Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

15 Pages·2000·0.16 MB·English

Compositio Mathematica123: 209^223, 2000. 209
# 2000 Kluwer Academic Publishers. Printed inthe Netherlands.
Very Ampleness of Line Bundles and Canonical
Embedding of Coverings of Manifolds
SAI-KEE YEUNG*
Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A.
e-mail: [email protected]
(Received: 2 December1998; in ¢nal form: 20 April1999)
Abstract. LetLbeanamplelinebundleonaKa«hlermanifoldsofnonpositivesectionalcurvature
withKasthecanonicallinebundle.WegiveanestimateofmsuchthatK(cid:135)mLisveryamplein
termsoftheinjectivityradius.This impliesthat mcanbe chosenarbitrarilysmalloncewego
deep enough into atowerofcoveringofthe manifold.The same argumentgives an effective
Kodaira EmbeddingTheorem for compact Ka«hler manifolds in terms ofsectional curvature
and the injectivity radius. In case oflocally Hermitian symmetric space of noncompacttype
orifthesectionalcurvatureisstrictlynegative,weprovethatK itselfisveryampleonalarge
coveringofthemanifold.
Mathematics Subject Classi¢cations (2000). Primary14E25, 32J27, 32Q05,32Q40.
Key words: very ampleness, canonical embedding.
LetLbeanamplelinebundleonanalgebraicmanifoldM.LetK bethecanonical
line bundle of the manifold. It follows by de¢nition that there is a constantmsuch
that K (cid:135)mL is ample. It is natural to ask for the smallest value of such m. In case
that the line bundle is the canonical line bundle, the question is about the smallest
k such that kK is very ample. Let M be a nonpositively curved algebraic manifold
with a pro¢nite fundamental group so that there is a tower of coverings over M
corresponding to normal subgroups of ¢nite index. In this paper, we show that
for suf¢ciently large covering manifold in the tower, m can be taken to be 1. In
the particular case of a suf¢ciently large covering of the Hermitian symmetric
manifoldofnonpositivecurvature,weshowthatactuallyk(cid:136)1,thatis,thecanonical
linebundleK itselfisampleforacoveringmanifoldofasuf¢cientlylargecovering
index. The same conclusion holds for similar examples of Ka«hler manifolds with
negative Riemannian sectional curvature. An effective version of the Kodaira
embeddingtheoremwhichgivesanestimateofkormintermsofcurvaturebounds
and the injectivity radius of a general manifold is also obtained.
Here are the main results of this article.
*The authorwaspartially supported bygrantsfrom the National Science Foundation.
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210 SAI-KEEYEUNG
THEOREM 1. Let M be a complex manifold of complex dimension n. Suppose the
sectional curvature R of M satis¢es (cid:255)a2WRW0. Let the curvature R of the
L
holomorphic line bundle Lsatisfy c<R : Let the injectivity radius of the manifold
L
M beboundedfrombelowbyt:Then,K (cid:135)mLisveryampleformX2nd ;where
t;a;c
(cid:20) (cid:16) (cid:17)(cid:21)
1 16(cid:135)16log2 4log2 at
d (cid:136) (cid:135) acoth : (cid:133)1(cid:134)
t;a;c c t2 t 2
Moreover, K (cid:135)mL generates the Nth order jet of M at any point of M for
mX(cid:133)(cid:133)N=2(cid:134)(cid:135)n(cid:134)d :
t;a;c
Remarks.(1) We remark that d !0 as t!1:
t;a;c
(2)(EffectiveKodairaEmbeddingTheorem)Ifwerelaxthecurvatureconditionto
(cid:255)a2WRWb2; where b>0; the conclusion is that K (cid:135)mL is very ample for
mX2nd : Letting t (cid:136)min(cid:133)t;p=2b(cid:134), d is estimated by
t;a;b;c o t;a;b;c
(cid:20) (cid:16) (cid:17) (cid:21)
1 16(cid:135)16log2 4log2 at bt cot(cid:133)bt (cid:134)(cid:255)1
d (cid:136) (cid:135) acoth o (cid:255) o o ; (cid:133)2(cid:134)
t;a;b;c c t2 t 2 t2
o o o
Moreover, K (cid:135)mL generates the nth order jet of M at every point of M for
mX(cid:133)(cid:133)N=2(cid:134)(cid:135)n(cid:134)d : The number t is used instead of t so that cot(cid:133)bt (cid:134) will
t;a;b;c o o
not become too negative. This is an effective version of Kodaira embedding
Theorem.
AssomeapplicationsofTheorem1,weassumethatthefundamentalgroupp (cid:133)M(cid:134)
1
of M is pro¢nite in the sense that there exists a sequence of normal subgroups G
i
satisfying G <G, G (cid:136)p (cid:133)M(cid:134) and \1 G (cid:136);. Let M~ be the universal covering
i(cid:135)1 i 0 1 i(cid:136)0 i
of M. Then M (cid:136)M~ =G is a covering space of M with the covering map denoted
i i
by p:M !M: As \1 G (cid:136);, we conclude that the injectivity radius of M tends
i i i(cid:136)0 i i
to 1asi!1duetothediscretenessofp (cid:133)M(cid:134):WecallfMgatowerofcoverings
1 i
for M with the injectivity radius increasing to 1: The following is an immediate
corollary of Theorem 1:
THEOREM 2. Let M be a nonpositively curved algebraic manifold with pro¢nite
fundamentalgroup. Thereexists i suchthatforalliXi ;K (cid:135)p(cid:3)Lisvery ample.
o o Mi i
In fact, there is i such that K (cid:135)p(cid:3)L generate the jth jet space of M for iXi:
j Mi i j
Furthermore, the same is true for K (cid:135)ep(cid:3)Lfor any small rational e>0such that
Mi i
ep(cid:3)L is a line bundle on M:
i i
Since the fundamental group of Hermitian symmetric manifolds of noncompact
typearediscretesubgroupsofgenerallineargroups,theyhavetobepro¢nite.Hence,
theaboveconclusionisreadilyapplicableinthiscase.However,thisissupersededby
the following theorem:
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AMPLENESSOFLINE BUNDLES 211
THEOREM3.LetfMgbeatowerofcoveringofHermitiansymmetricmanifoldsof
j
noncompact type. There exists a constant i X0 such that K is very ample for
0 M
j
jXi : Moreover, given any l >0; there exists i X0 such that K generates the
0 l M
j
kth jet Jk(cid:133)M(cid:134) of M for jXi:
j j l
SimilarstatementsforKa«hlermanifoldswithsectionalcurvaturepinchedbetween
two negative numbers are also true.
THEOREM 4.LetfMgbea towerofcoveringoveraKa«hlermanifoldM withsec-
j
tionalcurvatureRsatisfying(cid:255)a2 <R<(cid:255)b2 <0:Thereexistsaconstanti X0such
0
thatK isveryampleforjXi :Moreover,givenanyl >0;thereexistsi X0such
M 0 l
j
that K generates the kth jet Jk(cid:133)M(cid:134) of M for jXi:
M j j l
j
Followingfromthede¢nitionoftheSeshadriconstantforalinebundle,whichwill
be explained in Section 1, we get the following conclusion:
COROLLARY1.FortheexamplesinTheorems3and4,theSeshadriconstantforthe
canonical line bundle is at least 1.
Theorganizationofthearticleisasfollows.InSection1,we¢rstuseL2estimates
toshowthatthevalueofm,sothatK (cid:135)mLisample,canbeeffectivelyestimatedby
the injectivity of the manifold and the curvature form of L. In this way, we also
estimate the Seshadri constant of the line bundle. Then we apply the results to a
towerofcoveringsofpro¢nitenonpositivelycurvedmanifoldstogettheresultthat
K (cid:135)L is very ample after one goes deep enough into the covering space. In par-
ticular, this includes the class of Hermitian symmetric manifolds of noncompact
type. In Section 2, we relate the L2 geometry of the universal covering to conclude
that K is actually very ample for the covering of a suf¢ciently large covering index
for the manifolds stated in Theorems 3 and 4.
1. Some Criteria forVeryAmpleness of Line Bundles on General Manifolds
The main tool is the following L2-estimates due to Ho«rmander [Ho].
LEMMA1.LetMbeacompactKa«hlermanifoldwithaKa«hlermetricoandletK be
M
the canonical line bundle. Let j be a function on M. Let (cid:133)L;h(cid:134) be a Hermitian line
bundle on M. Assume that
p(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)
c (cid:133)L;h(cid:134)(cid:135) (cid:255)1@@(cid:22)j(cid:255)c (cid:133)K (cid:134)>co:
1 1 M
R
Letgbea@(cid:22)-closedL-valued(cid:133)0;1(cid:134)-formonMwith kgk2e(cid:255)j <0.Thentheequation
M h
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212 SAI-KEEYEUNG
@(cid:22)f (cid:136)g has a solution satisfying the L2- estimate
Z Z
kgk2e(cid:255)j
kfk e(cid:255)j < h :
h c
M M
WealsoneedthefollowingHessiancomparisontheoremasstatedin[G-W],p.19:
LEMMA 2.Let (cid:133)M ;o (cid:134)and (cid:133)M ;o (cid:134)be Riemannian manifolds with poles at o ;o
1 1 2 2 1 2
and of equal dimension. Suppose that the radial curvature of a point on a normal
geodesic g on M starting from o is at least the radial curvature of the point on
1 1 1
a corresponding normal geodesic g on M : Then for every increasing function f,
2 2
the following Hessian comparison is valid.
D2f(cid:133)r (cid:134)(cid:133)g (cid:133)t(cid:134)(cid:134)WD2f(cid:133)r (cid:134)(cid:133)g (cid:133)t(cid:134)(cid:134):
1 1 2 2
ProofofTheorem1.Weneedtoconsiderthelowerboundoftheeigenvaluesofthe
complex Hessian Lf(cid:133)X;Y(cid:134)(cid:136)D2f(cid:133)X;Y(cid:134)(cid:135)D2f(cid:133)JX;JY(cid:134), where J is the complex
structure involved. Since the injectivity radius of M is at least t; we can place a
geodesic ball B(cid:133)x;t(cid:134) of radius t centered at each point x of M within which there
isnocutlocusorconjugatelocus.Fora¢xede>0,letw(cid:133)t(cid:134)beaC1 bumpingfunc-
tion de¢ned on the interval (cid:137)0;1(cid:134), satisfying
t
w(cid:133)t(cid:134)(cid:136)1;tW ; w(cid:133)t(cid:134)(cid:136)0;tXt;
2
2(cid:135)e 4(cid:133)2(cid:135)e(cid:134)
(cid:255) Ww0(cid:133)t(cid:134)W0; jw00(cid:133)t(cid:134)jW :
t t2
Thenw(cid:133)t(cid:134)isadecreasingfunction,withsupportin(cid:137)0;t(cid:138):Thefunctionwcanbecon-
structed as follows. Construct a step function s(cid:133)t(cid:134),
(cid:18) (cid:19) (cid:18) (cid:19)
8 t 3t 8 3t
s(cid:133)t(cid:134)(cid:136)(cid:255) for t2 ; ; s(cid:133)t(cid:134)(cid:136) for t2 ;t
t2 2 4 t2 4
and s(cid:133)t(cid:134)(cid:136)0 outside the range. Let s (cid:133)t(cid:134) be the integral of s(cid:133)t(cid:134)with initial condition
1
s (cid:133)0(cid:134)(cid:136)0;ands (cid:133)t(cid:134)betheintegralofs (cid:133)t(cid:134)withs (cid:133)0(cid:134)(cid:136)1.Smoothings(cid:133)t(cid:134),theresulting
1 2 1 2
s (cid:133)t(cid:134)givesacandidateforw.Letr (cid:133)y(cid:134)bethedistanceofyfromxwithrespecttothe
2 x
Ka«hler metric. De¢ne a function c on M by c (cid:136)(cid:133)log(cid:133)4r2=t2(cid:134)(cid:134)w(cid:14)r : c is
x x x x x
supported only on B(cid:133)x;t(cid:134). For simplicity of notations, we will suppress x in the
formula below. Note that
D2f(cid:133)r(cid:134)(cid:133)X;Y(cid:134)(cid:136)f00(cid:133)r(cid:134)dr(cid:133)X(cid:134)(cid:10)dr(cid:133)Y(cid:134)(cid:135)f0(cid:133)r(cid:134)D2r(cid:133)X;Y(cid:134)
fortwovectorsX;Y onM:LetM bethespaceformofconstantRiemanniansec-
m
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AMPLENESSOFLINE BUNDLES 213
tional curvature m: We let X0 be the vector on M corresponding to vectors X:
m
(cid:20) (cid:18) (cid:19) (cid:21)
4r2
D2c(cid:133)X;X(cid:134)(cid:136)D2 log x w(cid:14)r (cid:133)X;X(cid:134)
t2 x
(cid:136)w(cid:14)r D2logr2(cid:133)X;X(cid:134)(cid:135)2Dlogr2D(cid:133)w(cid:14)r (cid:134)(cid:133)X;X(cid:134)(cid:135)
x x
(cid:18) (cid:19)
4r2
(cid:135)log x D(cid:133)(cid:133)w(cid:14)r (cid:134)(cid:133)X;X(cid:134)(cid:134)
t2 x
Applying Lemma 2 by comparing it with the £at space M and denoting the
0
restriction of g to the geodesic sphere perpendicular to the radial direction by h,
we get
Llogr2(cid:133)X;X(cid:134)(cid:136)D2logr2(cid:133)X;X(cid:134)(cid:135)D2logr2(cid:133)JX;JX(cid:134)
XD2logr2 (cid:133)X0;X0(cid:134)(cid:135)D2logr2 (cid:133)(cid:133)JX(cid:134)0;(cid:133)JX(cid:134)0(cid:134)
M M
(cid:255)b 0
2 2
X (cid:255) dr(cid:10)dr(cid:133)X;X(cid:134)(cid:255) dr(cid:10)dr(cid:133)JX;JX(cid:134)(cid:135) (cid:133)3(cid:134)
r2 r2
2 2
(cid:135) h(cid:133)X;X(cid:134)(cid:135) h(cid:133)JX;JX(cid:134)
r2 r2
X0:
ComparingwithM andusingthefactthattXr Xt=2intheregionwherew0 6(cid:136)0;
(cid:255)a x
we have
(cid:18) (cid:19) (cid:18) (cid:19)
4r2 4r2
log x D2(cid:133)w(cid:14)r(cid:134)(cid:133)X;X(cid:134)X log x D2(cid:133)w(cid:14)r (cid:134)(cid:133)X0;X0(cid:134)
t2 t2 M(cid:255)a
(cid:18) (cid:19)
4r2
(cid:136)log x w00(cid:133)r (cid:134)dr (cid:10)dr (cid:133)X0;X0(cid:134)(cid:135)
t2 M(cid:255)a M(cid:255)a M(cid:255)a
(cid:18) (cid:19)
4r2
(cid:135)log x w0(cid:133)r (cid:134)D2r (cid:133)X0;X0(cid:134)
t2 M(cid:255)a M(cid:255)a
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214 SAI-KEEYEUNG
(cid:18) (cid:19)
4r2
X log x w00(cid:133)r (cid:134)dr (cid:10)dr (cid:133)X0;X0(cid:134)(cid:255)
t2 M(cid:255)a M(cid:255)a M(cid:255)a
(cid:18) (cid:19)
4r2 2(cid:135)e
(cid:255)log x acoth(cid:133)ar(cid:134)g(cid:133)X;X(cid:134)
t2 t
4(cid:133)2(cid:135)e(cid:134)
X (cid:255)log4 dr(cid:10)dr(cid:133)X;X(cid:134)(cid:255)
t2
(cid:16) (cid:17)
2(cid:135)e at
(cid:255)log4 acoth g(cid:133)X;X(cid:134)
t 2
and
Dr
Dlogr(cid:10)D(cid:133)w(cid:14)r(cid:134)(cid:133)X;X(cid:134)(cid:136) (cid:10)D(cid:133)w(cid:14)r(cid:134)(cid:133)X;X(cid:134)
r
1
(cid:136) w0Dr(cid:10)Dr(cid:133)X;X(cid:134)j
r rXt2
2(cid:133)2(cid:135)e(cid:134)
X (cid:255) dr(cid:10)dr(cid:133)X;X(cid:134):
t2
Combining the above inequalities, we get
Lc (cid:133)X;X(cid:134)
M;x
4(cid:133)2(cid:135)e(cid:134)
X (cid:255)log4 (cid:137)dr(cid:10)dr(cid:133)X;X(cid:134)(cid:135)dr(cid:10)dr(cid:133)JX;JX(cid:134)(cid:138)(cid:255)
t2
(cid:16) (cid:17)
at 8(cid:133)2(cid:135)e(cid:134)
(cid:255)log42(cid:135)etacoth (cid:137)g(cid:133)X;X(cid:134)(cid:135)g(cid:133)JX;JX(cid:134)(cid:138)(cid:255) dr(cid:10)dr(cid:133)X;X(cid:134)
2 t2
(cid:20) (cid:16) (cid:17)(cid:21)
2(cid:135)e 2(cid:135)e at
X (cid:255) (cid:133)8(cid:135)4log4(cid:134) (cid:135)log4 acoth (cid:133)g(cid:133)X;X(cid:134)(cid:135)g(cid:133)JX;JX(cid:134)(cid:134):
t2 t 2
We can now apply the L2-estimates to construct sections which separate points
and generate the ¢rst jet of the tangent bundle. Let x;y be arbitrary points on
M. The functions c and c , as constructed above, are supported in B(cid:133)x;t(cid:134) and
x y
B(cid:133)y;t(cid:134), respectively. Note that for r(cid:133)x;w(cid:134) suf¢ciently small, r(cid:133)x;w(cid:134)2 is
jx(cid:255)wj2(cid:133)1(cid:135)O(cid:133)jx(cid:255)wj(cid:134); where O(cid:133)jx(cid:255)wj(cid:134) is a bounded term tending to 0 as w
approachesx:Hence,sodoes c . Letj(cid:136)n(cid:133)c (cid:135)c (cid:134):As histheHermitianmetric
x x y
for L and h (cid:136)detg(cid:255)1 is the metric for K , heh e(cid:255)j is a metric for K (cid:135)mL: It
1 M 1 M
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AMPLENESSOFLINE BUNDLES 215
follows from our choice of j that
p(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)(cid:129)
mc (cid:133)L;h(cid:134)(cid:135) (cid:255)1@@(cid:22)j(cid:135)c (cid:133)K (cid:134)(cid:255)c (cid:133)K (cid:134)>e o
1 1 M 1 M 1
with some positive e provided that the following inequality is satis¢ed:
1
(cid:20) (cid:16) (cid:17)(cid:21)
2(cid:135)e 2(cid:135)e at
e (cid:136)mc(cid:255)2n (cid:133)8(cid:135)4log4(cid:134) (cid:135)log4 acoth >0: (cid:133)4(cid:134)
1 t2 t 2
Let l(cid:136)min(cid:133)t;1r(cid:133)x;y(cid:134)(cid:134). The line bundle K (cid:135)mL is trivial on the support of
2 M
B(cid:133)x;l(cid:134) which is the ball of radius l centered at x: Let s be the canonical section
of the bundle (cid:133)K (cid:135)mL(cid:134)j : Consider now
M B(cid:133)x;l(cid:134)
(cid:18) (cid:19)
r(cid:133)x;w(cid:134)
z(cid:133)w(cid:134)(cid:136)w s(cid:133)w(cid:134)
1=2r(cid:133)x;y(cid:134)
as a C1 section of K (cid:135)mL, which is 1 in a small neighbourhood of x and 0 in a
M
small neighbourhood of y: @(cid:22)z is an integrable @(cid:22)-closed K (cid:135)eL-valued 1-form,
M
as@(cid:22)ziszeroaroundxandy:Hence,fromL2-estimatesasstatedinLemma1,there
is a solution of @(cid:22)f (cid:136)@(cid:22)z satisfying
Z Z
k@(cid:22)zk2e(cid:255)j
kfk2e(cid:255)j < h <1:
h e
M M 1
Fromthepoleorderofjatxandy;weconcludethatf(cid:133)x(cid:134)(cid:136)f(cid:133)y(cid:134)(cid:136)0:Hence,z(cid:255)f is
a holomorphic section of K (cid:135)mL which is 1 at x and 0 at y:
M
To prove that sections of K (cid:135)mL generate 1-jet at any x2M, let w be a
M 1
bumping function as w supported in a normal coordinate chart of x so that
z(cid:136)0 corresponds to x. Let z(cid:133)z(cid:134)(cid:136)zw (cid:133)z(cid:134)s(cid:133)z(cid:134) and extend by 0 so that z is a
i i 1 i
well-de¢nedC1 sectionofK (cid:135)mLonM.Letj(cid:136)(cid:133)n(cid:135)1(cid:134)c :Thenthesameargu-
M 2 x
ment as above shows that we can solve @(cid:22)f (cid:136)@(cid:22)z with f vanishing to order 2 at x
i
corresponding to our choice of (cid:133)n(cid:135)1(cid:134)c in j once the inequality (4) is satis¢ed.
2 x
Hence, z (cid:255)f is a holomorphic section of K (cid:135)mL satisfying @=@z(cid:133)z (cid:255)f(cid:134)(cid:133)x(cid:134)
i M i i
(cid:136)1: As z can be an arbitrary holomorphic coordinate function at x, this shows
i
that the sections of K (cid:135)mL generates the 1-jet and, hence, together with earlier
M
discussions the very ampleness of the line bundle if Equation (4) is satis¢ed. Note
that we can always ¢nd an e satisfying Equation (4) once equation (1) is true.
For the generation in the Nth order jet, it suf¢ces for us to consider
z (cid:133)z(cid:134)(cid:136)z (cid:1)(cid:1)(cid:1)z w (cid:133)z(cid:134)s(cid:133)z(cid:134) instead of z(cid:133)z(cid:134) in the earlier argument. This concludes
i1(cid:1)(cid:1)(cid:1)in i1 in 1 i
the proof of Theorem 1. &
ProofofRemarktoTheorem1.Weuset (cid:136)min(cid:133)t;p=2b(cid:134)insteadoftintheproof
o
of Theorem 1. The only modi¢cation to this case is the estimates for
L2logr2(cid:133)X;X(cid:134)inEquation(3).InsteadofcomparingwithM ;weneedtocompare
0
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216 SAI-KEEYEUNG
with M : Instead of Equation (3), we get the new estimate
b
L2logr2(cid:133)X;X(cid:134)
2
X (cid:255) (cid:137)dr(cid:10)dr(cid:133)X;X(cid:134)(cid:135)dr(cid:133)JX;JX(cid:134)(cid:138)(cid:135)
r2 (cid:133)5(cid:134)
2
(cid:135) bcot(cid:133)br(cid:134)(cid:137)h(cid:133)X;X(cid:134)(cid:135)h(cid:133)JX;JX(cid:134)(cid:138):
r
NotethatifX isaradialtangentvector,JX istangentialtothegeodesicsphere.We
now observe that (cid:133)(cid:133)rbcot(cid:133)br(cid:134)(cid:255)1(cid:134)=r2(cid:134)0W0 and, hence, the minimum of
(cid:133)rbcot(cid:133)br(cid:134)(cid:255)1(cid:134)=r2 is achieved at t : This concludes the proof of Remark 1. &
o
Asadetour,weconsidertheSeshadriconstantofalinebundleLwhichisde¢ned
as follows (cf [De]): For each x2M; let p:X~ !X be the blow-up of X at x
and E be the exceptional divisor. Let
s(cid:133)L;x(cid:134)(cid:136)supfeX0jp(cid:3)L(cid:255)eE is nefg
L(cid:1)C
(cid:136) inf
C2x n(cid:133)C;x(cid:134)
where n(cid:133)C;x(cid:134) is the multiplicity of C at x and the in¢mum is taken over all curves
passing through x. The relation between the Seshadri constant and very ampleness
is related by the following Lemma, which follows immediately from the de¢nition
of very ampleness (cf. [De], p.68).
LEMMA 3. Suppose mL is very ample. Then s(cid:133)L(cid:134)X 1:
m
As a corollary, we get
COROLLARY 2. AssumethatM isanalgebraicmanifoldwithsectionalcurvature
satisfying(cid:255)aWKW0andthecurvatureofLisatleastcwithrespecttotheKa«hler
metric.ThentheSeshadriconstantofanamplelinebundleLisboundedfrombelow
by c=(cid:133)2nd (cid:135)na2(cid:134); where d is the function considered in Theorem 1.
t;a;c t;a;c
This follows immediately from Theorem 1, Lemma 3 and the estimates
(cid:16) (cid:17)
na
(cid:135)m c (cid:133)L(cid:134)Xc (cid:133)K(cid:134)(cid:135)mc (cid:133)L(cid:134):
c 1 1 1
2. VeryAmpleness of Canonical LineBundlesin SomeHermitian Symmetric
Manifolds and Negatively Curved Manifolds
In the following, we ¢rst assume that M is a Hermitian symmetric manifold of
noncompacttypeandgiveaproofofTheorem3.Lateronwewillmodifytheproof
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AMPLENESSOFLINE BUNDLES 217
tohandlethecaseofnegativelycurvedKa«hlermanifolds.fMgisatowerofcovering
j
over M (cid:136)M.
0
Before we go to the proof of Theorem 3, we need some preliminaries. On a
compact manifold M, the space of L2 sections of K is ¢nite-dimensional. Let
M
s;i(cid:136)1;...;nbeanorthonormalbasiswithrespecttotheHermitianinnerproduct
i R P
(cid:133)s;s(cid:134)(cid:136) s ^s: The Bergmann kernel is de¢ned to be H (cid:133)x;y(cid:134)(cid:136) n s(cid:133)x(cid:134)
i j M i j M i(cid:136)1 i
^s(cid:133)y(cid:134)onM(cid:2)M andisindependentofthebasischosen.Fortheuniversalcovering
j
M~ , the space of L2-holomorphic section of K form a Hilbert space with respect
R M~
to a similar inner product (cid:133)t;t(cid:134)(cid:136) t ^t: Take an orthonormal basis t;i2N
i j M~ i Pj i
and form the Bergman kernel H (cid:133)x;y(cid:134)(cid:136) t(cid:133)x(cid:134)^t(cid:133)y(cid:134): It is well known that
M~ i2N i j
the L2-cohomology of a Hermitian symmetric space of noncompact type is trivial
except for those corresponding to holomorphic n-forms which are
in¢nite-dimensional. Hence, Theorem 1.1.1 of [Y] can be phrased as the following
lemma:
LEMMA 4. The dimension of the space of holomorphic n-forms on M is
j
asymptoticallyproportionaltothevolumeofM withtheproportionalconstantgiven
j
by the von Neumann dimension of L2-holomorphic n-forms on M~ .
Hence,bothH andH arenontrivial.Letusnowidentifyapointx2Mwitha
M~ Mj
pointx~ 2M~ inthefundamentaldomainofMinM~ .Letp :M !M bethecover-
j;0 j 0
ingmap.p(cid:255)1xconsistsofa¢nitenumberofpointsinM.Letx beoneofthepointsin
j;0 j i
p(cid:255)1x. H (cid:133)x;x(cid:134) is independent of the point chosen as representative since the
j;0 Mj j j
Bergman kernel is invariant under deck transformation which is an isometry.
The following result is essentially due to Donnelly [Do]:
LEMMA 5. H (cid:133)x;y(cid:134) converges pointwise to H (cid:133)x;y(cid:134) in a C1 way.
Mj j j M~
Donnelly stated in [Do] that H (cid:133)x;x(cid:134) converges pointwise to H (cid:133)x;x(cid:134)
Mj j j M~
uniformly.Asthekernelfunctionsareholomorphicwithrespecttothe¢rstvariable
and antiholomorphic with respect to the second variable, it follows easily from
power series expansion the uniform convergence of H (cid:133)x;y(cid:134) to H (cid:133)x;y(cid:134): Then
Mj j j M~
we conclude the convergence in a C1 way from Schauder estimates.
Proof of Theorem 3.
Base point freeness
Letus¢rstprovethattheglobalsectionsG(cid:133)M;K (cid:134)generatesjforsuf¢cientlylarge
j Mj i
M:Letsj;i(cid:136)1;...;N W1beanorthonormalbasisofG(cid:133)M;K (cid:134),here0WjW1
j i j j Mj
with M (cid:136)M~ , and B be the base locus of G(cid:133)M;K (cid:134). As p :M !M is a
1 j j M j(cid:135)1;j j(cid:135)1 j
j
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218 SAI-KEEYEUNG
holomorphic covering map, p (cid:3)(cid:133)sj(cid:134) is a holomorphic section of K for each
j(cid:135)1;j i Mj(cid:135)1
sectionsj ofK .LetDbeafundamentaldomainofM (cid:136)M intheuniversalcover-
i Mj 0
ing M~ and p:M~ !M as before. It follows that p(cid:255)1(cid:133)B (cid:134)\D(cid:26)p(cid:255)1(cid:133)B(cid:134)\D:
j j j(cid:135)1 j(cid:135)1 j j
Hence, p(cid:255)1(cid:133)B(cid:134)\D(cid:136)\j p(cid:255)1(cid:133)B (cid:134)\D is a decreasing set. We claim that
j j k(cid:136)0 j k
\1 p(cid:255)1(cid:133)B(cid:134)\D(cid:136); so that from the relative compactness of D, p(cid:255)1(cid:133)B (cid:134)\D is
j(cid:136)0 j j j j(cid:135)1
emptyforallsuf¢cientlylargej.Toprovetheclaim,notethattheBergmannkernel
function speci¢ed at x(cid:136)y can be expressed as
H (cid:133)x;x(cid:134)(cid:136) sup jf(cid:133)x(cid:134)j2
M
j
f2G(cid:133)M(cid:134);kfk(cid:136)1
j
H (cid:133)x;x(cid:134)(cid:136) sup jf(cid:133)x(cid:134)j2:
M~
f2G(cid:133)2(cid:134)(cid:133)M~(cid:134);kfk(cid:136)1
ThisfollowsfromthefactthattheBergmannkernelisindependentofthechoiceof
base and, hence, we may choose s with maximal value at the point x. Suppose
1
x2\1 p(cid:255)1(cid:133)B(cid:134)\D6(cid:136); so that H (cid:133)x(cid:134)(cid:136)0 for each j. From the above lemma, it
j(cid:136)0 j j Mj
follow that H (cid:133)x(cid:134)(cid:136)0 as well. However, since M~ is homogeneous, the base locus
M~
of K is empty and, hence, such a x does not exist. This concludes the proof of
M~
the claim and, hence, the statement that the global sections generate the bundle.
Separation of points
LEMMA6.AssumethattheL2-canonicalsectionsoftheuniversalcoveringseparates
points.Alsoassumethatforanyc>0;thereexistsanumberk>0suchthatforevery
pairofpointsx;y2M~ ofdistanced(cid:133)x;y(cid:134)Xc,thereisalwaysaholomorphicsection
s2G(cid:133)2(cid:134)(cid:133)M~ ;K(cid:134) satisfying ksk (cid:136)1; s(cid:133)x(cid:134)(cid:136)0;ks(cid:133)y(cid:134)kXk: Then K separates M
L2 Mj j
for all suf¢ciently large j:
Proof. Take a nested sequence of domains D on M~ so that each D is a funda-
j j
mental domain of M: From the above discussion on base-point freeness, we
j
may assume that sections of G(cid:133)M;K(cid:134) is base point free for all jX0. Let
j
t ;...t be a basis of G(cid:133)M ;K(cid:134):
1 N 0
Consider ¢rst the case that x;y2M both lying in some fundamental domain of
j
M whenpulledbacktoM~ :Wemayassumethatx;y2D afterabiholomorphism
o P 0
if necessary. Since H (cid:133)w;z(cid:134)(cid:136) sj(cid:133)w(cid:134)sj(cid:133)z(cid:134) converges to H (cid:133)w;z(cid:134) uniformly on
Mj i i i M1
anyrelativelycompactsetcontainingw;zaccordingtoLemma1,andfor0WjW1,
X
(cid:133)sj(cid:133)x(cid:134)(cid:255)sj(cid:133)y(cid:134)(cid:133)sj(cid:133)x(cid:134)(cid:255)sj(cid:133)y(cid:134)(cid:134)
i i i i
i
(cid:136)H (cid:133)x;x(cid:134)(cid:255)H (cid:133)y;x(cid:134)(cid:255)H (cid:133)x;y(cid:134)(cid:135)H (cid:133)y;y(cid:134);
M M M M
j j j j
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