Bergman 核の問題 (複素幾何学の諸問題)

Bergman

核の問題

大沢健夫 (名大多元数理)

Tableof Contents

Introduction

\S 1. Prehmmaries–before and after theBergmankemel \S 2. Studiesontheboundarybehavior

\S 3.Asymptotic expansionin tensorpowers

\S 4.Variations inanalytic families.

\S 5.Bergmankemel and$L^{2}$extension

References

Introduction

The Bergmankemel,_{named after Stefan Bergman}(1895-1977), isby

definition thereproducingkernel of the

space

of$L^{2}$holomorphic n-forms

on any

connected n-dimensional complex manifold. Its significance in

complex geometry hasbeen$\mathscr{X}^{adual1}y$understoodthrough

many

spectacularworks in the lastcentury. Forinstance, C. Fefferman [F-1]

analyzed theboundary behavior of the Bergmankernel onstrongly

pseudoconvexdomains with $C^{\infty}$ boundary, and provedthatany

biholomorphic map between suchbounded domains in $C^{\mathfrak{n}}$ extends

smoothly to the closure. Recently, methods for analyzingthe Bergman

kemelbrought new insights into algebraic geomehy anddifferential

geometry (cf. [Siu-2-5], [Brn-P], [D] md[Mab-1,2]). Thepurpose ofthis

articleis to review some of the results onthe Bergmankemel with

geometric $back_{\mathscr{X}}ounds$, presenting

open

questions onthe

way.

\S 1. Preliminaries–beforeand afterthe Bergman kernel

The circle divisiontheory of C. F. Gauss (1777-1855),whichwas

discovered on1796/3/30, is a giantleap in mathematics and the first step

towards complex geometry. In theearly 19thcentury, it broughta new

progress

in the _theoryof elliptic integrals,which had been developed by

the work ofGauss,N.H.Abel (1802-29) was led atfirst toalgebraic insolvabilityof equations ofdegree 5, subsequently discovered thatthe

inversefunctions of elliptic

integrals are

nothin$g$butdoubly

periodic

analyticfunctions inone complexvariable (i.e. elliptic functions), and

eventually arrived at a remarkable characterization ofprincipaldivisors inthe theory of algebraicfunctions ofonevariable (Abel’s theorem). The

latter is now regarded as the startingpointof algebraic geometry.

As a generalization ofAbel’s theory

on

ellipticfunctions, thetheory of

multiply periodicfunctions was developed in several

variables

by G.

Jacobi

(1804-51), K. Weierstrass (1815-97) and B. Riemann (1826-66).

Onthe otherhand, inspite $of’m$importmt contribution ofH. Poincare

(1854-1912) onnormalfunctions and a subsequent workof S. Lefschetz

(1884-1972), it

was

notbefore the

appearance

of the celebrated

theory

of

W. V D. Hodge (1903-75) $[Ho]$, of harmonic

integrals on

K\"ahler

mamifolds, that Abel$|s$ theorem on algebraicfunctionsfound a

proper

contextin several variables. This delayis mainlybecause ofthe rackof

the viewpointof orthogonalprojectioninHilbert

spaces.

Recall that it

was onlyin 1899thatD. Hilbert (1862-1943) awoke Riemam$|s$ idea of

Dirichlet‘$s$ principlefrom a deep sleep (cf. [R] md [H]) and thatthe basic

representation theoremof F. Riesz (1880-1956) wasnotavailable until

1907. Another historicalremark is that sucha systemization of abstract

mathematics emerged only after detailed studies oforthogonal

polynomials in the 19thcentury.Anyway, it culmuinated ina general,

method of _{orthogonal projection by}H. Weyl (1885-1955). Weyl$|s$method

(cf. [W-1]) became the analytic base of the Hodge theory,which

was

later

combined with analytic sheaftheory by Kodaira (1915-97) [K-1,2]. That

Weyl anticipated a lot in this method had beenmodestly suggested in

[W-2]. The Bergmankernelwas bom around1922 (cf. [B] and [Bo]) in

such acircumstance.

To be more explicit about the orthogonalprojection andthe Bergman

kernel, let $D$ be the unitdisc centered at theorigin in thecomplex plane

with coordinate $z$, and let $L^{2}(\partial D)$ be the Hilbert

space

of $L^{2}$

complex-valuedfunctions on $\partial D$. Thentheintegral transform

$f(z)$ – _{$\frac{\{}{2\pi\Gamma-1}\int_{\partial D}\frac{f(\zeta)}{\zeta-z}d\zeta$}

of A.-L. Cauchy (1789-1857) gives an orthogonal projectionfrom$L^{2}(\partial D)$

onto the subspace offunctions whichare theboundaryvalues of

holomorphic functions on $D$ in the $L^{2}$ sense _$(i$

.

_$e$. $L^{2}$functions with

vanishingFouriercoefficients in thenegative

powers

of $\exp(i\arg z))$. Replacingthe integral along $\partial D$ by the integral on _$D$, oneis

led to the representationof the orthogonal projection from$L^{2}(D)$, the

space

of the complex-valued $L^{2}$ functions on $D$, onto the subspace

consisting of $L^{2}$holomorphicfunctions on D. Thecorresponding integral

transformis

$f(z)-$

$\frac{1}{\pi}\int_{D}\frac{f(\zeta)}{(|-\overline{\zeta}z)^{2}}d\lambda_{\zeta}$

.

Here $d\lambda_{\zeta}$ denotes the

Lebesgue measure.

Thefunction $Jt^{-1}(1-\overline{\zeta}z)^{-2}$ is

the Bergmankernelof $D$,where holomorphicl-forms on $D$ are

naturally identified withholomorphic functions on D.

Thus, fromtheviewpointof orthogonalprojection, theBergmankernel

isa brother of theCauchy kernel. Anadvantage of the Bergmankernelis

thatit

naturally

encodes geometric information. Letus recall how itdoes.

Let $M_{j}(\dot{|}^{=}1,2)$ be two complexmanifolds withBergmankemels $\iota$

$M’$

md let $0$ : $M_{1}arrow M_{2}$ be a biholomorphic

map.

Thenone has _$m$

$j$

equality

(1)

which follows easilyfrom the $def\ddot{m}tion$

.

Wenote thatthe equality (1)

already suggests ahnk between theboundarybehavior ofBergman

kemels andbiholomorphic

maps.

To

see

it

more

explicitly, taking

as

$M_{1}$

any

$s$imply connected

proper

subdomain $\Omega$ of _$C$, let _{$z_{0}\in\Omega,$ $M_{2}=$} D,

o

_{$(z_{0})=0$} and _{$Q^{1}(z_{0})>0$}, basedon

Riemann’s mapping theorem. Then, _letting -,$b^{=}*(\zeta,z)d\zeta$dz, itfollows

mmediatelyfrom (1) that

(2) $\sigma(z)=\sqrt{\frac{\pi}{k^{(z_{0f}z_{o})}}}\int_{z_{0}}^{z}*(\zeta, z_{0})d\zeta$

holds truefor

any

$z\in\Omega$

.

It is obviousfrom (2) that theboundary

regularity of $K_{\Omega}(\zeta, z_{0})$ implies that of $0$.

Efficiency

ofthis reduction lies

inthat, _{as we} shall see later, theregularity question on $K_{\Omega}$ canbe

transformed into a questiononthe canonical solution operatorfor the

complex Laplacian. This observationmightalready suggest the reader the

validity, andeven the method ofproof, of $Fefferman^{1}s$ theorem which was

mentioned in the introduction.

That$|s$ all for preliminaries. We shallnow

go

into the substantial

\S 2.

Studies

on

the boundarybehavior

Fromnow on, let $\Omega$ be

any

bounded domainin $C^{n}$ and let

$?\iota_{\Omega}=K_{\Omega}(z,w)2^{-n}dz_{1}\wedge\cdots\wedge dz_{r\iota}\otimes d\overline{w}_{1}\wedge\cdots\wedge d\overline{w}_{\mathfrak{n}\prime}$

where $z=(z_{1}, \ldots, z_{\mathfrak{n}})$ and $w=(w_{1}, \ldots, w_{VL})$. $K_{\Omega}(z,w)$ willbe referred

to as theBergman kernelfunction of $\Omega$.

An alternate definitionof $K_{\Omega}(z,w)$ is givenby the formula

(3) $K_{\Omega}(z,w)=\sum_{\simeq j1}^{\infty}e_{j}(z)\overline{e_{j}(w})$

where $\{e_{1}, e_{2}, \ldots\}$ is

any

complete orthonormal system of the

space,

say

$A^{2}(\Omega)$, of $L^{2}$holomorphicfunctions on $\Omega$ withrespect to the Lebesgue

measure.

For simplicity we put

(4) $K_{\Omega}(z)=K_{\Omega}(z,z)$.

Clearly $K_{\Omega}(z)$ is strictly plurisubharmonic and strictlypositive. It is

also

easy

to verifythat $\log \mathscr{K}(z)$ is strictly plurisubharmonic. The

complexHessianof log&(z), denotedby $\partial\overline{\partial}\log I\langle\Omega(z)$ by anabuse of

notation,is called theBergman metric of $\Omega$. Afact of basic importanceis

thatbiholomorphic maps areisometries with respectto theBergman

metric.

For thecase $\Omega=B^{\mathfrak{n}};=\{z,\cdot|z|<1\}$, where _{$|z|^{2}:=|z_{1}|^{2}+\cdots+|z_{n}|^{2}$},

onehas

(5) $K_{\Omega}(z,w)=\pi^{-n}n!(1-\prec z,w>)^{-7\iota-\{}$

where $<z,w>;=z_{1}\overline{w}_{1}+\cdots+z_{\mathfrak{n}}\overline{w}_{\tau\iota}$

.

The expression (5) is animmediate

consequence

of(1) oncethe

biholomorphic automorphisms of $B^{r\iota}$ are explicitlyknown.Although it is

usually difficulttocompute the Bergmankernels, it is obviousthatthe

Bergmanmetrics onbounded homogeneous domains are complete. We

note that thereexists a completeK\"ahler metric on $B^{n}-\{0\}$ andthatthe

Bergmanmetric on $B^{n}-\{0\}$ is not complete (cf. [G-1]).

One

way

ofdescribingthe boundarybehavior of $K_{\Omega}(z,w)$ is to

express

the singularityof $K_{\Omega}(z)$ as $zarrow\partial\Omega$ in terms of thefunction $6_{\Omega}(z)$ $:=$

$id\{| z- w | ; w\not\in\Omega\}$ and geometric invariants on $\partial\Omega$. Here, by

geometric

invariants on $\partial\Omega$, wemean locally defined

systems offunctions satisfying

under CR diffeomorphisms (cf. [Wk]).

Forthat

purpose,

thefollowingformulaisuseful.

(6) $K_{\Omega}(z)=\sup\{|f(z)|^{2};f(\zeta)\in A^{2}(\Omega)$ and $||f||=1\}$.

Here $||f||$ denotes the $L^{2}$

norm

of $f$

.

Note that the

supremum

is attained

bythe function $Kfi(\zeta,z)/\sqrt{K_{\Omega}(z)}$

.

Basically;what canbe done is to

approximate this function from the geometric databy employingthe

techniques of producing$L^{2}$holomorphic functions on $\Omega$.As sucha

technique, there is a method, dueto L. H\"ormander [H\"o-l], of solvingthe

inhomogeneous Cauchy-Riemann equation$\overline{\partial}u=v$with$L^{2}$normestimates

(see also [H\"o-2]). AsimilarmethodofA. Andreotti (1924-80) and E.

Vesentini in [A-V-1,2] is alsouseful.

Inthe

case

where $\Omega$ is

a

strongly pseudoconvex domain, it

was

proved

in [H\"o-l] that

(7) $z>z_{0}1\dot{\underline{u}}nK_{\Omega}(z)6_{\Omega}(z)^{\tau\iota+t}=n!\pi^{-\mathfrak{n}}L(z_{0})$

holds for

any

$z_{0}\cdot\in\partial\Omega$,where

(8) $L(z_{0})=\Omega\ni 2arrow Z_{0}$

Recall that $\Omega$ is calleda strongly pseudoconvexdomain iflocally $\partial\Omega$ can

be mappedto $C^{2}$ strictlyconvex hypersurfaces by appropriatechoices of

biholomorphic

maps.

$L(z_{0})$ is a geometric invarianton $\partial\Omega$ inthe above

mentioned sense.

Actually, (7) holds for

any

bounded pseudoconvexdomain with

$C^{2}$ boundary. Infact, the lefthand side of (7) vanishes if _{$L(z_{0})=0$}. This

canbe seenfromCauchy$|s$ estimate applied on a sequenceofpolydiscs in

$\Omega$ convergin

$g$to $z_{0}$.

Hence, _{strongpseudoconvexityof} $\partial\Omega$ at

$z_{0}$, i.e. the conditionthat $\partial\Omega$

becomes strictly convex at $z_{0}$ after some biholomorphiccoordinate

change,is characterizedby the condition that

(9) $\lim_{zarrow}\inf_{z_{0}}Q(z)6(z)^{\tau\iota+4}>0$

holds true.

Onthe other hand, it isimplicitly contained in [Oh-2] that theLevi

flatness of $\partial\Omega$ on aneighbourhood

$U\ni z_{0}(U\subset\partial\Omega)$, i.e. theproperty

(10) $\lim_{\xiarrow}\sup_{z}K_{\Omega}(\zeta)@(\zeta)^{2}<\infty$

holds for

any

$z\in$ U.

For the general smoothpseudoconvex domains, $||$

buildingblocks ofthe

singularity“ of $K_{\Omega}(z)$ have been studied case by

case

(cf. [Oh-l],

[D-H-Oh], [D-H], [B-S-Y], [Km]; [Ch-Km-Oh]$)$.

Nextwe shall discussthe boundarybehaviorof the Bergmankernel

on

pseudoconvex domains fromslightly

more

analytic viewpoint.

Although the motivation ofBergman$|s$ thesis was to introduce a

new

method inthe

theory

ofpotentials andconformal mappings, it

was soon

recognized thatanalysis of theBergman kernel wouldplay animportant

role in severalcomplex variables, too,forinstance to solve the Levi

problem(cf. [H\"o-3]). (Recall that the Levi problemasks whether

or

not

every

pseudoconvex domain isholomorphicallyconvex.)

Indeed, it is

easy

to

see

that $\Omega$ isholomorphically convex if

(11) $\lim_{z\Rightarrow\partial\Omega}K_{\Omega}(z)=\infty$

holds true. The converse is false becausethe punctured disc $D-\{0\}$ is a

counterexample. Forthe domains in $C$, itrecently turned outthat (11) is

equivalentto certaingrowth property of thelogarithmic capacity function on $\Omega$ (cf. [Zw-2]).

Concerningthe Levi problem, whichwas the principal questionin

severalcomplex variables for some time, _KiyoshiOka (1901-78) firstcame

up

with a solutionby the strategy of exhaustin$g$pseudoconvex domains

by stronglypseudoconvex ones, _{constructingholomorphic} functions on

strongly pseudoconvex domains by patching locallydefined ones by

solving $Cou\sin^{I}s$problem, and approximating themby globally defined

ones by a theoremofRunge type (cf.[O]). However, all these arguments

areindependent of theBergmmkernel.

Acounterpart of Oka$|s$ theorem oncompact manifolds was established

by Kodairaby the method ofharmonic integrals (cf. [K-1.2]).

After animportmtworkof C. B. Morrey (1907-84) (cf. [M]), the method

of Okawas extendedby H. Grauert[G-2] to

prove

that strongly

pseudoconvex domains incomplex manifoldsareholomorphically

convex, and Kodaira$|s$ methodwas extended in [A-V-1,2], [Kh] md

[H6-1] to yield a powerful methodof directlyand effectivelyreaching

the basic existencetheorems in severalcomplexvariables. Especially, itis

remarkable that [H\"o-l]

gave

asimple alternateproof toOka$\dagger_{S}$ theorem

byestablishinga quantitative solutionto the additive Cousinproblem by

the method of$L^{2}$estimates for the $\overline{\partial}$

-operator. Here the $\overline{\partial}$

a

closedlinear operator from$L^{2}(\Omega)$ to $\oplus^{\iota}L^{2}(\Omega)\gamma$ defined

by

$\overline{\partial}f=(\partial f/\partial\overline{z}_{1},$ . .

., $\partial f/\partial\overline{z}_{n})$ on Dom $\overline{\partial}=\{f, \partial f/\partial\overline{z}_{j}\in L^{2}(\Omega),j=1,2,\ldots,n\}$ .

Since $A^{2}(\Omega)=Ker\overline{\partial},$ $\mathscr{K}$ involves operatortheoreticinformation on

$\overline{\partial}$ as we shall see later.

The $\overline{\partial}$

-operator isnaturally extendedto $L^{2}$differential forms givingrise

to a complex. Generalizingthe situation tothe $L^{2}$

spaces

withrespectto

arbitrary measures, $L^{2}$ estimates

are

formulated

as inequalities

of theform

(12) $||u||\leq$ const $(||\overline{\partial}u||+||\overline{\partial}^{\star}u||)$,

where $\overline{\partial}^{\star}$

denotes the Hilbert

space

adjoint

of $\overline{\partial}.L^{2}$

estimates that work

inseveral complex variables

were planned

byP. R. Garabedian

(1927-2010) and D. C. Spencer (1912-2001) [G-S]. Based onthe idea of

orthogonal projectionandpushed by thecomplete solution ofthe Levi

problem for the domain$s$ over $C^{\mathfrak{n}}$(cf. [O], [Br] and [Ng]), theplan was

realized inthe abovementioned

papers.

Anadvantage ofthis methodisthat the

passage

tohimits isquite

easy,

so thatone has effectiveexistencetheorems on general pseudoconvex

domains. (7) was obtainedas an application ofthismethod.

Inspiredbythe success of this approach, Skoda [S] and

Ohsawa-Takegoshi [Oh-T] establishedrespectively the $L^{2}$varimtsof Oka’s

division theorem and extensiontheorem. The method of [Oh-T]

_was

fluencedby [D-F] and [Wi].

Skoda$|sL^{2}$ division$\backslash theorem$was applied by Pflug [P] to show that (11)

holds if $\Omega$ is a pseudoconvex domain satisfyingthe “generalized cone

condition“ (see [P] for the definition). Moreoverit tuned outlater thatthe

same

technique is availableto show,under the assumptionthat $\partial(\Omega\cup\partial\Omega)$

$=\partial\Omega$, that $\Omega$ is pseudoconvex ifand only ifit carries a completeK\"ahler

metric (cf. [D-P]).

Onthe otherhand, _{by applying}the $L^{2}$ extensiontheoremin [Oh-T],it

was shown in [Oh-3] that (11) holds if $\Omega$ is hyperconvex,i.e. if $\Omega$ admits

a bounded plurisubharmonic exhaustion function.

Itis wellknown thata bounded domainin $C$ ishyperconvex ifand

onlyifits

boundary

points

are

regularwith respectto the $D\ddot{m}chlet$

problem (forthe regularity of theboundary pointsinthis sense,

_see

[Kishi] forinstance). Wenotethat (11) holds onsomenon-hyperconvex

domains, e.

g.

on $\{(z,w)\in C^{l};|z|<1,$ $|w|\prec 1$ and $|z|\prec|w|\}$,

so

that hyperconvexityis considered to bea morenatural condition than (11).

Pluripotential theory, including the existence ofpluricomplexGreen

function andLelong-Jensen measure, hasbeen developed on

hyperconvex domains. Here, to be analyzed as the severalvariables

versionoftheLaplace operator is theMonge-Amp\‘ere operator(cf. [Klm]

discussed on hyperconvex domains (cf. [P-S]).

The condition (11) is

very

close to the completeness of theBergmm

metric on hyperconvex manifolds. Such a linkwasfirst observed in [Kb]

by identifyingthe Bergmanmetric with the pull-back of theFubini-Study

metric

on

the projectivizationof the

topological

dual$A^{2}(\Omega)^{\star}$of $A^{2}(\Omega)$,

say

$P(A^{2}(\Omega)^{\star})$, by thecanonically defined holomorphic embedding

(13) $\iota:\Omega-P(A^{2}(\Omega)^{\star})$

$w$ $(U$

$z|arrow\{m\in A^{2}(\Omega)^{\star}-\{0\};m(f)=0 if f(z)=0\}$

.

By thisidentification, _denotingthe distmce between $\iota(z)$ and $\iota(w)$ by

$|$z,w $|$, onehas

(14) z,w $|=$ Arctm $\frac{\sqrt{K_{\Omega}(Z)K_{\Omega}(W)-|K_{\Omega^{(z_{\text{ノ}}w)1^{2}}}}}{|K_{\Omega^{(Z,W)|}}}$

Thefollowing estimate,whichis essentially equivalenttoKobayashi’s

criterion forthe completeness of theBergmanmetric, follows from (14).

(15) $|$z,w$| \geq\min(1/2, \sup\{|f(z)|^{2}/K_{\Omega}(z),\cdot f\in A^{2}(\Omega), ||f||=1 and f(w)=0\})$.

(See also [Oh-8]).

Combinin$g(15)$ witha recently developed technique of estimatin$g$

integrals oftype $\int_{\Omega_{\sim}}|u|^{\tau\iota}(\partial\overline{\partial}v)^{\mathfrak{n}}$, Bfocki-Pflug [B-P] and Herbort [Hb]

independently proved thatthe Bergmanmetric is completeif $\Omega$ is

hyperconvex. Itis known that there existnon-hyperconvex domain$s$in $C$ ,

whose Bergmanmetrics are complete (cf. [Zw-l,Theorem 5]).

The Bergmm metric ona connectedn-dimensionalcomplex mamifold

$M$ is defined in the same

way

as above via the

map

(13),by takin$g$the

space

of$L^{2}$ holomorphicn-forms instead of $A^{2}(\Omega)$, as long as the

map

corresponding to $\iota$ is animmersion. Acomplex manifoldis called

hyperconvexif itcarries a bounded strictlyplurisubharmonic exhaustion

function. Itis easily seen bythe $L^{2}$method that

every

hyperconvex

manifold carries a Bergmanmetric. In [Ch], thecompleteness result of $[B-$

$P]$ and [Hb] was generalized tohyperconvexmanifolds.

Inview of thefact that singularities of$L^{2}$

holomorphic

functions

are

negligible if their Hausdorff dimensionis not greaterthan $2n-2$, it

$Q1_{--}^{*\underline{*}}1)Let\Omega be$

_{a$propersubdomainofB^{n}$}

_.

_{$How^{\dagger t}smal1^{\dagger I}(inthesenseof$}

Hausdorffdimension,forinstance)can $\partial\Omega\cap B^{\mathfrak{n}}$beif theBergman

metric of $\Omega$ is complete?

2$)$ Is thereapropersubdomain $D$ of acompact complex manifold $M$

without nonconstant boundedplurisubharmonicfunctions such that

theBergmanmetric of $D$ is complete?

As for 1),

case

studies based

on

the analysis of Cauchykernel shouldbe

possible at leastfor$n=1$. As arelatedresult, see [An].

Boundarybehavior ofthe Bergmanmetric on strongly pseudoconvex

domains wasfirst described byK. Diederich (cf. [Di]). As wellas the

Bergmanmetric onthe model domain $B^{\mathfrak{n}}$, the

Bergmanmetrics

on

strongly pseudoconvex domains are completeK\"ahlermetrics. Afamous

result of Lu Qi-Keng [L]

says

that $\Omega$ is biholomorphically equivalentto

$B^{\mathfrak{n}}$ if the

Bergmanmetricon $\Omega$ is complete and ofconstantholomorphic

sectional curvature. Anaturalquestion askedby S.-Y.

Cheng

[Chg] is

whether ornot $\Omega$ is equivalentto $B^{\tau\iota}$ if the Bergmanmetric on $\Omega$ is

K\"ahler-Einstein. _{ByFu andWong} [F-W], this

was

answeredaffirmatively

when $\Omega$ is simply connected and $n\leq 2$ . Recently, itwas pointedoutby

Nemirovski and Shafikov [N-S] that Cheng$\{s$ conjecturefollows bomthe

Ramadmov conjecture (see Q5 below), so thatthe resultof Fu andWong

holds without assuningthat $\Omega$ is simplyconmected.

When $\Omega$ is not strongly pseudoconvex, more

case

studies seemtobe

necessary

in ordertofind how the geometry of $\Omega$ and $\partial\Omega$ determines

the Bergmankernel. For instmce, in viewof the fact thatonecan

characterize the strongpseudoconvexity of $\Omega$ in terms oftheboundary

behavior of the Bergman metric (cf. [Kl] and [Di-Oh-l]), the author would like to askthe following question.

$Q—————–Let\Omega beabo\overline{undedpseudoconvexdomaininC^{\mathfrak{n}}withC^{2}boundarymd}$

let $z_{0}\in\partial\Omega$

.

Isittruethat $\partial\Omega$ is Leviflatnear

$z_{0}$ ifand only if there exists

aneighbourhood $U$ of $z_{0}$ in $\partial\Omega$ such that

(16) linsup $|\xi|^{-2}|<\partial\overline{\partial}\log \mathscr{K},\xi\otimes\xi>-\delta(\zeta)^{-a}|\xi\delta|^{2}|<\infty$

$\zeta|arrow z$

holds forany $z\in U$ and forany nonzeroholomorphic tangentvector $\xi$ of $C^{\iota}$

This should notbe too difficult because itis already knownby [D-F-H] and [C-1] that (16) doesnotholdif $\partial\Omega$ is offinite type at _$z$.

Arelated questionwas raised in [Di-Oh-2] and [Oh-7] on theeffective

estimate ofthe distance function. Letus putit here ina more idealized

form;

$Q3(conjecture)————–$

-$**\kappa Let\Omega beaboundedp$

seudoconvexdomainin $C^{11}$ with $C^{1}$ boundary,

let d(z,w) be the distancebetween $z$ and $w$ withrespectto $\partial\overline{\partial}\log \mathbb{R}$, and let $z_{0}\in\Omega$ beany point. Then

(17) $\lim d(z0,w)/|\log\delta(w)|=1$

.

$w+\partial\Omega$

An estimate obtained in [Di-Oh-2] is weaker than (17),but stillgives a

quantitative completeness result for theBergmanmetric. Itwas improved

by Bfocki [Bf]. We note that the infimitesimal variant of (17) askedin

[Oh-7] _{was negatively} solved (cf. [D-H-2]).

Note 1. Ifwerestrict ourselves to aclass ofboundedhomogeneous

domains, itwas shownby Nomura [Nm] that bounded symmetric

domains canbe characterizedby a property of theBergman kernel,

_e.g.

the commutativity ofthe Laplacianwithrespecttothe Bergmanmetric

and the Berezin transform. (See also [En].) It is known thata complex

mamifold equippedwith the Bergmmmetric is homogeneous if and only

ifit is equivalentto a bounded homogeneous domain (cf. [PS]).

Exploitingthe fact that

every

boundedhomogeneous domainis

equivalentto a domain onwhich a setof affine trmsformations acts

transitively (cf. [V-G-PS]), it is

easy

to see thatboundedhomogeneous

domains arehyperconvex (cf. [K-Oh]. See also [Dn-3] and [Is]). A

longstanding open questioniswhether ornot,for the n-dimensional

bounded homogeneous domains, the $L^{2}\partial$-cohomology

groups

of

type

(p,q) withrespect tothe Bergmanmetricare allinfinite dimensional if

$p+q=n$. For the bounded symmetricdomains, the assertionwas

verified by Gromov [Gm]. Recently, Ishi-Yamaji [I-Y] showed that the

Bergmanmetric ofa bounded homogeneous domain is thepull-back of

that ofa bounded symmetric domainby a canonically defined

embedding.

Goingbackto thetheory of Oka andGrauert, the Levi problem on

modifications because otherwisethere existcounterexamples (cf. [G-3], [Siu-l] and [Oh-10]$)$. Generally speaking,with all conceivablynatural

settings, the Levi

problem

is stillfar reaching

on complex

manifolds. Amongthetough questions of this kin$d$, the Shafarevich conjectUre is

most attracting. Itasks whether or nottheuniversalcovering

space

of

my

compact Kffier manifoldis holomorphically

convex.

It is remarkable that

arecent partial

answer

to itby Robert Treger [Tr-1,2] is based onthe

analysis ofthe Bergmankernel.

optimuistically speakin$g$, it is not only challengingbutalsoprofitableto

exploremethods tocharacterize the domain$s$ ofexistence ofmalytic

functions on complex manifolds,because theywilllead usto

new

boundary valueproblems withrich contents.

Inthis $sp\ddot{m}t$, it

may

be alsoworthwhile to considerrefined Levi

problems oncomplex manifolds. Forinstance, let $M$ be a complex

manifold equippedwithavolume form $dV$, let $D$ beabounded domain

in $M$, and let $\Delta$ be theembeddingof $D$ into $D\cross D$ as the diagonal.

$Q4^{*_{---}}---$

Does $\lim_{z\cdot r\partial \mathcal{D}}\Delta^{*}\%/dV=\infty$ hold if

$D$ ishyperconvex?

Let

CP“

denote thecomplex projective

space

of dimension $n$

.

For $CP^{n}$,

itis knownthat

every

pseudoconvex

proper

subdomainishyperconvex

(cf. [Oh-S]). The abovequestionis

open

evenin such arestrictive

situation. The maindfficultyis thatthe hyperconvexity of the domain $D$

does notimplythe existence of a strictly plurisubharmonic functionon a

neighbourhood of $\partial D$. More precisely, it isfalsein general (cf. [Di-Oh-4])

andnotknownevenif $D$ is a domainin CP$\mathfrak{n}(n\neq 1)$. Nevertheless, itis

known that the $\overline{\partial}$

-equationsfor (n,q)-forms

are

solvablewith $L^{2}$norm

estimates for all $q$ on

any

pseudoconvex

proper

subdomainwith $C^{2}$

boundary

in $CP^{\eta,}$(cf. [c-aw]. See also [H-I] and [Brn-Cha]). So, the

solution forthe

case

$M=$

CP’

shouldnotbe too difficultand

may

clarify

the essential part of Q4. Concerming the related questions,

see

also

[Di-Oh-3] and [M-Oh].

More intricaterelationship between $K_{\Omega}(z)$ and geometric invariants

on $\partial\Omega$ canbe explored when $\partial\Omega$ is $C^{\infty}$ and everywhere strongly

pseudoconvex. A_{groundbreaking}resultinthis directionwas a theorem

ofFefferman [F-1] assuring that there existtwo $C^{\infty}$ functions

defined

on a neighbourhood

of $\partial\Omega$ such that

(18) $K_{\Omega}(z)=\varphi(z)@(z)^{-\uparrow 1-\{}+\psi(z)\log 6(z)$

holds

near

$\partial\Omega$. (7) implies that $\varphi(z_{0})=n!\pi^{-t1}L(z_{0})$ for

any

$z_{0}\in\partial\Omega$.

Geometric invariants besides $L(z_{0})$ areinvolved inthe coefficients of

the asymptotic expmsions of $\varphi$ and $\psi$ in

6

(their expression canbe made

simpler after some rescaling),which have beeninvestigated byFefferman

[F-2], Bailey-Eastwood-Graham [B-E-G] and Hirachi [Hr-l].

The

following was

a

very

famous questionknown

as

the Ramadanov

conjecture (cf. [Rm]).

$Q\not\in*_{-}\underline{\lambda}\underline{*})_{---}Let\Omega beastrong1ypseudoco$

nvexdomain with $C^{\infty}$ boundary,$md$let

$z0\in$

$\partial\Omega$

.

Suppose that there exists aneighbourhood _{$V\ni z_{0}$} in $C^{\tau\iota}$ such

that $\psi=0$ on V.Then,is $\partial\Omega$ spherical around

$z0$ ?Namely,is therea

neighbourhood $W\ni z0$ anda $C^{\infty}$diffeomorphism $\Phi$ from $\Omega$

nw

onto

$B^{n}\cap[z,$ ${\rm Re} z>1-\epsilon\}$ forsome $\epsilon$ suchthat $\Phi|\Omega\cap W$ isholomorphic?

The

answer

is

yes

if $n\leq 2$ (cf. [BM], [G] and [Bu]) andturned out to be

no if $n\geq 3$ (cf. [E-Z] and [Hr-2]). However,it is notknownwhether

or

not the conclusionholds ifone strengthensthe assumptionto $|\dagger_{\psi=0}$ on a

neighbourhood of $\partial\Omega^{\dagger\dagger}$

.

Fefferman applied (18)in [F-1] to analyzethe geodesicswithrespectto

the Bergmanmetric, which is infact a

very

hard work.

Another effective

way

of describingthe boundarybehavior of $K_{\Omega(z,w)}$

is in terms of the operatortheoreticproperties ofthe orthogonal

projection,

say

$P_{\Omega}$ ,from the

space

$L^{2}(\Omega)$ onto $A^{2}(\Omega)$. $P_{\Omega}$ is calledthe

Bergmm projection.

The principal questionin this settingis whether ornot

(19) $P_{\Omega}(C^{\infty}(\overline{\Omega}))\subset C^{\infty}(\overline{\Omega})$ holds hue,where $\overline{\Omega}$ _{$:=\Omega\cup\partial\Omega$}

and $C^{\infty}(\overline{\Omega})$ denotes the setof $C^{\infty}$

functions on $\overline{\Omega}$.

(19) is called “condition $R$“ byS. Bell [Bl].

Itturned out that the property (19) is directly linkedtothe smooth

extendibilityofbiholomorphic

maps.

Itis actually

very

efficient,because

bythis method itispossible to generalize theresults to

proper

$C]$ and [B-B-C]. See also [Oh-7]$)$.

If $\Omega=B^{\mathfrak{n}},$ (19)

can

beverifieddirectly

by

using (5) (cf. [Cha] and

[L-$M])$. Inorder to

generalize

this to

more

generalclasses ofpseudoconvex

domains with $C^{\infty}$ boundary, anatural method is toconvert (19) into the

property of mother operator $N_{\Omega}$ by usingKohn$|s$formula

(20) $E=Id-\overline{\partial}^{*m5}$.

Here $N_{g}L$ denotes the inverse of

$\overline{\partial}\overline{\partial}^{\star}$

on the imageof $\overline{\partial}.N_{\Omega}$ iscalled the

Neumann operator. The Neumamoperator existsbecause $\Omega$ is

pseudoconvex (cf. [C-1]. See also [Oh-9]).

By such

an argument,

(19)

can

beverified for the domains of finite

type.

The point isthatsubelliptic estimates hold onthem (cf. [C-1]).

Onthe other hmd;it is known that (19) is satisfiedby certaindomain$s$

of infinitetype. Forimstance, (19) holds whenever $\Omega$ is

a

complete

Remhardtpseudoconvexdomain with smoothboundary (cf. [B-B]).

By [K-N], itisknown that (19) is a

consequence

ofthecompactness of

$N_{\Omega}$. If $\Omega$ is convex, Fu andStraube [F-S] provedthat thecompactness of

$N_{\Omega}$ is equivalenttothe condition that $\partial\Omega$ does notcontain

any

complex

curve.

For theproof, theboundarybehavior of $\mathscr{W}(z)$ is analyzed.Inthis

context, domains for which (19) doesnothold arealso ofconsiderable

interest (cf. [Ba] md [Chr]).

Thus,

_{as a}

state ofart,

we

understand

a general tendency

thatthe

existence ofa complex curve intheboundary destroys the

regularity

properties of $P_{\Omega}$ and $N_{\Omega}$ . So, it

may

be worthwhileto extend

Fu-Straube$|s$ theoremto moregeneral domains. Acandidate is the class of

lineally convex domains. Recallthat $\Omega$ is said to be lineally convexif

every

point $z_{0}\in\partial\Omega$ is containedin a complex hyperplane $H=H(z_{0})$

whichdoes notintersectwith $\Omega$.

$Q6^{\underline{*}}$

_Supposethat $\Omega$ is lineallyconvex. Isithuethat

$N_{\Omega}$ iscompact if

and onlyif $\partial\Omega$ does notcontainanycomplexcurve ?

\S 3. Asymptotic expansion in tensor powers

We shall nowreview some results on the asymptotics of the

generalized Bergmankernels fortensor

powers

ofpositive linebundles,

as the power tendsto infinity. Motivationfor consideringsucha question

[Dn-l] and a supersymmetric fieldtheory [Wi] (see [Dm-3] forinstance).

Let $M$ be a connected complexmamifold of dimension $n$ and let $E$ be

a holomorphicline bundle

over

M. The cmonical line bundle of $M$ will

be denotedby $0$)

$M^{\cdot}$ Givena fiber metric

$h$ of $E$,we denoteby _{$A^{2}(E\otimes(0_{N})$}

the space of $L^{2}$ holomorphic sectionsof $E\otimes Q$₎

$M$ withrespectto

$h$. The

reproducingkernel of $A^{2}(E\otimes(0_{M})$ willbedenotedby $7t_{k}$

.

Let $\Delta$ be the diagonal embeddingfrom $M$ into $M\cross$M. Then $\Delta^{*}\uparrow\{h$

is a sectionof$E\otimes 0)_{MM}\otimes\overline{E\otimes 0)}.\uparrow\{\}_{t}$ and $\Delta^{*}n_{h}$ are calledthe

weighted

Bergmankernels, whereweshall allow $h$ to be locally of the form $e$

$-\varphi$

for a locally integrablefunction $\varphi$

.

Such ageneralized fiber metric is

called asingular fiber metric. Similarly as theBergman kemelfunction,

the weighted Bergmankernelfunctionis defined whenever

a

trivialization of the canonical bundle exists andis fixed.We shall denote

itby $K_{\varphi}$ if $h=e^{-\varphi}$ . Generally, the product $h\cdot\Delta^{*}\iota_{1_{1}}$

, , being a section of $0)\otimes\overline{(o}M\triangleright t$, canbe writtenas $\rho_{k}dV$, where $dV$ is a volume form and $Q_{k}$ is

a nonnegative function. $\rho\}_{\iota}$ measures the size of the Bergman kernelwith

respectto $dV$

.

Asinthecase of the Bergmankernelfunction, the value of $\rho_{h}$ at $z_{0}$

is characterized as the

supremum

of the squared

length

at $z_{0}$ bf$L^{2}$

holomorphic sections of $E\otimes\omega_{M}$ with$L^{2}$norm one.

Similarly as inthe proofof (7),Bouche [Bou] provedthat

(21) $\lim_{\ln\Rightarrow\infty}g_{k^{\mathfrak{n}\iota}}^{t/m}=1$

holds if $M$ is compactand $h$ is $C^{\infty}$ md ofpositivecurvature, by

extending awork of Tian[Ti], wheretheHodge metric is approximated

by $1/m$ timesthe curvature formof $(\Delta^{\star}7t_{\iota^{m}})^{-\tau}$. The model casefor (21) is the anti-tautological line bundle over CP“ equippedwith the fiber metric

induced from the euclidean metric of $C^{\mathfrak{n}+t}$

. Although themethod is

similar as inthe estimate of the Bergmankernel,whatis approximatedis

reversed here. Namely, thefiber metric is approximated bythe m-throots

of the Bergmmkernels.

In the

same

spirit, Demailly [Dm-2] applied the $L^{2}$ extension theorem to

approximate

any

plurisubharmonic function andits Lelongnumber in

terms ofthe weightedBergman kernels. Recall that,for

any

plurisubharmonic function $\varphi$ on a domain

$\Omega$ and for

any

point _{$z_{0}\in\Omega$},

the Lelongnumber $v(\varphi,z_{0})$ of $\varphi$ at $z_{0}$ is defined by

Theorem 1. (cf.[Dm-2]) Let $\Omega$ be a boundedpseudoconvex domain

in $C^{\mathfrak{n}}$. Thenthere existconstants _{$C_{1}$} and

C2

dependin

$g$only

on

$n$ and

the diameter of $\Omega$ such that thefollowinghold for

my

plurisubharmonic

function $\varphi$ on

$\Omega$ md for

any

positiveinteger _$m$.

(22) $\varphi(z)-C_{1}/m\leq(2m)^{-1}\log K_{2m\varphi}(z)\leq\sup\varphi(\zeta)+m^{-1}\log(C_{2}/r^{\tau\iota})$ $|(-z|<r$

if$z\in\Omega$ and_$r<6(z)$.

(23) $v(\varphi,z_{0})-n/m\leq v((2m)^{-1}\log K_{2m\varphi}(z), z_{0})\leq v(\varphi,z_{0})$, $z_{0}\in\Omega$.

Since (23) is a comparisonbetween $2m\varphi$ md $\log K_{2m\varphi}$ near $\varphi=-\infty$,

one

may

naturally

askits counterpart

near

$\varphi=\infty$

.

$Q————-$

Let $\varphi$ be a plurisubharmonic functionon aboundedpseudoconvexdomain

$\Omega$ in $C^{\mathfrak{n}}$

.

Istherea constant _$C$ such

$that\int_{oe\varphi}(i\partial\partial\varphi(z))^{n}<RcJ(i\partial\overline{\partial}\log \mathscr{K}(z))^{t}$ forall $R$?

$———————-arrow—\underline{0}\leq\varphi_{-}\sigma_{-}\underline{\hslash}_{---arrow---}$

Cathn [C-2] and Zelditch [Z] reversed againtherole of $h$ and $\iota_{h}$ in

the approximation and established the asymptotics of $\Delta^{\star},\iota_{h^{m}}$ in _$m$ as a

counterpartof (18). The spirit isto construct theorthogonalprojection

explicitlyfrom thegeometric data. Itwill be statedbelow, wherethe

factor $0)_{M}$ is notexplicitlyinvolved, for

simplicity.

Let $M$ and (E,h) be as above and let $dV$ be

my

$C^{\infty}$volume formon

M. By A(E) wedenote the

space

ofholomorphic sections ofE. Thelength

of$s\in A(E)$ withrespectto $h$ willbe denotedby $|s|_{h}$ .

Thenwe put

11

$s||^{2}=\int_{M}|s|_{k}^{2}dV$,

$\rho(dV,h)(x)=\sup\{|s(x)|^{2}/||s||^{2}, s\in A(E)-\{0\}\}$,

$\beta(dV,h)=Q(dV,h)dV/d\dot{m}A(E)$, whenever $A(E)\neq\{0\}$

and

$\Theta=$ thecurvatureform of $h$.

$\beta(dV,h)$ is a probabihty

measure

on $M$ whichis cmonically associatedto

Theorem 2. (cf. [C-2] and [Z]) Inthe above situation

suppose

that $h$

is $C^{\infty}$ and of positive curvature, then there exist $C^{\infty}$ functions

$b_{j}$ on $M$

such that

(24) $\beta(dV,h^{m})=\sum^{\infty}b_{j}m^{n-j}dV$, _{$b_{0}dV=$$MA$$(h)/V$}

j-holds asymptoticallyin $m$, where MA(h) $:=(\sqrt{-1}\Theta)^{\mathfrak{n}}$ and V:

$= \int_{M}(\sqrt{-1}\Theta)^{\mathfrak{n}}$

.

Catlin$|s$proof depends onthe malysis ofthe Bergmankernel of the disc

bundle associated to $E^{\star}$, the dual of $E$, and Zelditch$|s$onthat of theSzeg\"o

kernel. Their approaches

are

both naturalbecause, _letting $T$ be theumit

discbundle associated to $E^{\star}$,which is a tubularneighbourhood ofthe

zero section, $\rho(dV,h^{m})$ are naturallyidentifiedwith the diagonalized

reproducingkernels of (relatively small) subspaces of $L^{2}(T)$ or $L^{2}(\partial T)$ consistingofholomorphic functions ofrestricted typeon T. However,

their

common

toolis the microlocal method of Boutetde Monvel and

Sj\"ostrand [BM-S] which extends [F-1]. For$m$elementary proofof

Theorem2, _see [B-B-S].

Inview of such a closerelationshipbetween (18) and (24), itseems

natural to ask acounterpart of the Ramadanov conjecture in this context.

For instance, if $M=$ CP 1 _and $E$ is the anti-tautologicalbundle, itis

easy

to see that the fiber metric $h$ is determinedby $b_{0}$

up

toa constant

factor. When $M=$ CP” and$n>1$, itbecomesmore difficulttoformulate

the question. Of courseit willbe even more difficultwhen $M$ and $E$

are

notfixed in advance. Wenote that, accordingtothework ofZiqinLu

[Lu], $\beta(dV,h^{\pi\iota})$ looks like the stress

energy

in Einstein’s equation.

In sucha

way,

theBergmankernelis related toalgebraic geometry and

differential geometry through(pluri-)potential theory. Itis remarkable

that Theorem 2was appliedby Donaldson [D] to the stabihty theory of

projectively embedded manifolds (see also [Mab-1,2]). Motivatedby

Donaldson$|s$ work, Theorem 2

was

extended to a

more

generalcontext of

symplecticmamifolds and orbifolds (cf. [D-L-M]). But letuswaitfor

mother opportunityto enterthis fancy topic.

\S 4. Variations in analyticfamilies

Returningto theformula (2), itsuggests, as wellas Fefferman’s

theorem, _that

_any

$C^{r}$family ofbiholomorphic maps say

$\{\alpha_{t}\}_{0<t<t}$ , from

a $C^{\infty}$family of $C^{\infty}$ stronglypseudoconvex domains

$\{\Omega_{t}\}_{0<t<I}$ in $C^{\mathfrak{n}}$to

mother $C^{\infty}$famuily

$\{\Omega_{t}^{I}\}_{0<t<1}$ in $C^{\tau\iota}$, extends to the boundaries also

smoothlyin $t$.

extendibihity

of $\{\alpha_{t}\}$ is

reduced

tothe smoothness in $t$ of the

Bergman

projections, whichcan beverified,via Kohn’s formula, _{by checking}the

corresponding property of the family ofNeumannoperators (cf. [G-K]).

Hence, (18) relates theBergman kernel of $\Omega$ notonlyto geometric

invariants of $\partial\Omega$,but also to their variations. So does (24) similarly.Thus

the Bergman kernelis linked notonly to the Levi problem,but alsoto the

moduliproblem. Inthis sense, variational questionsfor the Bergman

kernels on complex analyticfamilies

are

particularly interesting. Let us

review someresults in this direction.

Let $\mathcal{J}4$ bea connected

complex

manif\‘old, let $U$ be

a

domainin

$C^{\tau \mathfrak{n}}$

and let $\pi:\mathcal{J}4arrow U$ be a $su\dot{\eta}$ective

holomorphic

map

such that $d\pi$ is

everywhere of maximal rank. Thefamily of the Bergmankemels onthe

fibers of $\pi$ is called the relativeBergman kernel

on

$B4$. For

any

$\zeta\in U$

weput $n_{\zeta}=\Delta^{\star}u_{\pi^{-I}(\zeta)}$. We shallassume, for simplicity, that $n_{\zeta}$ isnot

everywhere zero. Thenthe collection $\{n_{\zeta}^{-r}\}_{\zeta\epsilon U}$ isnaturally regarded as a

singular fiber metric of $\omega_{\mathcal{M}/U}$ , where weput

%/U

$:=0_{\mathcal{M}}$)

$\otimes(\pi^{\star}\omega_{U})^{\star}$. We put $\beta_{\backslash N/U}=\{,c_{\zeta}^{-1}\}_{\zeta\in U}$.

There are two modelcases :

1$)$ $J4=B^{n+m},$ $U=B^{m}$ and $\pi(z)=z^{||}$, where $z=(z^{1},z^{||})$. Inthiscase, the

curvature form of $\beta_{\mathcal{M}/U}$is

$-\partial\overline{\partial}(\log(1-|z^{||}|^{2})^{\mathfrak{n}}+\log(1-|z^{1}|^{2})^{n+1})$.

Obviouslyit ispositive on $M$.

2$)$ $M=\perp_{z}\perp C^{n}/\Gamma_{Z}$ (disjoint union), where $Z$

runs

throughthe set

$End^{\star}C^{\mathfrak{n}}$ _{$:=\{Z\in EndC^{\mathfrak{n}}$}, det ImZ _$>0\}$ and $\Gamma_{Z}$ stands for the lattice in $C^{n}$

generated bythe columns of the$n\cross n$unit matrix and those of Z. Here,

End $C^{\mathfrak{n}}$, the set of

complex endomorphisms of $C^{n}$, is naturally identified

with the set of$n\cross n$matrices whose entries are complex numbers. Thenwe

put $U=End^{\dagger}C^{\mathfrak{n}}$and

$\pi(q)=Z$ for

my

$q\in C^{n}/\Gamma_{Z}$

.

Inthis case,the

curvature formof $\beta_{\mathcal{M}/U}$is $-\partial\overline{\partial}\log(\det{\rm Im} Z)$, whichis easily seento be not

semipositive on $f4$ if $n>1$.

As is wellknown, $-\log$(det Im Z) becomes strictly plurisubharmonic

when itis restricted to the set of those $Z$ for which ${}^{t}Z=Z$ md${\rm Im} Z$is

$Q8_{---}^{*}Compute$

the signatureof $\partial\overline{\partial}\log(\det{\rm Im} Z)$

.

It issurprising thatnothing general about $\beta_{MU}$was knownin the last century, althoughthe semipositivity properties of the directimage sheaf

$\pi_{\#^{\omega_{t}}\lambda 1/U}$ had been knownin thecontextofvariation ofHodge structures

md its application totheclassificationtheory of algebaricvarieties (cf. [Gr] and [Fj]$)$. (See also [Oh-l].) The first result in this direction,extending

the model

case

1), was obtained byMaitani-Yamaguchi [M-Y] in the

case

where $g4$ is a Steinmanifold of dimension2. Namely, by combinin

$g$the

analysis ofvariationof Green functions (cf. [L-Y]) with

a

characterization

of theBergman kernelbyN. Suita (1933-2002) as thesecond derivative of

the Green function(cf. [Sui]), they provedthat $\Re_{t/u}$ is of semipositive

curvature in this situation. (See also [Mt].)

Itis aninterestingcoincidence that Suita$|s$workwas motivated by an

open

question raisedin atreatise byK. Oikawa (1927-92) andL. Sario,

where the comparison betweenthe Bergman kernel andthe capacity of

$\backslash the$boundary ofan arbitrary domain $W$ in $C$ was asked (cf. [O-S] p.342,

7$)$. Here, the capacity _{$c_{\beta}(z)$} oftheboundary $\beta=\partial W$ isdefined by

(25) $\log c_{\beta}(z)=\lim_{w*z}(g_{W}(z,w)-\log| z- w |)$,

where $g_{W}$ denotes the (negativevalued) Greenfunctionof $W(c_{\beta}:\equiv 0$ if

$g_{W}\equiv 0)$. Itis clearthat sc$K_{W}=c_{\beta}^{2}$ holds if $W=$ D. In [O-S],itwas shown

that $K_{W}\underline{=}0$ifand onlyif $c_{\beta}\equiv 0$

.

Suita showedthat

(26) $\pi K_{W}(z)>c_{\beta}(z)^{2}$ for

any

$z\in W$

holds if $W$ is anannulus.

$Q9$$Suita^{\uparrow}s**$( conjecture)——————————————————————–

(27) $\pi K_{W}(z)>c_{\beta}(z)^{2}$ holds forany $z\in W$

if $g_{W}\not\equiv-\infty$ and $W$ is not equivalenttotheunit disc.

The reader

may

noticethat Q9 couldhavebeen included in \S 2. Infact,

afterprovingthe divergence of $K_{\Omega}(z)$ at $\partial\Omega$ for hyperconvex $\Omega$ in

[Oh-2], _{the author refined [Oh-T] in [Oh-5] md memwhile found its}

applicationto Q9 (cf. [Oh-6,7]), inwhich an inequality $750\pi K_{W}>c_{\beta}^{2}$ was

$2\pi K_{W}>q^{2}$ holds in the situation ofQ9.

After [M-Y], Berndtsson [Bm-2] and Tsuji [Tj] succeededin

generalizing

the resultto Steinmamifolds of arbitrary dimension, _{by directly exploiting}

thereproducingproperty ofthe Bergmankernel. On the otherhand,the

method of [M-Y]

was

extendedto explore variational properties of

families of harmonic functions withprescribed

singularities

and Dirichlet

orNeumanntype boundary conditions (cf. [Hm] md [Hm-M-Y]).

Recently, $Bemdtsson-P\dot{a}un$[Bm-P] obtained a result whichisalso

relatedto 2). Motivatedby applications to algebraic geometry, they

consider asurjective projective morphism

say

$p$ : $Xarrow Y$ between

complex

mamifolds X and $Y$, and

a

holomorphic lin$e$ bundle $(L, h)$

over

X endowed witha singularfibermetric $h$. Let _$I(h)$ denote the

multiplierideal sheafof $h$(cf. [Dm-3]). Let $Y^{0}\subset Y$ bethe Zariski

open

setofpoints that

are

notcritical values of $P$ in $Y$, and let

$X^{0}\subset X$ be the

inverse image of $Y^{0}$ withrespectto

$p$

.

As $y$ varies in

$Y^{0}$, therelative Bergmanmetric on $\omega_{x/\gamma}$\copyright L

over

$X^{0}$ is defined similarly

as

$\beta_{y/U}$, only it

is allowed to beidentically $\infty$.

Theorem3. In the above situation, assumethefollowing.

i$)$ thecurvature current of $(L, h)$ issemipositive on X.

$)$ $H^{0}(Xox,b_{y}\otimes L\otimes I(h))\neq 0$ for

some

$y\in Y^{0}$, where $X_{\gamma}=p^{-l}(y)$

.

Then the relativeBergmankernelmetric of the bundle $\omega_{X/\vee}\otimes L|X^{0}$ is not

identically $\infty$

.

Ithas semipositivecurvature current and extends

across

$X-X^{}$ _to a_metric withsemipositive curvature current

on

all of X.

For theproof,the assumptionthat $p$ isprojective iscmcial. Thepoint

is that

every

point $y\in Y$ admits aneighbourhood V such that $P^{-1}(V)$

contains a divisor whosecomplementis Stein.Theorem 3 has interesting

applications to pluricanonical

maps

viaan inequality for the $||$

restricted

volume“ (cf. Theorem0.3 in [B-P]. See also [Tk]). For theasymptotics of

the restrictedvolume, see [Hs].

$Ql0_{-arrow---arrow---arrow--}^{W}$

Forwhich morphismis Theorem3 valid?

Any extension of the model

case

2) towards this direction will be quite

interesting andfruitful. Berndtsson [Brn-2] hasproved

Nakano-semipositivity of the direct images for $Khler$_{morphisms and appliesthe}

result in [Brn-3] to studyvariations of K\"ahler_metrics. Moreover, _{the deep}

work ofFang-Lu-Yoshikawa [F-L-Y]

on

thefamily of Calabi-Yau

threefolds

seems

to be closelyrelated to this question.

Note. If OV4 is aHartogs domain over $U$, there is a formula which

relates theweighted Bergman kernels on $U$ to

$\{M$ (cf. [Li]), whichis

usefulto derive explicit formulae (cf. [Ym]). Computationfor Hartogs

domains is done also in[M-Y] for the relative Bergmankernels.

\S 5. BergmanKernel and $L^{2}$ Extension

As before, let $\Omega$ be a domain in C’ md let

$A^{2}(\Omega)$ be the Hilbert space

of$L^{2}$holomorphic functions on $\Omega$ withrespectto theLebesgue measure.

Let $z=$ $(z , . . . , z)$ bethe coordinate of $C^{\mathfrak{n}}$

For

any

pseudoconvex domain $\Omega$ in $C^{\iota}$

; for

any

plurisubharmonic

function cp on $\Omega$, and for anynonnegativenumber

$\epsilon$, weput

$A_{\phi,\epsilon}^{2}(\Omega)=$

{

$f|f$ is holomorphic on$\Omega$ and $\int_{\Omega}e^{-\varphi}(1+|z_{n}|^{2})^{-l-\epsilon}|f|^{2}<\infty$

}

and, _{by letting} $\Omega‘=\{z\in\Omega|z_{\mathfrak{n}}=0\}$, put

$A_{\varphi}^{2}(\Omega^{I})=$

{

$f|f$ is holomorphic on $\Omega$‘ and

$\int_{x1}e^{-\varphi}|f|^{2}<\infty$

}.

Thenwe have

Theorem4. (cf. [Oh-T], [Oh-3]) Suppose that $\Omega$ is pseudoconvex.

Then, for

any

$\epsilon>0$, thereexists

a

bounded linear operator

$I_{\epsilon}:A_{\varphi}^{2}(\Omega^{1})arrow A_{\varphi\epsilon}^{2}(\Omega)$

whosenormdoes notexceed aconstant $C_{\epsilon}$ depending onlyon $\epsilon$, such

that $I_{\mathcal{E}}(f)|\Omega‘=f$ holds for any $f\in A^{2}(\Omega^{1})$.

Obviously Theorem 4 does nothold for $\epsilon=0$. Thebestconstantfor $c_{\epsilon}$

is notyet known (cf. [Bf]). As was mentioned in \S 3 and \S 4,Theorem4was

appliedto plurisubharmonic functions md to theBergmankernels.

holomorphicvectorbundle

over

$M$, and let $\omega_{M}$ be thecanonical line

bundle ofM. Let $dV$ be a $C^{0\circ}$volumeform on _$M$ and let $h$ be a $C^{\omega}$fiber

metric of E. For

any

reduced analytic set$S\subset M$ equippedwitha

measure

$\mu,$ $A^{2}(s,E\otimes m^{h\otimes(dV)^{-1},d\mu)}$ will stmd for the

space

of

$L^{2}$holomorphic

sections of $E\otimes\omega_{M}$ over $S$ withrespectto $h\otimes(dV)^{\dashv}$ and $\mu$

.

Since

$A^{2}(M,E\otimes\omega_{M},h\otimes(d\eta^{1},dV)$ isindependent of $dV$, we shall denoteitby

$A^{2}(M,E\otimes Qh^{h)}$ forbrevity.

Givenlocallyintegrablefunctions $\psi:Marrow[-\infty,\infty)$, the

spaces

$A^{2}(S,E\otimes\omega_{M},e^{\sim\psi}h\otimes(dV)^{\dashv},d_{[\lambda)}$ md $A^{2}(M,E\otimes w^{e^{-\psi}h)}$

are

defined similarly.

Weshall call $e^{-1}\gamma h$

a

singular fibermetric of E. Given

any

singular

fiber

metric $\hat{h}$

of $E$,

an

$L^{2}$extensionoperator for $(E\otimes\omega,\hat{h}\otimes(dV)^{-1})$ from $(S,\mu)$

is definedtobe abounded linear operator

I: $A^{2}(s,E\otimes\%^{\hat{h}\otimes(dV)^{-(},d\mu)}arrow A^{2}(M,E\otimes W^{\hat{h})}$

satisfying $I(f)|S=f$ for

my

$f$. Let $\Phi:Marrow[-\infty,0)$ be

any

continuous function. We shall

say

that $\Phi$ is oflogarithmictype along $S$ if

the following aresatisfied.

$\Phi^{-1}(-\infty)=$S.

$\Phi|(M\backslash S)$ is $C^{\infty}$

$e^{arrow\Phi}$

is notintegrable on an

open

subset $U\subset M$ whenever

$U\cap S\neq\otimes$.

Givena function $\Phi$whichis oflogarithmic type along$S$, wesaythat $\mu$

is aresidualmajorant of $(dV,\Phi)$ if the inequality

$\lim_{\gamma-\prime}\sup_{\infty}$ $\zeta$

$pe^{\sim\Phi}dV$ _$\leq\int$ pdpt

$-r<\Phi<-r+1$ $S$

holds for anynonnegative continuous function $p$ withcompact support

on M.

$suchthat(E,he^{-(|\gamma\tau})areNakmosemipositiveonM\backslash Sformy\tau\in wesaythat(E_{d^{is\Phi-positiveifthereexistsapositivenumber\tau_{0}}},h$

$[0, \tau_{0}]$. We shall denote the supremum of such $\tau_{0}$ by $\tau(h,\Phi)$.

Let $T$ be a closed subset of M. We

say

that $T$ is $L^{2}$-negligible if, for

any

point $p\in T$ and for my neighbourhood $W\ni p$,

every

$L^{2}$

Intheseterms, the main result of [Oh-5] is expressed as follows.

Theorem 5. Let $M$ be a complex manifold with a $\sigma_{volume}$ form $dV$,

let $E$ be a holomorphicvector bundle over $M$ with a $C$fiber metric $h$,

let $S$ be areduced analytic subset of $M$ equipped withameasure $\mu$,

md let $\Phi$ : $Marrow[-\infty, 0)$ be a continuous function which is of

logarithmic type alongS. Suppose that $\mu$ is a residual majorantof $(dV$,

$\Phi),$ $h$ is $\Phi$-positive, and thatthere exists an $L^{2}$-negligibleset $T\subset M$ such that $M\backslash T$ is Stein and $S\cap T$ is nowhere dense in S. Then, for any

plurisubharmonic function $\psi$ on $M$, there exists $mL^{2}$extension operator

for

_$(h$

, from $(S,\mu)$ whose normis boundedby a

constmt depending only

on

$\tau(h,\Phi)$.

In [Oh-5, Theorem4], the result is stated for amore restricted class of $\Phi$,

but it is easy to seethat theproofofthis generalizedversion is completely similar.

ThepointofTheorem5 aswell as Theorem4is thatthenormofthe$L^{2}$

extension operatoris estimated by a relatively simple geometric quantity.

Therefore it seemsto make senseto askthe following.

$Q11^{\underline{**}}-$

_{Findareasonablegeneralization of Theorem 5 forthe}$\partial$closedformsoftype(O,q)for$q\succeq 1$.

For anicebutpartial answer, see [Kz] for instmce.

Finally, let’s see how one canderive a division theoremfrom an

extensiontheoremin such a

way

that Theorem5 yields $mL^{2}$ division

theorem.

Let $E^{\star}$ denote the dualbundleof _$E$, let _{$P(E^{\star})$} be the projectivization

of $E^{\star}$, i.e.

$P(E^{\star})=\cup(E^{\star}-\{0\})/(C-\{0\})$, and let $\varpi:P(E^{\star})arrow M$ be the

bundleprojection.

Recallthat, in thepresenceof sucha fiberstructure, the sheaf

cohomology

groups

of $P(E^{\star})$ and those of $M$ arerelatedby the$I_{\lrcorner}eray$

spectral sequence. Based onthis, onehas a canonicalisomorphism

between the E-valued cohomology

groups

of $M$ and cohomology

groupsof $P(E^{\star})$ with valuesina certainline bundle. More precisely, one

has thefollowing.

Theorem 6. (cf. [LP]) Let $L(E^{*})$ denote thetautological line bundle

over$P(E^{*})$, i.e. $L(E^{\star})=\cup L(E_{x}^{\star})x\epsilon M$,where $L(E_{x}^{\star})$ denotes the tautological line

(28) $I\mathscr{K}^{q}’(M, E)\cong H^{p,q}(P(E^{\star}), L(E^{\star})^{\star})$.

Here $H^{\rho q}(\cdot,.)$ denotes the Dolbeaultcohomology

group

of type (p,q).

We notethat L(E$*$

)$*$

and $E$ are relatedby the followingcommutative

diagram.

$L(E^{\star})^{\star}<-\varpi^{*}Earrow E$

1

$\Downarrow$ $P(E^{\star})->M$

Herethe morphism $\varpi^{\star}Earrow L(E^{*})^{\star}$ is defined over $y\in P(E^{*})$ as the

naturalprojectionto the quotient

space

of $E_{\alpha(y)}$bythekernel of $y$.

Now, _{by applying} (28), one

can

transform a divisionproblem

on

$M$ to

an extension problemon $P(E^{\star})$ asfollows.

Let $Y$ : $Earrow Q$ bea $su\dot{\eta}$ectivemorphism betweenholomorphic

vector bundles $E$ md $Q$ over M. A (generalized) divisionproblemasks

for conditions for theinduced morphisms from $H^{p,q}(M, E)$ to $H^{p,\mathfrak{q}}(M, Q)$

to besurjective. In view of (28), this $su\dot{\eta}ectivity$ is equivalentto thatof

$H^{t,\uparrow}(P(E^{\star}), L(E^{\star})^{\star})arrow H^{P,t}(P(Q^{\star}), L(Q^{\star})^{\star})$,

which isnothingbut theextendibilitybecause $P(Q^{\star})$ is naturally

identified witha complex submanifold of $P(E^{\star})$ by $\gamma^{*}$ and a cmonical

isomorphism between $L(E^{\star})^{\star}|Q$, and $L(Q^{\star})^{\star}$ isinduced by $\gamma$.

Thus,_{by interpretingthe conditions in Theorem 5in this situation,we}

shall obtainan$L^{2}$ divisiontheorem.

Infact, _given $Earrow Q$

as

above,

any

$C$ fiber metric $h$ of $E$ and a

point $v\in P(E^{\star})$, let $8_{h}(v)$ denote the fiberwise distancefrom $v$ to $P(Q^{*})$

with respecttotheFubini-Study metric associatedto $h$, normalized in

such a

way

that $\sup\{6_{h}(v);v\in P(E^{*})\}=1$ for

every

$x\in$ M. In this

situationwe have thefollowing.

Theorem7. Let (E,h) and $Q$ be as above. Assumethatthere exists an

$L^{2}$-negligibleset _{$T\subset M$} such that M-T is Stein$md$, with respecttothe

fiber metric of $L(E^{\star})^{\star}$ induced from h, L$(E^{\star})^{\star}$ is log6-positive. Thenthe

natural homomorphism

$A^{2}(M, E\otimes_{0}b, h)arrow A^{2}(M, Q\otimes_{0}b, h_{Q})$

Corollary1. (cf. [Oh-7]) Let $\Omega$ be a boundedpseudoconvex domain

in $C^{\tau t}$ . Then there exists a

constant $C$ dependingonly the diameter of $\Omega$

such that, for

any

plurisubharmonic function $\varphi$ on

$\Omega$ and for

any

holomorphic function $f$ on $\Omega$ satisfying

$\int_{\Omega}|f(z)|^{2}ed\lambda<\infty-\varphi(z)-2\mathfrak{n}|og^{|_{Z}|}$

there exists a vector valuedholomorphic function $g=(g_{4},\ldots, g_{\iota})$ on $\Omega$

satisfying

$f(z)=\sum_{\sim,j-1}^{n}z_{j}g_{j}(z)$

and

$\int_{\Omega}|g(z)|2^{-\varphi(Z)^{-2(n-1)|_{0}}g}ed\lambda|z|\leq Cb|f(z)|^{2}e^{-\varphi \mathfrak{n}}(Z)-2|_{\circ g|z|}d\lambda$ .

Here $d\lambda$ denotes the Lebesguemeasure.

Remark. Itdoes not seemtobe easy to derive Corollary1 just by

applyingthe result of [S].

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