22 the concept of a height [Chap. I
i) The Fubini–Study metric is given by
+ . . . +
, O(1) ).
ii) The minimum metric is given by
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 ∈
) where vanishes. Simi-
larly, in the definition of
(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
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).