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)
.
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