27r 7r
On the other hand one deduces easily from (5.19) that
Tf(r; D)=O(re).
Hence,ms (7-; D)
(f(f; D) = ~--,~li---mm ~ ~-~ > 0. []
We will now give an explicit example with S t ( D ) = { 0 } .
H O L O M O R P H I C C U R V E S I N T O S E M I - A B E L I A N V A R I E T I E S 155 Example 5.22. Let A be the semi-Abelian variety A = C * x C * , compactified by p I ( C ) x P I ( C ) with a pair of homogeneous coordinates, ([x0:xx], [Y0:Yl]). For a pair of natural numbers (m, n) with r e < n , let D be the divisor given by
D={([g0:xl], [Y0:yl]):
y~jxl§ .... y~Zxo§Set D = D n A . Note that St(D)={0}. Moreover, D violates condition 4.11, since D ~ ([1:0], [1:0]). Let e be a positive irrational real number such that
0 < c m < 1 < cn. (5.23)
Let f : C - + A be the holomorphic curve given by f : z ~-+ ([1: eZ], [1: ec~]).
Let fti, i = 1 , 2, be the Fubini Study metric forms of the two factors of
(pI(c))2.
Then c l ( D ) = f t l + n f ~ 2 . By an easy computation one obtainsTf(I';
e l ( D ) ) =l§247 (5.24)
?r
Thus, L)f=l,
and the image f ( C ) is Zariski dense in A, because c is irrational.We compute N ( r ; f ' D ) as follows. Note the following identity for divisors on C:
f * D = (eZ + e . . . . +e'~Z)o.
(5.25)
We consider a holomorphic curve g in p 2 ( C ) with the homogeneous coordinate system [wo : wl: w2] defined by
9: z E C --+ [e~: e . . . . : e ncz] E p2(C).
By computing the Wronskian of e z, e mcz and e ncz one sees that they are linearly inde- pendent over C; i.e., g is linearly non-degenerate. Let Tg(r) be the order function of g with respect to the Fubini Study metric form on P2(C). It follows that
1
f{
l o g ( l e Z l 2 + l e ~ . ~ , 1 2 § e ...12)d0+0(1)
%(,-)
= W ~z~=~}1 f { log(l+[e(mC_l)~[2+[e(n~_l)z[2)dO+O(1).
47r Izl=r}
(5.26)
If R e z ) 0 (resp. ~<0), then
le(mc-1)zl~<l
(resp. ~>1) and [e(~C-1)zl~>l (resp. ~<1). There- fore, if z = r e w and Rez>~0,log(l+ le(m=-l>12+le(~-l>12 ) =21og § le(~=-X)zl+O(1 ) =2(ne-1)rcosO+O(1).
If z=re i~ and Rez~<0,
l o g ( l + le (m~-l)z 12 + le (n~-a)z]2) = 2 log + le ( . . . . 1)z]+o(1) = 2 ( m c - 1) r cos 0+O(1).
Combining these with (5.26), we have
T A r ) - _ _ (5.27)
7f
We consider the following four lines Hi, l~<j~<4, of P2(C) in general position:
Hj = {wj_l = 0 } , l~<j~<3, H4 = { w 0 + w l - ~ w 2 = 0}.
Noting that 9 is linearly non-degenerate and has a finite order (in fact, o g = l ) , we infer from Cartan's S.M.T. [Ca] that
4
Tg(r) <<. E N2(r; 9*Hj)+O(log r). (5.28)
j = l
Since N2(r;g*Hj)=O, 1<~j~<3, we deduce from (5.28), (5.27) and (2.1) that N(r; g'H4) = ( n - m ) c r + O ( l o g r).
7r
By (5.25), N(r; g*H4)=N(r; f ' D ) , and so N(r; f ' D ) -- - - It follows from (5.24) and (5.29) that
( n - m ) c
r+O(logr). (5.29)
By elementary calculations one shows that ordz f*D>~2 implies ( m c - 1)(e~Z)"~ + ( n c - 1) (e~) n = O.
Furthermore, f(z) C D if and only if e z + e mcz +e ncz =0. Combined, these two relations imply that there is a finite subset S c C 2 such that ordz f*D>~2 implies (e z, e Cz) ES. Since z~-+(e z, e cz) is injective, it follows that {z :ordz f*D~>2} is a finite set. Therefore,
N1 (r, f ' D ) = N(r, frO) +O(log r),
l + m c (5.31)
5 1 ( f ; D ) = 6 ( f ; D ) - l + n c "
5(f;
D) = l +mc.(5.30)
l +nc
H O L O M O R P H I C C U R V E S I N T O S E M I - A B E L I A N V A R I E T I E S 157 Let c ' > 1 be an irrational number, and set
c=l/c', m=[c'], n=[c']+l,
where [c'] denotes the integral part of
c'.
T h e n m, n and c satisfy (5.23), and by (5.30) a ( f ; D ) = l + [ c ' ] / c ' ~ 1 (c' --+ oo).l+([c']+l)/c'
Thus (~(f; D ) ( = ( ~ l ( f ; D) by (5.31)) takes values arbitrarily close to 1.
Remark
5.32. In [Noh], the first author proved t h a t for D without condition 4.11 a holomorphic curve f : C--+A, omitting D, has no Zariski dense image, and is contained in a translate of a semi-Abelian subvariety which has no intersection with D. W h a t was proved in [Noh] applied to f: C - + A with Zariski dense image yields that there is a positive constant ~ such thatx T / ( r ; cl (D)) ~< N1 (r;
f'D) + S/(r;
cl (D)), (5.33) provided that S t ( D ) = {0}. T h e above ~ may be, in general, very small because of the m e t h o d of the proof. One needs more detailed properties ofJk(D)
to get the best bound such as in the Main Theorem than to get (5.33); this is the reason why we need boundary condition 4.11 for D.Remark
5.34. In [SY2] Siu and Yeung claimed (1.2) for Abelian A of dimension n.The most essential part of their proof was L e m m a 2 of [SY2], but the claimed assertion does not hold. T h e cause of the trouble is due to the application of the semi-continuity theorem to a non-fiat family of coherent ideal sheaves. But, it is a bit delicate, and so we give a counter-example to their lemma.
Let f : C--+A be a one-parameter subgroup with Zariski dense image. Let D be an ample divisor on A containing 0EA such that f ( C ) is tangent highly enough to D at 0 so that
Jkf(O) E Jk (D),
but f ( C ) g D. Let m0 be the maximal ideal sheaf of the structure sheaf OA at 0. Since Wk consists of only one point, we can identify A x Wk with A. Then,OEJk(D)A(AxWk).
LetZk=Z(Jk(D)A(AxWk))
be the ideal sheaf ofJk(D)A(AxWk).
T h e n :/:k Cm0. If the claimed L e m m a 2 of [SY2] were correct, it should follow that
H~ (9((L(D))~)| D H~
(9((L(D))~)| q) • {0} for all q >~ 1.Thus we would obtain that dim H~ O ( ( L ( D ) ) ~ ) ) = o c ; this is a contradiction.
Remark
5.35. It is an interesting problem to see if the truncation level k0 of the counting functionNko(r; f'D)
in the Main Theorem can be taken as a function onlyin dim A. By the above proof, it would be sufficient to find a natural number k such t h a t
~ r 2 ( J k ( D ; l o g A A O A ) ) A W k # W k . Note t h a t
dim 7r2 (dk (D; log 4 ) ) ~< dim dk (D; log A) = ( n - 1)(k + 1).
Thus, if d i m W k > ( n - 1 ) ( k + l ) we m a y set ko=k. For example, if J ~ ( f ) ( C ) is Zariski dense in Jn(A), then d i m W n = n 2. Since dimTr2(Jn(D;logOA))=n2-1, we m a y set ko=n. In general, it is impossible to choose ko depending only on n because of the following example.
Example 5.36. Let E = C / ( Z + i Z ) be an elliptic curve, and let D be an irreducible divisor on E 2 with cusp of order N at 0 E E 2. Let f: zEC--+(z, z2)EE 2. T h e n f ( C ) is Zariski dense in E 2, and
Tl(r, L(D)) ~ r 4 ( l + o ( 1 ) ) .
Note t h a t f - 1 (0) = Z + iZ and f*D ~> N (Z + iZ). For an a r b i t r a r y fixed k0, we take N > k0, and then have
g ( r , f*D)-Nko(r, f ' D ) >. ( g - k o ) r 2 ( l +o(1) ).
T h e above left-hand side cannot be bounded by Sy(r, Cl (D))=O(logr). This gives also a counter-example to [Krl, L e m m a 4].
6. Applications
Let the notation be as in the previous section. Here we assume t h a t A is an Abelian variety and D is reduced and hyperbolic; in this special case, D is hyperbolic if and only if D contains no translate of a o n e - p a r a m e t e r subgroup of A. Cf. [NO], [L] and [Ks] for the theory of hyperbolic complex spaces.
THEOREM 6.1. Let DC A be hyperbolic and let do be the highest order of tangency of D with translates of one-parameter subgroups. Let 7r: X--+ A be a finite covering space such that its ramification locus contains D and the ramification order over D is greater than do. Then X is hyperbolic.
Proof. By B r o d y ' s t h e o r e m (cf. [Br], [NO, T h e o r e m (1.7.3)1) it suffices to show t h a t there is no non-constant holomorphic curve g: C ~ X such t h a t the length [[g'(z)11 of the derivative g'(z) of g(z) with respect to an arbitrarily fixed Finsler metric on X is bounded.
Set f(z)=Tr(g(z)). T h e n the length IIf'(z)ll with respect to the flat metric is bounded, too, and hence if(z) is constant. Thus, f(z) is a translate of a o n e - p a r a m e t e r subgroup.
By definition we m a y take k 0 = d o + l in (5.12). Take d (>do) so t h a t X ramifies over D
H O L O M O R P H I C C U R V E S I N T O S E M I - A B E L I A N V A R I E T I E S 159 with order at least d. Then we have t h a t Nl(r; f*D)<.(1/(d+l))N(r; f ' D ) . Hence it follows from the Main Theorem t h a t
T I (r; L (D)) = Ndo+l (r, f ' D ) + O(log r) ~< (do + 1) N1 (r; f ' D ) + 0 (log r)
do+ 1~ , do +1 Tl(r; L(D))+O(log r).
<~ - ~ l v (r; f*D)+O(logr) <<. d + l
Since Tf(r; L(D))>~cor 2 with a constant e0>0, d<~do; this is a contradiction. []
Remark. In the special case of dim X = d i m A = 2 , C.G. Grant [Gr] proved that if X is of general type and X - + A is a finite (ramified) covering space, then X is hyperbolic.
When dim X = d i m A=2, D is an algebraic curve, and hence the situation is much simpler than the higher dimensional case.
THEOREM 6.2. Let f: C--+ A be a 1-parameter analytic subgroup in A with a = f f ( 0 ) . Let D be an effective divisor on A with the Riemann form H ( . , . ) such that D ~ f ( C ) .
Then we have
N(r; f ' D ) = H(a, a)~r 2 +O(log r).
Proof. Taking the Zariski closure of f ( C ) , we may assume that f ( C ) is Zariski dense in A. Note that the first Chern class cl(L(D)) is represented by iOOH(w, w). It follows from (2.1) and Lemma 5.1 that
N (r; f ' D ) = T/(r; L( D ) ) +O(log r) r dt
= fo -t f~,(t) iH(a'a) dzAdz +O(l~ = H(a'a)~rr2 +O(l~
[]Remark 6.3.
estimate
which is equivalent to
In the case where f ( C ) is Zariski dense in A, Ax ([Ax]) proved the 0 < lim
n(r,f*D) ~ ~ n(r,f*D)
r 2 r--+oc r 2
< (:x:),
0 < lim N ( r , f * D ) ~ ~ N ( r , f * D ) <co.
r 2 r - ~ c c r 2
[Ah]
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