20 the concept of a height [Chap. I

2.8. For most theoretical investigations and, perhaps, some practical ones,

it is actually better to work with the logarithm of the absolute height.

2.9. Definition. For (x0 : . . . : xn) ∈ Pn( ), put

hnaive(x0 : . . . : xn) := log Hnaive(x0 : . . . : xn) .

The height function hnaive is called the logarithmic height.

ii. Application: The height defined by an ample invertible sheaf.

2.10. Definition. Let X be a projective variety over a number field K, and

let L be an ample invertible sheaf on X.

Then a height function hL induced by L is given as follows. Let m ∈ be such

that L

⊗m

is very ample. Put

hL (x) :=

1

m

hnaive

(

iL

⊗m

(x)

)

for x ∈ P a closed point. Here, iL

⊗m

: P →

PN

denotes a closed embedding defined

by L

⊗m.

2.11. Lemma. Let X be a projective variety over a number field K, and let

L be an ample invertible sheaf on X. If hL

(1)

and hL

(2)

are two height functions

induced by L , then there is a constant C such that

| hL

(1)

(x) − hL

(2)

(x)| ≤ C

for every x ∈ X( ).

Proof. First, it is clear that the k-tuple embedding fulfills

hnaive

(

ρk(x)

)

= k·hnaive(x)

for every k ∈ and x ∈

PN

( ). Hence, if hL is defined using L

⊗m,

then the

same height function may be defined using L ⊗km.

We may therefore assume that hL

(1)

and hL

(2)

are defined using the same tensor

power of L . Then the two embeddings

i(1)⊗m

L

and

i(2)

L ⊗m

differ by an automorphism

ι:

PN

→

PN

.

We have to verify that

| hnaive

(

ι(x)

)

− hnaive(x)| ≤ C

for every x ∈ PN ( ). ι is explicitly given by

(x0 : . . . : xN ) → (x0 : . . . : xN )

:= ((a00x0 + . . . + a0N xN ) : . . . : (aN0x0 + . . . + aNN xN ))

for some a = (aij)ij ∈ MN+1(K).