24 the concept of a height [Chap. I

ii) Let X =

Pn

, and let L be the invertible sheaf O(1) equipped with the Fubini–

Study metric. Then the height with respect to L is the l2-height, which is given by

hl2 (x0 : . . . : xn) := log x02 + . . . + xn2

for

(x0 : . . . : xn) = (x0 : . . . : xn)

projective coordinates such that all xi are integers and gcd(x0, . . . , xn) = 1.

3.14. Lemma. Let f : X → Y be a morphism of arithmetic varieties, and

let L be a hermitian line bundle on Y .

Then, for every x ∈ X ( ),

hf

∗L

(x) = hL (f(x)) .

Proof. Let x be the extension of x to Spec . Then f ◦ x is the extension of f(x)

to Spec . Thus,

hf

∗L

(x) = deg

(

x∗(f ∗L

)

)

= deg

(

(f◦x)∗L

)

= hL (f(x)) .

3.15. Proposition. Let X be an arithmetic variety.

a) Let .

1

and .

2

be hermitian metrics on one and the same invertible sheaf L .

Then there is a constant C such that

|h(L

, .

1)(x) − h(L

, .

2)(x)| C

for every x ∈ X ( ).

b) Let L1 and L2 be two hermitian line bundles. Then, for every x ∈ X ( ),

h(L1⊗L2)(x) = hL1 (x) + hL2 (x) .

c) Let L be a hermitian line bundle such that the underlying invertible sheaf is am-

ple. Then, for every B ∈ , there are only finitely many points x ∈ X ( ) such that

hL (x) B .

Proof. a) For x the extension of x to Spec , we have

h(L

, . 1)

(x) − h(L

, . 2)

(x) = deg

(

x∗L

,

x∗

.

1

)

− deg

(

x∗L

,

x∗

.

2

)

.

Working with a non-zero section s ∈ Γ(Spec ,

x∗L

), we see that the latter differ-

ence is equal to

log

#(x∗L /sx∗L

) − log s 1(x) −

(

log

#(x∗L /sx∗L

) − log s 2(x)

)

= log

.

2

(x)

.

1

(x)

.