22 the concept of a height [Chap. I

i) The Fubini–Study metric is given by

FS

:=

1

(

X0

)2

+ . . . +

(

Xn

)2

for ∈

Γ(Pn

, O(1) ).

ii) The minimum metric is given by

min

:= min

i=0,...,n

Xi

.

In the case of the Fubini–Study metric, the formula should be interpreted in

such a way that FS(x) = 0 at all points x ∈

Pn(

) where vanishes. Simi-

larly, in the definition of

min

(x), the minimum is actually to be taken over all i

such that xi = 0.

The Fubini–Study metric is smooth (C∞). The minimum metric is continuous but

not even

C1.

3.6. Definition (Arakelov degree). For a hermitian line bundle (L , . )

on Spec , define its Arakelov degree by

deg (L , . ) := log #(L /sL ) − log s

for s ∈ Γ(Spec , L ) a non-zero section.

3.7. Remark. L is associated with a free -module L of rank one. L is

the -vector space L⊗ . Thus, we work with a non-zero element s ∈ L and with

the norm s ⊗ 1 .

The definition is independent of the choice of s since

#(L /nsL ) = n·#(L /sL ) ,

therefore log #(L /nsL ) = log #(L /sL ) + log n. On the other hand,

log ns = log s + log n.

3.8. Fact. Let L be a hermitian line bundle on Spec . If there is a sec-

tion s ∈ Γ(Spec , L ) of norm less than one, then deg L 0.

Proof. This follows immediately from the definition.

3.9. Fact. For two hermitian line bundles L1, L2 on Spec , one has

deg (L1 ⊗ L2) = deg L1 + deg L2 .

Proof. Apply the definition to arbitrary non-zero sections s1 ∈ Γ(Spec , L1),

s2 ∈ Γ(Spec , L2), and to their tensor product s1 ⊗s2 ∈ Γ(Spec , L1⊗L2).