yields two natural homomorphisms

iX : Pic(X ) → Pic(X) ,

aX : ker(Pic(X ) → Pic(X)) ⊗ → Pic(X) .

Further, one has the forgetful homomorphism

v : Pic(X) → Pic(X) .