4

SANDOR J. KOVACS

A line bundle

£

on X is called big if X is proper and the global sections of

,em

define a birational map for some m 0.

£

is called nef if deg(Cic)

~

0 for every

proper curve

CCX.

In particular every ample line bundle is nef and big.

In this paper a log variety (X, D) consists of a proper variety X and an effective

divisor D, called the boundary. A log morphism takes boundary to boundary, in

particular the image of the boundary is a divisor. (X, D) is of log general type if

Kx +Dis big. It is log canonically polarized if Kx + D is ample. Note that in this

case

X

is necessarily projective.

For a log variety,

(X,

D),

Aut(X,

D) (resp. Bir(X, D)) denotes the set of au-

tomorphisms (resp. proper birational automorphisms) of

X

that leave

D

fixed. If

X

~

lP'N, then Lin(

X, D)

= Lin(

X)

n

Lin(

D)

denotes the set of linear automor-

phisms that leave both

X

and

D

fixed. A smooth log variety will mean simply that

X is smooth. Note that this is very different from the notion of log smooth.

Let G be a finite group and

g

E

G. Then IGI (resp.

o(g))

denotes the order of

G (resp. the order

g).

§1. Log varieties

§§1.1 Log canonical embeddings.

Let (X, D) be a log canonically polarized

smooth log variety of dimension n. Let m

E

N be such that the complete linear

system of (m + 1)Kx + mD gives a birational morphism ¢ : X

-+

lP'N for some

N 0 such that

D

= ¢(D) is still a divisor in

X

= ¢(X), i.e.,

(X, D)

is a log

variety and ¢ is a log morphism. This can be achieved for instance by requiring

that ¢ separate points.

1.1.1 FACT. In characteristic 0, l(m + 1)Kx + mDI separates the points of X

as soon as m ~ (n!

2)

by [Angehrn-Siu95] ( cf. [Kollar97, 5.8]).

Observe that any linear automorphism of lP'N leaving the pair

(X, D)

fixed

induces a proper birational automorphism of (X, D). In fact ¢* gives an isomor-

phism between Lin(

X, D)

and Bir(X, D). Now Bir(X, D) is finite by [Iitaka82,

11.12], hence so is Lin(

X, D)

and in order to estimate I Aut(

X, D)

I it is enough to

estimate ILin(X,D)I.

1.1.2 LEMMA. Let (X, D) be a log canonically polarized smooth log variety of

dimension n and

¢ :

X

-+

lP'N a birational log morphism given by the complete

linear system l(m + 1)Kx + mDI. Let

X=

¢(X) and

D

=¢(D).

(a) If D is nef, then

deg

X:::;

((m + 1)(Kx +

D))n

deg

D:::;

((m + 1)(Kx +

D))n.

(b) If K x is nef, then

degD:::; degX = ((m + 1)Kx + mD)n.

PROOF.

(a) In this case we actually prove that

degX + degD:::; ((m + 1)(Kx +

D))n.

(m + 1)(Kx +D), D, and (m + 1)Kx + mD are nef, so for all i = 1, ... , n- 1,

((m + 1)Kx + mD)i · D · ((m + 1)Kx + mD)n-i-l

~

0,