10 2. KOBAYASH I HYPERBOLICIT Y
THEOREM 2.1. (Brody) Let M be a compact complex manifold. Suppose
that M is not Kobayashi hyperbolic. Then there exists a non constant holomorphic
map f : C i— M.
Pick a smoot h Hermitia n metri c ds o n M. Suppos e tha t M i s no t Kobayash i
hyperbolic. The n ther e exist s a sequenc e o f holomorphic map s {/n}£Li » / n ^ n
M, /
n
(0) = p
n
, |/;(0)| : = ^ ( p
n
, f' nm{dldz)) . - (X) .
We may assum e tha t /
n
i s holomorphic i n a slightly large r disc .
Define H
n
: A H R + ,H
n
(z) - |f^(z)|(l-|z 2|). The n there exists z
n
G A, #
n
(z
n
) =
maxA ff
n
. Le t g
n
: A H- M, #
n
= /
n
o 0n(iu) wher e 0
n
(w) = ^ " ^ . The n
K H K i -
h2|)
= |/;(z)iKMi( i
-1^2|)
= |/;(z)|( i
-|*2|)
(the las t equalit y follow s b y a short computation) .
So now , |#^(w) | fa^2i Rescal e th e disc : Le t R
n
: = |^(0) | (not e tha t
|7n(0)|•— oo , n i— oo) an d defin e fc
n
: A(0, Rn)i— M, fc
n
(^) 9n(z/Rn)- The n
say o n A(0 , Rn/2). Als o |fc^(0) | = 1, s o usin g a norma l familie s argumen t w e ca n
find a holomorphic ma p / : C •—• M suc h tha t |/'(0) | 1, hence / i s non constant .
Next w e mentio n a fe w fact s whic h follo w directl y fro m th e definitio n o f th e
Kobayashi metric .
THEOREM 2.2 . Holomorphic maps are distance decreasing, i.e., if F : M —»
N is a holomorphic map, then
ds(F(p),Fi(t))ds{p,t).
Hence invertible maps are isometries. Covering maps are also isometries. Bounded
open set in C
n
are Kobayashi hyperbolic
Developing th e techniqu e use d i n th e proo f o f Brody' s Theorem , w e obtai n
higher dimensiona l generalization s o f th e fac t tha t P
x
\ thre e point s i s Kobayash i
hyperbolic, whic h i s another wa y t o stat e Montel' s Theorem .
Green ([Grl] ) an d ([Gr2] ) prove d th e followin g theorem .
THEOREM 2.3 . LetXi,..,X
m
be compact complex hypersurfaces in P n . Then
P n \ ((J - Xj) is Kobayashi hyperbolic if
(i) there is no non-constant holomorphic map from C to P
n
\ (|J . Xj) ,
(ii) there is no non-constant holomorphic map from C to (X^ D D Xik) \
(Xjl U ' - U X j , ) for any {zi,.. . ,ijfe,ji, - - - ,iz} = {l,...,ra} .
Assume that the image of a holomorphic map f : C P does not intersect any
of k + 2 different complex hypersurf aces. Then /(C ) is already contained in some
compact complex hypersurf ace.
A holomorphic map s / o n P
k
ca n alway s be lifte d t o a homogeneous holomor -
phic polynomia l F o n C
fc+1:
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