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
is very ample. Put
hL (x) :=
for x ∈ P a closed point. Here, iL
: P →
denotes a closed embedding defined
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
are two height functions
induced by L , then there is a constant C such that
(x) − hL
(x)| ≤ C
for every x ∈ X( ).
Proof. First, it is clear that the k-tuple embedding fulfills
for every k ∈ and x ∈
( ). Hence, if hL is defined using L
same height function may be defined using L ⊗km.
We may therefore assume that hL
are defined using the same tensor
power of L . Then the two embeddings
differ by an automorphism
We have to verify that
− 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).