Continuity of volumes on arithmetic varieties

0.39 MB·English

by  Atsushi Moriwaki

CONTINUITY OF VOLUMES ON ARITHMETIC VARIETIES
7
ATSUSHIMORIWAKI
0
0
2 ABSTRACT. WeintroducethevolumefunctionforC∞-hermitianinvertiblesheaveson
n anarithmeticvarietyasananalogueofthegeometricvolumefunction.Themainresultof
a thispaperisthecontinuityofthearithmeticvolumefunction. Asaconsequence,wehave
J thearithmeticHilbert-SamuelformulaforanefC∞-hermitianinvertiblesheaf. Wealso
giveanotherapplications,forexample,ageneralizedHodgeindextheorem,anarithmetic
6
Bogomolov-Gieseker’sinequality,etc.

]
T
N
.
h INTRODUCTION
t
a
Let X be a d-dimensionalprojectivearithmetic variety and Pic(X) the group of iso-
m
morphismclassesofC∞-hermitianinvertiblesheavesonX.ForL Pic(X),thevolume
[ ∈
vol(L)ofLisdefinedby c

2 c
v log# s H0(X,mL) s 1
9 c vol(L)=limsup { ∈ |k ksup ≤ }.
md/d!
6 m→∞
2 For example, ifcL is ample, then vol(L) = deg(c(L)·d) (cf. Lemma 3.1). This is an
2
1 arithmeticanalogueof the volumefunctionfor invertiblesheaves on a projectivevariety
6 overafield.Thegeometricvolumefcunctionpladysabcrucialroleforthebirationalgeometry
0 viabiginvertiblesheaves. Inthissense,tointroducethearithmeticanalogueofitisvery
/
h significant.
t ThefirstimportantpropertyofthevolumefunctionisthecharacterizationofabigC∞-
a
m hermitianinvertiblesheaf by the positivity of its volume(cf. Theorem4.5). The second
oneisthehomogeneityofthevolumefunction,namely,vol(nL)= ndvol(L)forallnon-
:
v negativeintegersn(cf. Proposition4.7). Bythisproperty,itcanbeextendedtoPic(X)
i ⊗
X Q. Fromviewpointofarithmeticanalogue,themostimpcortantandfuncdamentalquestion
r isthecontinuityof c
a
vol:Pic(X) Q R,
⊗ →
thatis,thevalidityoftheformula:
c c
lim vol(L+ǫ A + +ǫ A )=vol(L)
1 1 n n
ǫ1,...,ǫn∈Q ···
ǫ1→0,...,ǫn→0
c c
for any L,A ,...,A Pic(X) Q. The main purpose of this paper is to give an
1 n
∈ ⊗
affirmativeanswerfortheabovequestion(cf. Theorem5.4). Asaconsequence,wehave
thefollowingarithmeticHilcbert-SamuelformulaforanefC∞-hermitianinvertiblesheaf:
Date:5/January/2007,17:30(JP),(Version2.0).
1991MathematicsSubjectClassification. 14G40,11G50.
1
2 ATSUSHIMORIWAKI
TheoremA(cf. Corollary5.5). LetLandN beC∞-hermitianinvertiblesheavesonX.
IfLisnef,then
deg(c (L)·d)
log# s H0(X,mL+N) s 1 = 1 md+o(md) (m 1).
sup
{ ∈ |k k ≤ } d! ≫
Inparticular,vol(L)=deg(c (L)·d),andLisbidgifabndonlyifdeg(c (L)·d)>0.
1 1
Inamoregeneralsetting,wehavethefollowinggeneralizedHodgeindextheorem:
c d b d b
Theorem B (cf. Theorem 6.2). Let L be a C∞-hermitian invertible sheaf on X. We
assumethefollowing:
(i) L isnefonX .
Q Q
(ii) c (L)issemipositiveonX(C).
1
(iii) Lhasmoderategrowthofpositiveevencohomologies,thatis,thereareageneric
resolutionofsingularitiesµ:Y X andanampleinvertiblesheafAonY such
→
that,foranypositiveintegern,thereisapositiveintegerm suchthat
0
log#(H2i(Y,m(nµ∗(L)+A)))=o(md)
forallm m andforalli>0.
0
≥
Thenwehaveaninequalityvol(L) deg(c (L)·d).
1
≥
TheoremB impliesthatif L is nefoneverygeometricfiberof X Spec(Z), c (L)
1
issemipositiveonX(C),andcdeg(c (Ld)·d)b> 0,thenLisbig(cf. Cor→ollary6.4). Thisis
1
ageneralizationof[17,Corollary(1.9)].MoreoverwecanseethearithmeticBogomolov-
Gieseker’sinequalityasanappdlicatbionofTheoremB(cf. Corollary6.5).
In the geometric case, the above Theorem A can be proved by using the Riemann-
Roch formulaand Fujita’s vanishingtheorem. In the arithmetic case, the proof in terms
ofthearithmeticRiemann-Rochtheoremseemstobedifficult. Insteadofit,weprovethe
continuityofthevolumefunctionbydirectestimates. Forthispurpose,thetechnicalcore
isthefollowingtheorem,whichwasinspiredbyYuan’spaper[16].
TheoremC(cf. Theorem3.4). LetX beaprojectiveandgenericallysmootharithmetic
varietyofdimensiond 2. LetLandAbeC∞-hermitianinvertiblesheavesonX. We
≥
assumethefollowing:
(i) AandL+AareveryampleoverQ.
(ii) ThefirstChernformsc (A)andc (L+A)onX(C)arepositive.
1 1
(iii) There is a non-zero section s H0(X,A) such that the vertical component of
∈
div(s) iscontainedin theregularlocusofX andthatthe horizontalcomponent
ofdiv(s)issmoothoverQ.
Thentherearepositiveconstantsa ,C andDdependingonlyonX,LandAsuchthat
0
log# s H0(X,aL+(b c)A) s 1
sup
{ ∈ − |k k ≤ }
log# s H0(X,aL cA) s 1
sup
≤ { ∈ − |k k ≤ }
+Cbad−1+Dad−1log(a)
forallintegersa,b,cwitha b c 0anda a .
0
≥ ≥ ≥ ≥
Inordertoexplainthetechnicalaspectsoftheabovetheorem,letusconsideritinthe
geometriccase,namely,weassumethatX isaprojectivesmoothvarietyoverC,andwe
trytoestimate
∆=h0(X,aL+(b c)A) h0(X,aL cA).
− − −
CONTINUITYOFVOLUMESONARITHMETICVARIETIES 3
The first elegantway: Let us choose an infinite sequence Y ∞ of distinct smooth
{ i}i=1
membersof A suchthat
| |
h0(Y ,nL+mA )=h0(Y ,nL+mA )
i |Yi j |Yj
foralli,jandallintegersn,m. Thenanexactsequence
b
0 H0(X,aL cA) H0(X,aL+(b c)A) H0(Y ,aL+(b c)A )
→ − → − → i − |Yi
i=1
M
givesriseto∆ b h0(Y ,a(L+A) ). Thisargumentdoesnotworkinthearithmetic
≤ · 1 |Y1
situation.
Thesecondway: Inthepaper[16],forafixedsmoothmemberY A,Yuanconsid-
∈ | |
eredanexactsequence
0 aL+(k 1 c)A aL+(k c)A aL+(k c)A 0
→ − − → − → − |Y →
foreach1 k b,whichyields
≤ ≤
b
∆ h0(Y,aL+(k c)A ) b h0(Y,a(L+A) ).
≤ − |Y ≤ · |Y
k=1
X
This second way works if we consider the arithmetic χˆ instead of the number of small
sections. In this way, Yuan [16] obtained an arithmetic analogue of a theorem of Siu.
However,ifweestimatethenumberofsmallsectionsbyusingtheaboveway,thegrowth
ofthecontributionfromerrortermsislargerthanthemainterm.
Thethirdway:Anexactsequence
0 aL cA aL+(b c)A aL+(b c)A 0
→ − → − → − |bY →
givesriseto
∆ h0(bY,aL+(b c)A ).
≤ − |bY
Ontheotherhand,usingexactsequences
0 aL+(b c k)A aL+(b c)A aL+(b c)A 0,
→ − − |Y → − |(k+1)Y → − |kY →
wehave
b−1
h0(bY,aL+(b c)A ) h0(Y,aL+(b c k)A )
− |bY ≤ − − |Y
k=0
X
b h0(Y,a(L+A) ).
≤ · |Y
Inthearithmeticcontext,thebehavioroftheerrortermsbythiswayisbetterthanthesec-
ondway,sothatwecouldgetthedesiredestimate. Ofcourse,thiswayisverycomplicated
becauseitinvolvesnon-reducedschemes.
Thepaperisorganizedasfollows: InSection1,weprepareseveralestimatesofnorms
oncomplexmanifolds. InSection2,manyformulaeconcerningthenumberofsmallsec-
tionsarediscussed.ThroughSection3,wegivetheproofofthemaintechnicalestimateof
thenumberofsmallsections. InSection4,weintroducethevolumefunctiononanarith-
meticvarietyandconsiderseveralbasicproperties. InSection5,weprovethecontinuity
ofthevolumefunctionandthearithmeticHilbert-SamuelformulaforanefC∞-hermitian
invertiblesheaf. Finally, inSection6, we considerthegeneralizedHodgeindextheorem
andthearithmeticBogomolov-Gieseker’sinequality.
FinallywewouldliketothankProf. Mochizukiforvaluablecorrespondences.
4 ATSUSHIMORIWAKI
Conventionsandterminology. Wefixseveralconventionsandterminologyofthispaper.
1. For a real numberx R, the round-up x , the round-down x and the fractional
∈ ⌈ ⌉ ⌊ ⌋
part x aredefinedby
{ }
x :=min k Z x k , x :=max k Z k x and x =x x .
⌈ ⌉ { ∈ | ≤ } ⌊ ⌋ { ∈ | ≤ } { } −⌊ ⌋
2. Foracomplexvectorz =(z ,...,z ) Cn,twonorms z and z ′aredefinedby
1 n
∈ | | | |
z = z 2+ + z 2 and z ′ = z + + z .
1 n 1 n
| | | | ··· | | | | | | ··· | |
Notethat z z ′ p√nz forallz Cn.
| |≤| | ≤ | | ∈
3. Let(V,σ)beafinitedimensionalnormedvectorspaceoverR. Thenormσ issome-
timesdenotedby . Letf : W V beaninjectivehomomorphismofvectorspaces
k·k →
overR. Thenthenormσ onV yieldsanormσ′ onW givenbyσ′(x) = σ(f(x)). This
norm σ′ is denoted by σ and is called the subnorm of σ. Let g : V Q be a
W֒→V
→
surjectivehomomorphismofvectorspacesoverR. Thenanormσ′′ onQisdefinedby
σ′′(y)=inf σ(x) x g−1(y) .
{ | ∈ }
Thisnormσ′′ isdenotedbyσV։Qandiscalledthequotientnormofσ. Let
0 V′ V V′′ 0
→ → → →
beanexactsequenceoffinitedimensionalvectorspacesoverR. Letσ′,σandσ′′benorms
ofV′,V andV′′ respectively.Wesay
0 (V′,σ′) (V,σ) (V′′,σ′′) 0
→ → → →
isanexactsequenceofnormedvectorspacesifσ′ = σV′֒→V andσ′′ = σV։V′′. LetV∨
bethedualspaceofV,thatis,V∨ =Hom (V,R). Thedualnormσ∨ ofV∨ isgivenby
R
σ∨(φ)=sup φ(x) x V andσ(x) 1 .
{| || ∈ ≤ }
4. LetX be eithera scheme ora complexspace. LetL ,...,L be invertiblesheaves
1 n
onX andm1,...,mn integers. Inthis paper,the tensorproductL⊗1m1 ⊗···⊗Lmnn of
invertiblesheavesisusuallydenotedby
m L + +m L
1 1 n n
···
intheadditivewaylikedivisors.
5. LetX beacompactcomplexmanifoldandΩavolumeformonX.LetL=(L, )
L
|·|
be a C∞-hermitian invertible sheaf on X. Then the natural L2-norm L and the
k·kL2,Ω
sup-norm L onH0(X,L)aredefinedby
k·ksup
1/2
s L = s2Ω and s L =sup s (x) x X
k kL2,Ω | |L k ksup {| |L | ∈ }
(cid:18)ZX (cid:19)
fors H0(X,L).Forsimplicity, L (resp. L )isoftendenotedby L or
∈ k·kL2,Ω k·ksup k·kL2 k·kL2
(resp ). Forarealnumberλ,aC∞-hermitianinvertiblesheaf(L,exp( λ) )is
sup L
k·k − |·|
denotedbyLλ. LetAbeapositiveC∞-hermitianinvertiblesheafonX. Thenormalized
volumeformΩ(A)associatedwithAisgivenby
c (A)∧d
1
Ω(A)= ,
c (A)∧d
X 1
wherec1(A)isthefirstChernformofAanRdd=dimX. Notethat XΩ(A)=1.
R
CONTINUITYOFVOLUMESONARITHMETICVARIETIES 5
6. A quasi-projective scheme over Z is called an arithmetic variety if X is an integral
scheme and flat over Z. We say X is generically smooth if X is smooth over Q. By
Hironaka’s resolution of singularities [9], there is a projective birational morphism µ :
X′ X ofarithmeticvarietiessuchthatX′ isgenericallysmooth. Thisµ : X′ X is
→ →
calledagenericresolutionofsingularitiesofX.
7. LetX beaprojectivearithmeticvarietyandLaC∞-hermitianinvertiblesheafonX.
Accordingto[14],wedefinethreekindsofthepositivityofLasfollows:
ample:LisampleifLisampleonX,thefirstChernformc (L)ispositiveonX(C)
1
•
andnAisgeneratedbysectionss H0(X,nA)with s < 1forasufficientlylarge
sup
∈ k k
n.
nef:LisnefifthefirstChernformc (L)issemipositiveanddeg(H ) 0forany
• 1 Γ ≥
1-dimensionalclosedsubschemeΓinX.
(cid:12)
big : L is big if LQ is big on XQ and there are a positive intdegern(cid:12)and a non-zero
•
sectionsofH0(X,nL)with s <1.
sup
k k
By[17,Corollary(5.7)],ifLisample,then,forasufficientlylargeintegern,H0(X,nL)
hasabasiss ,...,s asaZ-modulewith s <1foralli=1,...,N.
1 N i sup
k k
8. LetX beaprojectivearithmeticvariety,andletLandM beC∞-hermitianinvertible
sheaveson X. We say L is less than or equalto M, denotedby L M, if there is an
≤
injectivehomomorphismφ:L M suchthat φ () onX(C),where and
C M L L
→ | · | ≤|·| |·|
are hermitiannormsof L andM respectively. Thefollowingpropertiesare easily
M
|·|
checked(fortheproof,seeRemark5.3):
(1) L M ifandonlyif M L.
≤ ′ −′ ≤− ′ ′
(2) IfL M andL M ,thenL+L M +M .
≤ ≤ ≤
1. SEVERALESTIMATESOF NORMS ON COMPLEX MANIFOLDS
1.1. Gromov’s inequality. In this subsection, we consider Gromov’s inequality and its
variants.LetusbeginwiththelocalversionofGromov’sinequality.
Lemma 1.1.1 (Local Gromov’s inequality). Let a,b,c be real numbers with a > b >
c > 0. We set U = z Cn z < a , V = z Cn z < b and W = z
{ ∈ | | | } { ∈ | | | } { ∈
Cn z < c . Let Ω be a volume form on U, and let H ,...,H be C∞-hermitian
1 l
| | | }
invertiblesheavesonU. Letω ,...,ω befreebasesofH ,...,H overU respectively.
1 l 1 l
Then there is a constant C dependingonly on H ,...,H , ω ,...,ω , Ω, a, b, c and n
1 l 1 l
such that, for any positive real number p, all non-negative integers m ,...,m and all
1 l
s H0(U,m H + +m H ),
1 1 l l
∈ ···
max sp (x) C( p )2n(m + +m +1)2n sp Ω ,
x∈W{| |(m1,...,ml) } ≤ ⌈ ⌉ 1 ··· l (cid:18)ZV | |(m1,...,ml) (cid:19)
where isthehermitiannormofm H + +m H and p istheround-up
|·|(m1,...,ml) 1 1 ··· l l ⌈ ⌉
ofp(cf. Conventionsandterminology1).
Proof. Let be the hermitian norm of H and u = ω on U. Considering an
i i i i i
|·| | |
upperboundofthepartialderivativesofu overV,wecanfindapositiveconstantK such
i i
that
u (x) u (y) K x y ′
i i i
| − |≤ | − |
6 ATSUSHIMORIWAKI
forallx,y V (forthedefinitionof ′,seeConventionsandterminology2). Weset
∈ |·|
K K 1
D =max max 1 ,...,max l , and R=1/D.
u (x) u (x) b c
(cid:26)x∈V (cid:26) 1 (cid:27) x∈V (cid:26) l (cid:27) − (cid:27)
Then,forx ,x V,
0
∈
K
u (x) u (x ) K x x ′ =u (x ) 1 i x x ′
i i 0 i 0 i 0 0
≥ − | − | − u (x )| − |
(cid:18) i 0 (cid:19)
u (x )(1 D x x ′).
i 0 0
≥ − | − |
We set B(x ,R) = x Cn x x ′ R . Then 1 D x x ′ 0 for all
0 0 0
{ ∈ | | − | ≤ } − | − | ≥
x B(x ,R). Moreover,ifx W,thenB(x ,R) V because
0 0 0
∈ ∈ ⊆
x x x x ′ R b c.
0 0
| − |≤| − | ≤ ≤ −
Hereweclaimthefollowing:
Claim1.1.1.1. Foranon-negativerealnumberm,
1 1 1 m 1
x x 1 (x + +x ) dx dx .
··· 1··· n − n 1 ··· n 1··· n ≥ ( m +1)n( m +2)n
Z0 Z0 (cid:18) (cid:19) ⌈ ⌉ ⌈ ⌉
First let us consider the case where m is an integer. If m = 0, then the assertion is
obvious,sothatweassumem 1. Since
≥
m n m
1 1
1 (x + +x ) = (1 x )
− n 1 ··· n nm − i !
(cid:18) (cid:19) i=1
X
1 m!
= (1 x )m1 (1 x )mn
nm m ! m ! − 1 ··· − n
1 n
mm11+≥·0·X,·.+..m,mnn=≥m0 ···
and
1 1
x(1 x)ddx=
− (d+1)(d+2)
Z0
foranon-negativeintegerd,theintegralI intheclaimisequalto
1 m! 1
.
nm m ! m !(m +1)(m +2) (m +1)(m +2)
1 n 1 1 n n
mm11+≥·0·X,·.+..m,mnn=≥m0 ··· ···
Thus
1 m! 1
I = .
≥ (m+1)n(m+2)nnm m ! m ! (m+1)n(m+2)n
1 n
mm11+≥·0·X,·.+..m,mnn=≥m0 ···
Ifmisnotinteger,then
m ⌈m⌉
1 1
1 (x + +x ) 1 (x + +x )
1 n 1 n
− n ··· ≥ − n ···
(cid:18) (cid:19) (cid:18) (cid:19)
because0 1 1(x + +x ) 1. Thustheclaimfollows.
≤ − n 1 ··· n ≤
WechooseapositiveconstantewithΩ eΩ onV,where
can
≥
n
√ 1
Ω = − dz dz¯ dz dz¯ .
can 1 1 n n
2 ∧ ∧···∧ ∧
(cid:18) (cid:19)
CONTINUITYOFVOLUMESONARITHMETICVARIETIES 7
Let s be an element of H0(U,m H + +m H ). Then we can find a holomorphic
1 1 l l
functionf overU withs=fω⊗m1 ···ω⊗ml. Wealsochoosex W suchthatthe
1 ⊗···⊗ l 0 ∈
continuousfunction s onW takesthemaximumvalueatx . Then
| |(m1,...,ml) 0
sp Ω e sp Ω
| |(m1,...,ml) ≥ | |(m1,...,ml) can
ZV ZB(x0,R)
=e f pupm1 upmlΩ
| | 1 ··· l can
ZB(x0,R)
eu (x )pm1 u (x )pml f p(1 D x x ′)mΩ ,
1 0 l 0 0 can
≥ ··· | | − | − |
ZB(x0,R)
wherem=p(m + +m ). Moreover,ifweset
1 l
···
x x =(r exp(√ 1θ ),...,r exp(√ 1θ )),
0 1 1 n n
− − −
then
f p(1 D x x ′)mΩ
0 can
| | − | − |
ZB(x0,R)
2π 2π
= f pdθ dθ r r (1 D(r + +r ))mdr dr .
1 n 1 n 1 n 1 n
Zrr11≥+0··,·.+..r,rnn≤≥R0(cid:18)Z0 ···Z0 | | ··· (cid:19) ··· − ··· ···
Since f pissubharmonic,wehave
| |
2π 2π
f pdθ dθ (2π)n f(x )p.
1 n 0
··· | | ··· ≥ | |
Z0 Z0
Therefore,usingClaim1.1.1.1,
f p(1 D x x ′)mΩ
0 can
| | − | − |
ZB(x0,R)
(2π)n f(x )p r r (1 D(r + +r ))mdr dr
0 1 n 1 n 1 n
≥ | | r1+···+rn≤R ··· − ··· ···
Zr1≥0,...,rn≥0
(2π)n f(x )p r r (1 D(r + +r ))mdr dr
0 1 n 1 n 1 n
≥ | | ··· − ··· ···
Z[0,R/n]n
(2π)n f(x )p 1
0
| | .
≥ (nD)2n ( m +1)n( m +2)n
⌈ ⌉ ⌈ ⌉
Gatheringallcalculations,ifwesetC′ =e(2π)n/(nD)2n,then
C′ s(x )p
sp Ω | 0 |(m1,...,ml) .
| |(m1,...,ml) ≥ ( m +1)n( m +2)n
ZV ⌈ ⌉ ⌈ ⌉
Further,since m p (m + +m ),
1 l
⌈ ⌉≤⌈ ⌉ ···
( m +1)n( m +2)n ( p (m + +m )+1)n( p (m + +m )+2)n
1 l 1 l
⌈ ⌉ ⌈ ⌉ ≤ ⌈ ⌉ ··· ⌈ ⌉ ···
( p (m + +m +1))n(2 p (m + +m +1))n
1 l 1 l
≤ ⌈ ⌉ ··· ⌈ ⌉ ···
=2n( p )2n(m + +m +1)2n.
1 l
⌈ ⌉ ···
Thuswegetthelemma. 2
Thepartialresultsofthefollowingcorollaryarefoundin[12]and[11].
8 ATSUSHIMORIWAKI
Corollary 1.1.2 (Gromov’s inequality). Let M be an n-dimensional compact complex
manifold,ΩavolumeformonM,andletH ,...,H beC∞-hermitianinvertiblesheaves
1 l
onM. ThenthereisaconstantC dependingonlyonH ,...,H ,ΩandM suchthat,for
1 l
any positive real numberp, all integers m ,...,m with m 0,...,m 0, and all
1 l 1 l
≥ ≥
s H0(M,m H + +m H ),
1 1 l l
∈ ···
max sp (x) C( p )2n(m + +m +1)2n sp Ω .
x∈M{| |(m1,...,ml) } ≤ ⌈ ⌉ 1 ··· l (cid:18)ZM| |(m1,...,ml) (cid:19)
Proof. Wetakeafinitecovering U ofM withthefollowingproperties:
i i=1,...,m
{ }
(1) U is isomorphic to z Cn z < 1 by using a local coordinate z (x) =
i i
{ ∈ | | | }
(z (x),...,z (x)). WesetV = x U z (x) <1/2 andW = x U
i1 in i i i i i
{ ∈ || | } { ∈ |
z (x) <1/4 .
i
| | }
(2) Therearelocalbasesω ,...,ω ofH ,...H overU respectively.
i1 il 1 l i
(3) m W =M.
i=1 i
ThenourcorollaryfollowsfromthelocalGromov’sinequality. 2
S
Corollary1.1.3. LetMbeann-dimensionalcompactcomplexmanifold,andletH ,...,H
1 l
beC∞-hermitianinvertiblesheavesonM. LetV beaclosedcomplexsubmanifoldofM.
LetΩ and Ω be volumeforms onM andV respectively. Thenthere is a constantC
M V
suchthat
C(m + +m +1)2n s2Ω s 2Ω
1 ··· l | | M ≥ | |V | V
ZM ZV
forallnon-negativeintegersm ,...,m andalls H0(X,m H + +m H ).
1 l 1 1 l l
∈ ···
Proof. Notethat
s 2Ω
s 2 s 2 V | |V | V .
k ksup ≥k |V ksup ≥ Ω
R V V
ThusthecorollaryfollowsfromGromov’sinequality.R 2
ThefollowinglemmaisduetoTakuroMochizuki,whokindlytellusitsproof. Thisis
avariantofGromov’sinequality.
Lemma 1.1.4. Let X be an n-dimensionalcompact complex manifold and ω a positive
(1,1)-formon X. Let H ,...,H be C∞-hermitianinvertiblesheavesonX. Then, for
1 l
anopensetU ofX,therearepositiveconstantsC,C′andD′suchthat
sup s (x) Cm1+···+ml sup s (x) .
{| |(m1,...,ml) }≤ {| |(m1,...,ml) }
x∈X x∈U
and
s2 ω∧n D′ C′m1+···+ml s2 ω∧n
| |(m1,...,ml) ≤ · | |(m1,...,ml)
ZX ZU
forallnon-negativeintegersm ,...,m andalls H0(X,m H + +m H ),where
1 l 1 1 l l
∈ ···
isthehermitiannormofm H + +m H .
|·|(m1,...,ml) 1 1 ··· l l
Proof. ShrinkingU ifnecessarily,wemayidentifyU with x Cn x < 1 . We
setW = x Cn x <1/2 .Inthisproof,wedefineaLaplac{ian∈(cid:3) by| |the| form}ula:
ω
{ ∈ || | }
√ 1
− ∂∂¯(g) ω∧(n−1) =(cid:3) (g)ω∧n.
ω
− 2π ∧
CONTINUITYOFVOLUMESONARITHMETICVARIETIES 9
Leta beaC∞-functiongivenbyc (H ) ω∧(n−1) = a ω∧n,wherec (H )isthefirst
i 1 i i 1 i
∧
ChernformofH . WechooseaC∞-functionφ onX suchthat
i i
a ω∧n = φ ω∧n
i i
ZX ZX
and that φ is identically zero on X W. Thus we can find a C∞-function F with
i i
(cid:3) (F )=a φ . Notethat(cid:3) (F )=\ a onX W.
ω i i i ω i i
− \
Lets H0(X,m H + +m H )andweset
1 1 l l
∈ ···
f = s2 exp( (m F + +m F )).
| |(m1,...,ml) − 1 1 ··· l l
Claim1.1.4.1. max f(x) =max f(x) .
x∈X\W x∈∂(W)
{ } { }
Iff isaconstantoverX W,thenourassertionisobvious,sothatweassumethatf is
\
notaconstantoverX W. Inparticular,s=0. Since
\ 6
√ 1
− ∂∂¯(log(s2 ))=c (m H + +m H )=m c (H )+ +m c (H ),
− 2π | |(m1,...,ml) 1 1 1 ··· l l 1 1 1 ··· l 1 l
wehave(cid:3) (log(f)) = 0onX (W Supp(div(s))). Letuschoosex X W such
ω 0
\ ∪ ∈ \
thattheC∞-functionf overX W takesthemaximumvalueatx . Notethat
0
\
x X (W Supp(div(s))).
0
∈ \ ∪
For,ifSupp(div(s)) = ,thenourassertionisobvious. Otherwise,f iszeroatanypoint
∅
ofSupp(div(s)).
Since log(f) is harmonicover X (W Supp(div(s))), log(f) takes the maximum
\ ∪
valueatx andlog(f)isnotaconstant,wehavex ∂(W)byvirtueofthemaximum
0 0
∈
principleofharmonicfunctions.Thustheclaimfollows.
Weset
d = min exp( F ) , D = max exp( F ) and C = max D /d .
i i i i i i
x∈X\W{ − } x∈∂(W){ − } i=1,...,l{ }
Then
dm1 dml s2 f
1 ··· l | |(m1,...,ml) ≤
overX W and
\
f Dm1 Dml s2
≤ 1 ··· l | |(m1,...,ml)
over∂(W). Hence
max s2 Cm1+···+ml max s2
x∈X\W{| |(m1,...,ml)}≤ x∈∂(W){| |(m1,...,ml)}
Cm1+···+mlmax s2 .
≤ {| |(m1,...,ml)}
x∈W
whichimpliesthat
max s2 Cm1+···+mlmax s2 .
x∈X{| |(m1,...,ml)}≤ x∈W{| |(m1,...,ml)}
Thisis the first partof the lemma. Note that ex x+1 forx 0. Thus, by the local
≥ ≥
Gromov’sinequality(cf. Lemma1.1.1),thereareconstantsC andD suchthat
1 1
max s2 D Cm1+···+ml s2 Ω
{| |(m1,...,ml)}≤ 1· 1 | |(m1,...,ml)
x∈W ZU
forallnon-negativeintegersm ,...,m andalls H0(X,m H + +m H ). There-
1 l 1 1 l l
forethesecondassertionfollows. ∈ ··· 2
10 ATSUSHIMORIWAKI
1.2. Distorsionfunctions. LetX beann-dimensionalprojectivecomplexmanifoldand
Ω a volume form of X with Ω = 1. Let H = (H,h) be a C∞-hermitian invertible
X
sheafonX. Fors,s′ H0(X,H),weset
∈ R
s,s′ = h(s,s′)Ω.
h iH,Ω
ZX
Lets ,...,s beanorthonormalbasisofH0(X,H)withrespectto , . Wedefine
1 N h iH,Ω
N
dist(H,Ω)(x)= h(s ,s )(x).
i i
i=1
X
Notethatdist(H,Ω)doesnotdependonthechoiceofanorthonormalbasis. Inthecase
of H0(X,H) = 0 , dist(H,Ω) is defined to be the constantfunction 0. The function
{ }
dist(H,Ω)iscalledthedistorsionfunctionofH withrespecttoΩ.
LetAbeapositiveC∞-hermitianinvertiblesheafonX. DuetoBouche[3]andTian
[15],weknowthat
dist(aA,Ω(A))(x)
sup 1 =O(1/a)
dimH0(aA) −
x∈X(cid:12) (cid:12)
(cid:12) (cid:12)
fora 1,whereΩ(A)isthe(cid:12)normalizedvolumeform(cid:12)associatedwithA(cf.Conventions
≫ (cid:12) (cid:12)
andterminology5). Usingthisresult,Yuan[16,Theorem3.3]provedthefollowing:
Theorem1.2.1. LetA= (A,h )andB =(B,h )bepositiveC∞-hermitianinvertible
A B
sheavesonX. ThentherearepositiveconstantsC andC suchthat
1 2
2C 3C
dist(aA bB,Ω(A))(x) dimH0(aA) 1+ 1 + 2
− ≤ a b
(cid:18) (cid:19)
forallx X,a 1andb 3C .
2
∈ ≥ ≥
Proof. Forreader’sconvenience,wereproveithere.ByBouche-Tian’stheorem,there
areconstantsC andC suchthat
1 2
C C
dimH0(aA) 1 1 dist(aA,Ω(A))(z) dimH0(aA) 1+ 1
− a ≤ ≤ a
(cid:18) (cid:19) (cid:18) (cid:19)
and
C C
dimH0(bB) 1 2 dist(bB,Ω(B))(z) dimH0(bB) 1+ 2
− b ≤ ≤ b
(cid:18) (cid:19) (cid:18) (cid:19)
for all z X, a 1 and b 1. By taking larger C and C if necessarily, we may
1 2
∈ ≫ ≫
assumethattheaboveinequalitiesholdforallz X andalla,b 1.
∈ ≥
Let us fix an arbitrary x X. Let us choose an orthonormalbasis of H0(bB) with
∈
respectto , suchthatonlyonesectionisnon-zeroatx. Wedenotethissection
h ibB,Ω(B)
bys(b). Then
h (s(b),s(b))(x)=dist(bB,Ω(B))(x) dimH0(bB)(1 C /b).
bB ≥ − 2
Ontheotherhand,
s(b) 2 supdist(bB,Ω(B))(z) dimH0(bB)(1+C /b).
k ksup ≤ ≤ 2
z∈X
Therefore
h (s(b),s(b))(x) 1 C /b
bB − 2 .
s(b) 2 ≥ 1+C /b
k ksup 2