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.