Noyau de Bergman sur une surface de Riemann épointée

(1)

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Differential geometry

Bergman kernels on punctured Riemann surfaces

Noyau de Bergman sur une surface de Riemann épointée

Hugues Auvray

^a

, Xiaonan Ma

^b

, George Marinescu

^c

aLaboratoiredemathématiquesd’Orsay,UniversitéParis-Sud,CNRS,UniversitéParis-Saclay,Départementdemathématiques,bâtiment425, 91405Orsay,France

bUniversitéParis-Diderot–Paris-7,UFRdemathématiques,case7012,75205Pariscedex13,France cUniversitätzuKöln,MathematischesInstitut,Weyertal86–90,50931Köln,Germany

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received8May2016

Acceptedafterrevision30August2016 Availableonline7September2016 PresentedbyJean-MichelBismut

Inthispaper,weconsiderapuncturedRiemannsurfaceendowedwithaHermitianmetric thatequalsthe Poincarémetricnear thepuncturesand aholomorphiclinebundle that polarizes the metric. We show that the Bergman kernel can be localized around the singularitiesanditslocalmodelistheBergmankernelofthepuncturedunitdiscendowed with the standard Poincaré metric. As a consequence, we obtain an optimal uniform estimateofthe supremum normof theBergman kernel function, involvinga fractional growthorderofthetensorpower.

©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

ré s u m é

On considère une surface de Riemann compacte munie d’une métrique hermitienne singulièreégaleà lamétrique de Poincarédudisqueépointé prèsd’un diviseurdonné.

Onconsidèreunfibréendroitesholomorpheavecunemétriquesingulièrequipolarisela métrique(singulière)surlasurfacede Riemann.On donnel’asymptotique expliciteprès dessingularitésdelafonctiondeBergman lorsquelapuissancedefibréendroitestend versl’infini.

©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Versionfrançaiseabrégée

Onconsidèreunesurface deRiemanncompacteetunsous-ensemble ﬁni D représentantdessingularités.Onmunitla surface d’unemétriquehermitiennesingulièreégaleàlamétriquedePoincaréprèsde D.Onconsidèreunﬁbréendroites holomorphe Lavecunemétriquesingulièrequipolariselamétrique(singulière)surlasurfacedeRiemann.Lafonctionde Bergman associéeest B_p

(

x

)

=

j|^sj

(

x

)|

²^,^avec {^sj}^une ^baseorthonorméede l’espace de sectionsholomorphes de carré intégrabledeL^p,lap-puissancetensorielledeL.

E-mailaddresses:[email protected](H. Auvray),[email protected](X. Ma),[email protected](G. Marinescu).

http://dx.doi.org/10.1016/j.crma.2016.08.006

1631-073X/©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccessârticleûnder^the^CC^BY-NC-ND^license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

(2)

Ondonnel’asymptotiqueexplicite(enparticulierprèsdessingularités)delafonctiondeBergmanBp quand ptendvers l’inﬁni.L’asymptotiqueuniformepourdesmétriquessingulièresest,ànotreconnaissance,unrésultatnouveau.Danslecas arithmétique,l’espacedesectionsholomorphesdecarréintégrabledeL^p estl’espacedeformesmodulairesparaboliquesde poids2p.

1. LocalizationofBergmankernelsnearthesingularities

In this note, we study the Bergman kernels of a singular Hermitian line bundle over a Riemann surface under the assumption that the curvaturehas singularitiesof Poincaré type at a ﬁniteset. Ourﬁrst resultshows that the Bergman kernelcanbelocalizedaroundthesingularitiesanditslocalmodelistheBergmankernelofthepunctureddiscendowed withthestandardPoincarémetric.TheprooffollowstheprinciplethatthespectralgapoftheKodairaLaplacianimpliesthe localizationoftheBergmankernel[9].Byadetailedanalysisofthelocalmodel,wededuceasharpuniformestimateofthe supremumnormoftheBergmankernels.Detailsandfurtherextensionsaredevelopedin[3].

Let

be a compact Riemann surface, D= {^a1

, . . . ,

aN}⊂

be a ﬁnite set and let

=

\^D ^be ^the corresponding punctured Riemannsurface. Weconsidera Hermitian form

ω

on

,aholomorphiclinebundle L on

,anda singular HermitianmetrichonLsuchthat:

(

α

) hissmoothover

,andforall j=¹

, . . . ,

N,thereisatrivializationofLinthecomplexneighborhoodV_jofa_j in

, withassociatedcoordinatez_j suchthat|¹|²_h

(

z_j

)

=log

(|

^zj|²

)

;

(

β

) thereexists

ε >

0 suchthatthe(smooth)curvatureR^L ofhsatisﬁesiR^L≥

εω

over

andiR^L=

ω

onVj:=^Vj\{^aj}^; inparticular,

ω

=

ω

_D^∗ inthelocalcoordinatez_jonV_jand

(, ω

)

iscomplete.

Here

ω

_D∗ denotesthePoincarémetriconthepuncturedunitdisc,normalizedasfollows:

ω

_D∗

:=

^{i dz}

∧

dz

|

^z

|

²^log²

(|

^z

|

²

) ·

⁽¹⁾

Forp≥^1,^let^h^p:=^h^⊗^p ^be^the^metric^induced^by^h^on ^L^p|,whereL^p:=^L^⊗^p^.^We^denote^by ^H⁰₍2)

(,

L^p

)

thespaceof L²-holomorphicsectionsofL^p relativetothemetricsh^p and

ω

,

H⁰₍₂₎

(,

L^p

) =

⎧ ⎨

⎩

^S

∈

H⁰

(,

L^p

) :

S

²_L2

:=

|

S

|

²_hp

ω

< ∞

⎫ ⎬

⎭ ,

(2) endowedwiththeobviousinnerproduct.Thesectionsfrom H₍⁰₂₎

(,

L^p

)

extendtoholomorphicsectionsofL^p over

,i.e.

H₍⁰₂₎

(,

L^p

)

⊂^H⁰

,

L^p

,(see[9,(6.2.17)]).Inparticular,thedimensiondp ofH₍⁰₂₎

(,

L^p

)

isﬁnite.

Wedenoteby Bp∈^C^∞

(,

R

)

theBergmankernelfunctionofthespaceH⁰₍₂₎

(,

L^p

)

,definedasfollows:if{^S^p}^d_≥^p₁ îsân orthonormalbasisofH⁰₍₂₎

(,

L^p

)

,then

Bp

(

x

) =

d_p

=¹

|

S^p

(

x

)|

²_hp

.

(3)

Notethat Bp isindependentofthechoiceofbasis (see[9,(6.1.10)]).Let B^D_p^∗ betheBergmankernelfunctionof D^∗

, ω

_D∗

,

C

,

_log

(|

z|²

)

^p|· |

.

ThemainresultofthispaperisaweightedestimateintheC^m-normnearthepuncturesfortheglobalBergman kernel function Bp comparedto theBergman kernelfunction B^D_p^∗ ofthe punctureddisc,uniformlyin thetensorpowers ofthe givenbundle.

Theorem1.1.Assumethat

(, ω

,

L

,

h

)

fulﬁllconditions(

α

)and(

β

).Thenthefollowingestimateholds:forany

,

m∈N^,^and^every

δ >

0,thereexistsC=^C

(,

m

, δ) >

0suchthatforallp∈N^∗^,^and^z∈^V1∪

. . .

∪^VNwiththelocalcoordinatezj,

^Bp

−

^B^Dp^∗

C^m

(

z_j

) ≤

^Cp⁻

log

(|

^zj

|

²

)

⁻^δ

.

(4)

HerethepointwiseC^m-normofafunctionisdeﬁnedbyusingtheLevi-Civitaconnectionon

(, ω

)

.

Remark1.Theorem 1.1admitsageneralizationtoorbifoldRiemannsurfaces.Assumethat

isacompactorbifoldRiemann surface,andtheﬁnitesetD= {^a1

, . . . ,

aN}⊂

doesnotmeetthe(orbifold)singularsetof

.AssumemoreoverthatLisa holomorphicorbifoldlinebundleon

.Let

ω

beanorbifoldHermitianformon

andhanorbifoldHermitianmetricon Linthesenseof[9,§5.4].TheproofofTheorem 1.1canbe modiﬁedtoshow: ifconditions

( α )

,

(β)

holdinthiscontext, then(4)holds.

(3)

By[9,Theorems 6.1.1, 6.2.3],foranycompactset K⊂

,wehavethefollowingexpansiononK inanyC^m-topology 1

pBp

(

x

) =

¹ 2

π +

^∞

j=1

bj

(

x

)

p⁻^j asp

→ ∞.

(5)

Theorem 1.1 givesaprecisedescriptionofBp nearthepunctures,intermsoftheBergmankernelfunctionofthePoincaré metriconthelocalmodelofthepuncturedunitdiscinC^.^Note^thatⁱⁿ^the^case^of^smooth^metrics^with^positive^curvature, theBergman kernelcanbelocalized,andits localmodelistheEuclideanspaceendowedwithatrivialbundle ofpositive curvature,see[9,Sections4.1.2–3].ThiskindoflocalizationisinspiredfromtheanalyticlocalizationtechniqueofBismut–

Lebeau[5]inlocalindextheory.Fortheproblemathandhere, wehavetoovercomediﬃcultieslinkedtothepresenceof singularities.

FromastudyofthemodelBergmankernelfunctionsB^D_p^∗ onthepuncturedunitdisc,wegetthefollowingratioestimate asacorollaryofTheorem 1.1andProposition 2.1.

Corollary1.2.Let

(, ω

,

L

,

h

)

beasinTheorem 1.1.Then

sup

x∈B_p

(

x

) =

^sup

x∈ σ∈^H⁰₍₂₎(,L^p)

| σ (

x

)|

²_hp

σ

²_L2

=

_p 2

π

3/2

+

O

(

p

)

as p

→ ∞.

⁽⁶⁾

It is,toourknowledge,the ﬁrstexampleofauniform L^∞ asymptoticdescriptionofthe Bergmankernelfunction ofa singularpolarization.Thisisofparticularinterestinarithmeticsituations.

Corollary 1.2 isalso quitestriking froma Kähler geometrypoint ofview, asthe supremumof theBergman kernel is equivalentto_p

2π

n

oncompactpolarizedmanifoldsofcomplexdimensionn.

2. Bergmankernelsonthepuncturedunitdisc Let p∈N^∗^,^and ^H₍^p₂₎

(

D^∗

)

:=^H₍⁰₂₎

D^∗

, ω

_D^∗

,

C

,

log

(

|^z|²

)

^p|· |

be the spaceofholomorphic functionsonD^∗ ^with^ﬁnite L²-norm.Forp≥^2,^the^set

^p 2

π (

p

−

¹

) !

1/2

z

: ∈ N , ≥

¹

(7)

formsanorthonormalbasisofH₍^p₂₎

(D

^∗

)

.WegetinparticulartheBergmankernelfunctionofH₍^p₂₎

(D

^∗

)

forall p≥^2,

B^D_p^∗

(

z

) =

log

(|

^z

|

²

)

^p 2

π (

p

−

²

) !

∞ =¹

^p⁻¹

|

^z

|

²

.

(8)

ThisreadilyprovidesthebehaviorofB^D_p^∗.

Proposition2.1.Forany0

<

a

<

1and0

< γ <

¹₂,thereexistsc=^c

(

a

, γ ) >

0suchthatform∈N^∗^,

^B^Dp^∗

(

z

) −

^p

−

¹

2

π

C^m({^ae⁻^p^γ≤|^z|<1},ω_D∗)

=

O

e⁻^cp¹⁻²^γ

as p

→ ∞ ,

(9)

andsup_z_∈D∗B^D_p^∗

(

z

)

= _p 2π

3/2

+O

(

p

)

asp→ ∞^. 3. Arithmeticcase

WegiveanimportantexamplewhereTheorem 1.1applies.If

isageometricallyﬁniteFuchsiangroupoftheﬁrstkind, without ellipticelements, then

:=

\H ^can ^by compactifiedby finitely manypoints D= {â1

, . . . ,

aN} ^into ^a ^compact Riemann surface

ofgenus g such that 2g−²+^N

>

0 and L=^K⊗O

(

D

)

isample,where K isthe canonicalline bundleon

.TheKähler–Einsteinmetric

ω

isinducedbythePoincarémetriconH^.^Let^h^K ^be^the^metric^on^Kinduced by

ω

.Let

σ

bethe canonicalsection ofO

(

D

)

. Thesingular metrich^O⁽^D⁾ onO

(

D

)

isdeﬁnedby |

σ

|²

h^O(D)=^1. ^The isomorphism

K

→

^K

⊗

O

(

D

) |

=

^L

|

,

s

→

^s

⊗ σ

over

andthe metrics h^K andh^O⁽^D⁾ inducethe metrich on L|.Then

(, ω

)

andthe formal square rootof

(

L

,

h

)

satisfyconditions

( α )

and

(β)

.

(4)

LetS₂_p bethespaceofcuspforms(Spitzenformen)ofweight2pof

endowedwiththePeterssoninnerproduct.From theabove construction,the isomorphism

:S₂_p→^H⁰

,

L^p

,

f→ ^f^dz^⊗^p ⁱⁿ ^[11,Proposition 3.3, 3.4(b)]isan isometry.

WecanformtheBergmankernelfunctionofSp asin(3),denotedbyB_p.WededucefromCorollary 1.2:

Corollary3.1.Let

⊂^PSL

(

2

,

R

)

beageometricallyﬁniteFuchsiangroupoftheﬁrstkindwithoutellipticelements.LetB_p bethe Bergmankernelfunctionofcuspformsofweight2p.If

iscocompactthenuniformlyon

\H^,

B_p

(

x

) =

^p

π +

O

(

1

),

as p

→ ∞ .

(10)

If

isnotcocompactthen

sup

x∈\HB_p

(

x

) =

_p

π

3/2

+

O

(

p

),

as p

→ ∞ .

(11)

Uniformestimatesforsup_x_∈_\HB_p

(

x

)

arerelevantinarithmeticgeometryandwereprovedinvariousdegreesofgener- alityandsharpnessin[1,10,8,7].In[7],itisprovedthat,inthecoﬁnitebutnon-cocompactcase,sup_x_∈\HB_p

(

x

)

=O(^p³^/²

)

andtheresultisoptimal,atleastuptoanadditivetermintheexponentoftheform−

ε

forany

ε >

0.Estimate(11)gives theprecisecoeﬃcientoftheleadingterm p³^/² andissharp(bykillingthe“

ε

frombelow”from[7]).Estimate(10)isthe consequenceofthegeneralexpansionoftheBergmankerneloncompactmanifolds[12] (cf.[9]foracomprehensivestudy andcompletereferences).

It turnsout that Corollary 3.1canbe formulated soasto underlinea certain uniformityin

,inthe samefashion as in[7]:

Theorem3.2.Let

0⊂^PSL

(

2

,

R)^beâ^fixed^Fuchsian^subgroupôf^the^first^kind^withoutêllipticêlementsând^let

⊂

0beany subgroupofﬁniteindex.If

0iscocompact,then

B_p

(

x

) =

^p

π +

O0

(

1

),

as p

→ ∞ .

(12)

If

0isnotcocompact,then

sup

x∈\HB_p

(

x

) =

_p

π

3/2

+

O0

(

p

),

as p

→ ∞ .

(13)

HeretheimpliedconstantsinO0

(

1

)

,O0

(

p

)

dependsolelyon

0.

Notethat(12)isaspecialcaseofamoregeneralresultimplicitin[9,§6.1.2].

WeconsiderafurtherextensionofTheorem 3.2tothecasewhenthegroup

0hasellipticelements.Thenthequotients

\Hâreⁱⁿ^generalôrbifolds.^Byûsing^the^resultôf^{Dai–Liu–Ma}^[6,^(5.25)]ôn^the^Bergman^kernelasymptoticsonorbifolds andtheorbifoldversionofTheorem 1.1,weobtainthefollowing.

Theorem3.3.Let

0⊂PSL

(

2

,

R)beaﬁxedFuchsiansubgroupoftheﬁrstkind.Let{xj}^q_j₌₁betheorbifoldpointsof

0\HandUxjbe asmallneighborhoodofxjin

0\H^.^Let

⊂

0beanysubgroupofﬁniteindexandπ:

\H→

0\H^be^the^naturalprojection.If

0iscocompact,thenasp→ ∞

B_p

(

x

) =

^p

π +

O0

(

1

),

uniformly on

(\H)

q j=¹

π

⁻¹

(

U_x_j

).

(14)

Oneachπ⁻¹

(

Ux_j

)

,wehave,asp→ ∞^, B_p

(

x

) =

1

+

γ∈_x j{1}

exp

ip

θ

_γ

−

^p

(

1

−

^eⁱ^θ^γ

) |

^z

|

²

_p

π +

O0

(

1

),

(15)

wherex_j ∈π⁻¹

(

x_j

)

isinthesamecomponentofπ⁻¹

(

U_x_j

)

asx,eⁱ^θ^γ istheactionof

γ

ontheﬁberofK_\Hatx_j,andz=^z

(

x

)

isthe coordinateofx innormalcoordinatesz centeredatx_j inH^,^and

y= {

γ

_∈

:

γ

y=^y}^the^stabilizer^of^y.

Inparticular,ifq₀=^lcm{|0,xj|:^j=¹

, . . . ,

q}^,ⁿ=^max{|y|:^y∈π⁻¹

(

x_j

),

j=¹

, . . . ,

q}^,^then sup

x∈\HB_q

0p

(

x

) =

ⁿ q₀p

π +

O0

(

1

).

(16)

(5)

If

0isnotcocompact,thenasp→ ∞

sup

x∈\HB_p

(

x

) =

_p

π

3/2

+

O₀

(

p

).

(17)

Hereagain,theimpliedconstantsinO0

(

1

)

,O0

(

p

)

dependsolelyon

0.

Theorems 3.2,3.3sharpeninanoptimalwaythemainresultof[7],whichstatesthat sup

x∈\HB_p

(

x

) =

O0

(

p

)

if

0is cocompact

,

O0

(

p³^/²

)

if

0is not cocompact

.

⁽¹⁸⁾

Weobtaininthiswaythe preciseleadingtermsin(18).

4. ProofofTheorem 1.1 Let

∂

^L

p∗

betheadjointoftheDolbeaultoperator

∂

^L

p

on

(

L^p

,

h^p

)

over

(, ω

)

.TheKodairaLaplacianon

⁽₀⁰^,⁰⁾

(,

L^p

)

is deﬁnedbyp:=

∂

^L

p∗

∂

^L

p

.Itisessentiallyself-adjointinL²

(,

L^p

)

andsatisﬁes(see[9,Theorem6.1.1])forsomec

>

0 and p^1,

Spec

(

p

) ⊂ {

0

} ∪ [

cp

,+∞) ,

(19)

andfor p^1,^ker

(

p

)

=^H⁰₍2)

(,

L^p

)

.

Togettheuniformestimatenearthesingularities,weneedtoestablishaweighted ellipticestimateforKodairaLapla- cianson

(

L^p

,

h^p

)

over

(, ω

)

suchthattheestimateisuniformon p.Thenwecancombinethelocalizationprinciple[9]

impliedbythespectralgap(19)andtheweightedSobolevembeddingtheoremin[4,§4]and[2,§4],toobtainthat,forany l∈N^∗^,

γ >

0,thereexistsC

>

0 suchthatforall p∈N^∗^,^and^z∈^V1∪

. . .

∪^VN withthelocalcoordinatezj,

^Bp

−

^B^Dp^∗

C^m

(

z_j

) ≤

^Cp⁻

log

( |

^zj

|

²

)

^γ

.

(20)

To ﬁnally get(4),we use thefollowing observation:the elements of H⁰₍₂₎

(,

L^p

)

are,for p≥^2, ^exactly^the holomorphic sectionsofL^poverthewhole

vanishingonthepuncturedivisorD= {^a1

, . . . ,

aN}^,ⁱⁿ^particular^Bp

(

z

)

=^{0 at}^D^.

Acknowledgements

H.A.ispartiallysupported byANR-14-CE25-0010andX.M.ispartiallysupportedby ANR-14-CE25-0012-01andfunded through theInstitutionalStrategyoftheUniversity ofColognewithin theGermanExcellenceInitiative.G. M.was visiting UniversitéParisDiderot–Paris-7duringthecompletionofthisproject.

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