6
SANDOR J. KOVACS
Furthermore in characteristic
0,
m:::;
(n~
2
) by
(1.1.1).
PROOF. (a) Using the birational morphism given by l(m
+
1)Kx
+
mDI one
obtains a projective log variety
(X,
D)
that satisfies the conditions of
(1.2.2).
Then by
(1.1.2),
(deg.X)I. (deg
tJ)n+l-l:::;
((m
+
1)n(Kx
+
D)n)n+l
for any 1 :::; l :::; n
+
1.
(b) If
Kx
is nef, then (degX)
1
·
(degtJ)n+l-l:::; (deg.X)n+l by
(1.1.2). D
1.2.4
COROLLARY. Under the assumptions of
(1.2.3),
I Aut(X, D) I is coprime
top as soon asp (
(m
+
1t
(Kx
+
D)n)n+l.
§§1.3 The general case. The general case, that is when Aut( X, D) is
not necessarily abelian, relies on group theoretical considerations. In particular
the results cited below use the classification of finite simple groups. I have used
[Szab696] as a reference, but many of the ideas behind his results already appear
in [Huckleberry-Sauer90] and possibly in the works of others.
1.3.1
THEOREM. Let (X, D) be a log canonically polarized smooth variety of
dimension n over an algebraically closed field of characteristic p ;:::
0,
and m the
smallest integer such that l(m
+
1)Kx
+
mDI separates the points of X. Let
1r
denote thep-part ofiAut(X,D)I, i.e., 1Aut(X,D)I/7r is an integer coprime top,
the characteristic of the ground field.
(a) If D is nef, then
I Aut(X,
D)
I:::; 7r3n ((m
+
1t
(Kx
+
D)n)l6n3n
(b) If K x is nef, then
I Aut(
X, D)
I :::;
7r3n
(((m
+
1)Kx
+ mD)n)l6n3n
Furthermore in characteristic
0, m:::;
(n~
2
) by
(1.1.1).
PROOF. Follows from
(1.2.3)
and [Szab696, Main bound].
D
REMARK. It is natural to allow some mild singularities when studying log
varieties. In fact the arguments of this section stay valid if (X, D) is only assumed
to be a klt pair. The estimates however become somewhat worse as one have to
include the index of X. (For the definition of klt see [Kollar97]).
One may also weaken the assumptions requiring
Kx
+
D be only semi-ample
and big. In that case however the Angehrn-Siu bound have to be replaced by
something larger, e.g., Kollar's effective base point freeness bound.
§2. Principally polarized abelian varieties
In this section the general results are applied for principally polarized abelian
varieties. It is very likely that by direct methods one can improve the actual
estimates. According to an argument of J.-P. Serre [Previato] a recent result of
Feit can be used to obtain significantly better estimates in characteristic 0. That
argument however does not work in positive characteristic.
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