PRINCIPALLY POLARIZED ABELIAN VARIETIES 5
and hence degX + deg D = ((m +
1)Kx
+ mD)n + D · ((m +
1)Kx
+ mD)n-l =
(m + 1)(Kx +D)· ((m +
1)Kx
+ mD)n-l
=
(m + 1)(Kx +D)· ((m +
1)Kx
+
mD) · ((m +
1)Kx
+ mD)n-
2
:$ ((m +
1)(Kx
+ D))
2

((m +
1)Kx
+ mD)n-
2
:$
... ::::; ((m + 1)(Kx + D))n.
(b)
If Kx
is nef, then so is (m +
1)Kx
+ (m -1)D =
2Kx
+ (m
-1)(Kx
+D),
hence degD = D · ((m +
1)Kx
+ mD)n-l :$ ((m +
1)Kx
+ mD)n = degX. 0
§§1.2 Abelian groups of automorphisms. Obtaining bounds for abelian
groups of automorphisms requires much less than the general case. The following
lemma is an adaptation of [Szab696, Bezout Lemma] to the log case.
1.2.1 LEMMA. Let
(X,
D) be a projective log variety of dimension n with a
fixed embedding
X
~
lP'N
and T a closed reduced subgroup of the projective linear
group. Assume that Lin(
X,
D) nTis finite and that there exists a point x EX such
that its stabilizer, Tx, is trivial and its orbit, Tx, is open and dense in
lP'N.
Then Lin(X,.D) n T has at most (degX)
1

(degD)n+l-l elements for some
1:$l:$n+l.
PROOF. Identify Tx with T. Let U =X n T and V = D n T. U is open and
dense in
X
and Vis open in
D.
Note that
V
may be empty.
An arbitrary t E T is in Lin(X, D) if and only if ta E U, i.e., t E U a-
1
for all
a E U and tb E V, i.e., t E Vb-
1
for all bE V. Equivalently,
Lin(X,D)nT=
n
ua-
1
n
n
Vb-
1

aEU bEV
Lin( X, D) n T is assumed to be finite so one can find
a0
, ... ,
az_
1
E U and
bz, ... ,bn E V for some 1::::; l::::; n
+
1 such that (
n~:;6
Uai
1)
n ( nj=z Vbj
1)
is
already finite. By Bezout's theorem this proves the statement. 0
1.2.2 THEOREM. Let
(X,
D) be a projective log variety of dimension n with a
fixed embedding
X
~
lP'N.
Assume that Lin(
X,
D) is finite. Let G :$ Lin(X, D) be
an abelian subgroup whose order is coprime to p, the characteristic of the ground
field and t E Lin(
X,
D) an arbitrary element. Then both jGj and o(t) is at most
(degX)
1

(degD)n+l-l for some 1::::; l :$ n
+
1.
PROOF. This follows easily by the arguments of [Szab696, Theorem 3 and
Abelian Lemma] using (1.2.1) in place of Szabo's Bezout Lemma. 0
1.2.3 THEOREM. Let (X, D) be a log canonically polarized smooth variety of
dimension n and m the smallest integer such that
I (
m
+
1
)K x
+
mDj separates the
points of X. Let G ::::; Aut(X, D) be an abelian subgroup and t E Aut( X, D) an
arbitrary element. Let
1r
denote the p-part of jGj, i.e.,
IGI/n
is an integer coprime
top, the characteristic of the ground field.
(a) If D is nef, then
jGj :$
7r
((m
+
1t
(Kx
+
D)n)n+l
o(t) :$
((m
+
1t
(Kx
+
D)nt+l.
(b) If K x is nef, then
IGI
:$
1r
(((m
+
1)
Kx
+
mD)nt+l
o(t) :$ (((m
+
1)
Kx
+
mD)nt+l.
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