数理解析研究所講究録 1211 巻 2001 年 129-132

129

SINGULARITIES OF THE BERGMAN KERNELAND NEWTON POLYHEDRAJOE KAMIMOTO$(\tau^{\backslash }\backslash \backslash \#.* \mathrm{x}_{-})$

1. INTRODUCTION
In this note, we announce aresult about the singularity of the Bergman kernel
for pseudoconvex domains of finite type. In the case of some class of pseudoconvex
domains, we show that the growth order of the Bergman kernel at the boundary isdetermined by the shape of the Newton polyhedron of the defining function of thedomain and that the boundary limit of the Bergman kernel takes the same value asthat in the case of local model.

2. BACKGROUND AND OUR RESULTS

and2.1. Background. Let 0be domain in
the set of the -holomorphic$B(z)$(ondiagonal) is defined by.
Bergmankernelof
theThefunctions on$L^{2}$$H^{2}(\Omega)$$\mathbb{C}^{n}$$\Omega$$\Omega$$B(z)= \sum|\phi_{\alpha}(z)|$,.
whereis acomplete orthonomal basis ofmanyofsingularity
the Bergman kernel at the boundThere arestudies about theary of pseudoconvex domains. Let us recall important results which are deeply
connected with our study.
$H^{2}(\Omega)$$\{\phi_{a}\}$-smoothly bounded strictly
2.1.1. Strictly pseudoconvex case. Assume that is a. Hormander [9] and Diederich $[3],[4]$ computed thepseudoconvex domain in
Bergmankernel as follows.
boundary limit of the$C^{\infty}$$\Omega$$\mathbb{C}^{n}$(2.1)$\lim_{zarrow \mathrm{p}}B(z)\cdot d(z-p)^{n+1}=\frac{n!}{4\pi^{n}}\cross$(Levi determinant at ),$p$

where means the distance. Later C. Fefferman [7] obtained the following strong
result of the asymptotic expansion of the Bergman kernel:$d$(2.2)$B(z)= \frac{\varphi(z)}{\rho(z)^{n+1}}+\psi(z)\log\rho(z)$whereandis the defining function (i.e.
are-smooth on andand$\Omega=$$\rho\in C^{\infty}(\overline{\Omega})$$\partial\Omega)$$\varphi(z)$$\psi(z)$$C^{\infty}$$\overline{\Omega}$,{z; $\rho(z)>0\}$ and $|d\rho|>0$ on$\varphi(z)$is positive on the boundary

130
JOE KAMIMOTO2.1.2. Semiregular ( -extendible)case. In the weakly pseudoconvex and of finite
type case, although there is not so strong result like asymptotic expansion (2.2),
many precise results have been obtained. In particular the following result due toBoas-Straube-Yu [2] (see also Diederich-Herbort [6]) is very important. Assume
that is abounded pseudoconvex domain inandis aboundary pointof semiregular ( -extendible)with the multi-type,) (see $[5],[12]$ for
precise definition). In $[2],[6]$ ,$h$$\Omega$$\mathbb{C}^{n+1}$$\mathrm{h}$$p\in\partial\Omega$$(1, 2\mathrm{m}\mathrm{n})\ldots$(2.3)$\lim B(z)\cdot d(z-p)^{2+\Sigma_{j=1}^{n}1/m_{j}}=B_{0}(\varpi)$$2m_{n}$,$z\in zarrow X$

where Ais anontangential cone,is the Bergman kernel of local model of at
and is some point in .
Recently the author [10] computed an asymptotic expansion of the Bergman kernelin the case of tube domains of finite type. Let be aconvex $C$ “-smooth function onsuch that $f(\mathrm{O})=df(0)=0$ and the set indefined by $y0+f(y_{1}, \ldots, y_{n})=$$yo+f(y’)<0$ . Let
be atube domain defined by. Assume that theorigin is apoint of finite type. This assumption implies that $f(y’)$ has an expression
near the origin: $f(y’)=P(y’)[1+h(y’)]$ where $P(y’)$ is aconvex polynomial such)that, )
and$\gamma\in(0,1]$$C>0$where,and
are some numbers. Here we set,
$B_{0}$$\Omega$$p$$\Omega$$\varpi$$f$$\mathrm{R}^{n+1}$$\mathbb{R}^{n}$$\omega_{f}$$\Omega_{f}=\mathrm{R}^{n+1}+\omega_{f}$$\Omega_{f}$$P(t^{1/2m_{1}}y_{1}, \ldots, t^{1/2m_{\hslash}}y_{n})=$$\mathrm{P}\{\mathrm{y}’$$\ldots$$y_{n}$$|h(y’)|\leq C\sigma(\tau)^{\gamma}$$(m_{\mathrm{j}}\in \mathrm{N})$$\sigma(\tau)=\sum_{j=1}^{n}y_{j}^{2m_{f}}$$\Delta_{P}=\{\tau\in \mathrm{R}^{n};P(\tau)<1\}$$\Gamma_{\delta}=\{(\tau, \rho^{1/m})\in\Delta_{P}\mathrm{x}[0, \delta);P(\tau)[1+C\rho^{\gamma}\sigma(\tau)^{\gamma}]<1\}$

Let : {v’where$\sigma$$arrow\Delta_{p}\mathrm{x}(0, \infty)$$\tau_{j}=-y_{j}\cdot y_{0}^{-1/2m_{f}}$be amapping defined by
and $\rho=-y_{0}$ .

$\sigma(y_{0}, y_{1},$Theorem 2.1 ([10]). The $Bergman^{\grave{\prime}}kemelB(z)$ of
neighborhood of the origin:(2.4)$\Omega_{f}$\ldots ,$\varphi(\tau,\rho^{1/m})\in C^{\infty}(\Gamma_{\delta})$$y_{n})=(\tau_{1},$\ldots ,has the form in$B(z)= \frac{\varphi(\tau,\rho^{1/m})}{\rho^{\Sigma_{f\approx 1}^{n}1/m_{f}+2}}+\psi(\tau, \rho^{1/m})\log\rho$

with
and
number, $m$ is the least common multiple of.

me small,, there
and$\psi(\tau,\rho^{1/m})\in C^{\infty}(\overline{\Delta_{P}}\cross[0, \delta))$$\{m_{1}, \ldots, m_{\mathrm{p}}\}$$s\dot{o}$$\tau_{n}, \rho)$$\delta>0$

is a small$\varphi(\tau, 0)=\varphi(\tau)>0$.

The aymptotic expansions (2.2) and (2.4) have very similar forms and the essential
difference between these asymptotic formulas only appears in the expansion variable,i.e. (2.2) takes the Taylor type while (2.4) takes the Puiseux type. But Herbort’sexample, below, asserts that an analogous asymptotic formula does not always holdin general case of finite type.

2.1.3. Herbort’s counterexample. Herbort [8] showed the Bergman kernel of the d0ma.
$\mathrm{n}\mathrm{y}$$\Omega_{HB}=\{z\in \mathbb{C}^{3};\Re(z_{0})+|z_{1}|^{6}+|z_{1}|^{2}|z_{2}|^{2}+|z_{2}|^{6}<0\}$131satisfies the inequalities near the origin on anontangential cone:$\frac{c_{1}}{\rho^{3}\log(1/\rho)}<B(z)<\frac{c_{2}}{\rho^{3}\log(1/\rho)}$,where , are positive constants. Note thatis not convex at the origin.
Althoughis apseudoconvex domain of finie type, the above inequarities imply
that the logarithmic function appears in the first term of the asymptotic expansionof its Bergman kernel. From Herbort’s exapmle, it seems difficult to imagin whatkind of pattern of the singularity of the Bergman kernel for general domains of finite
type.
$c_{1}$$\Omega_{HE}$$c_{2}$$\Omega_{HE}$2.2. The Newton polyhedron. Let. First let
and
us recall the definition of the Newton polyhedron of functions on the real space.
Let :be a “-smooth function in the neighborhood of the origin with$f(0)=0$ . Then $f(x)$ has the asymptotic expansion at the origin:$\mathbb{R}_{+}=[0, \infty)$$\mathrm{N}_{0}=\mathrm{N}\mathrm{U}\{0\}$

$f$

$C$

$\mathbb{R}^{n}arrow \mathbb{R}$$f(x) \sim\sum_{\alpha\in \mathrm{N}_{0}^{n}}c_{\alpha}x^{\alpha}$,. The Newton polyhedron
whereis the convex hull of the. The Newton diagram
union offor such thatis the unionof the compact faces of the Newton polyhedron. The pricipal partof$f(x)$ is defined by.
$\Gamma_{+}(f)$$x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$$\{\alpha+\mathbb{R}_{+}^{n}\}$$c_{\alpha}\neq 0$$\alpha$$\Gamma(f)$$f_{0}(x)$$\Gamma_{+}(f)$$f_{0}(x)= \sum_{\alpha\in\Gamma(f)}c_{\alpha}x^{\alpha}$

We generalize these concepts to the case of the function on the complex space.
Let $F$ :be a-smooth function in the neighborhood of the origin with
$F$
$F(0)=0$ . Then
has the asymptotic expansion at the origin:$C^{\infty}$$\mathbb{C}^{n}arrow \mathbb{R}$$F(z) \sim\sum_{\alpha,\beta\in \mathrm{N}_{\mathrm{O}}^{n}}C_{\alpha\beta}z^{\alpha}\overline{z}^{\beta}$,

where. The Newton polyhedronis the convex
hull of the union offor , such that
. The Newton diagramis the union of the compact faces of the Newton polyhedron. The$F(z)$
principal partof
is defined by$z^{\alpha}=z_{1}^{\alpha_{1}}\cdots z_{n}^{\alpha_{n}},\overline{z}^{\beta}=\overline{z}_{1}^{\beta_{1}}\cdots\overline{z}_{n}^{\beta_{n}}$$\{\alpha+\beta+\mathbb{R}_{+}^{n}\}$$\tilde{\Gamma}_{+}(F)$$\alpha$$\beta$$C_{\alpha,\beta}\neq 0$$\tilde{\Gamma}(F)$$\tilde{\Gamma}_{+}(F)$$F_{0}(z)$$F_{0}(z)= \sum_{\alpha+\beta\in\overline{\Gamma}(F)}C_{\alpha,\beta}z^{\alpha}\overline{z}^{\beta}$

The Newton distance$d_{F}$.

is defined by$d_{F}= \min\{d>0;(d, \ldots, d)\in\tilde{\Gamma}_{+}(F)\}$

Set $P=$ (faces on$\{$$d_{F}$$\tilde{\Gamma}(F)$,.

, )
. Let
be the number of the $(n-1)$ -dimensional$P$
containing . Then define.
$\ldots$$d_{F}$$\}\in\tilde{\Gamma}(F)$$\tilde{l}_{F}$$l_{F}= \min\{\tilde{l}_{F}, n\}$132
JOE KAMIMOTO
growth plurisub2.3. Boundary limit of the Bergman kernel. Let be aharmonic function on
satisfying that $F(0)=|\partial\nabla F(0)|=0$ . We consider thedomain:$F$$C^{\infty}$$\mathbb{C}^{n}$$\Omega_{F}=\{(z_{0}, z)\in \mathbb{C}\cross \mathbb{C}^{n};\rho=\Im(z_{0})-F(z_{1}, \ldots, z_{n})>0\}$We assume that$0\in\partial\Omega_{F}$.

is apoint of finite type and that

$F(e^{i\theta_{1}}z_{1}, \ldots, e^{i\theta_{n}}z_{n})=F(z_{1}, \ldots, z_{n})$for$\theta_{j}\in \mathbb{R}$.

Theorem 2.2. There is some positive constant $C(F)$ such that the Bergman kernelof the domainsatisfies
(2.5)
,
$B$

$\Omega_{F}$$(\begin{array}{l}0z_{0},z\end{array})\lim_{\Im(z)arrow 0}B(z_{0}, z)\cdot\rho^{2+2/d_{F}}(\log(1/\rho))^{lp-1}=C(F)$where Ais a nontangential cone. Moreover if let$C(F)=C(F_{0})$.

$F_{0}$

be a princepal part of$F_{f}$

thenREFERENCES[1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps,
Volume II, Birkhauser, 1988.
[2] H. P. Boas, E. J. Straube and J. Yu: Boundary limits of the Bergman kernel and metric,
Michigan Math. J. 42 (1995), 449-461.
[3] K. Diederich: Das Randverhalten der Bergmanschen Kernfunktion und Metrik in streng
pseudo konvexen Gebieten Math. Ann. 187 (1970), 9-36.
[4] –:Ueber die 1. und 2. Ableitungen der Bergmanschen Kernfunktion und ihr Randverhalten. Math. Ann. 203 (1973), 129-170.
[5] K. Diederich and G. Herbort: Pseudoconvex domains of semiregular type, Contributions toComplex Analysis and Analytic Geometry, Aspects of Mathematics E26, Vieweg 1994.
[6An alternative proof of atheorem of Boas-Straube Yu, Complex Analysis and Geometry, Pitman Research notes in Mathematics Series, vol. 366 (1997), 112-118.
[7] C. FefFerman: The Bergman kernel and biholomorphic mappings of pseudoconvex domains,
Invent. Math. 26 (1974), 1-65.
[8] G. Herbort: Logarithmic growth of the Bergman kernel for weakly pseudoconvex domains inof finite type, Manuscripta Math. 45 (1983), 69-76.
[9] L. Hormander:
estimates and existence theorems for the -operator, Acta Math. 113(1965), 89-152.
[10] J. Kamimoto: Asymptotic expansion of the Bergman kernel for tube domains of finite type,in preparation.
[11] A. N. Varchenko: Newton polyhedra and estimation of csillating integrals, Functional Anal.
APPI., 10-3 (1976) 175-196.
[12] J. Yu: Peak functions on weakly pseudoconvex domains, Indiana Univ. Math. J. 43 (1994),$\mathbb{C}^{3}$$L^{2}$$\overline{\partial}$1271-1295.

FACULTY OF MATHEMATICS, Kyushu UNIVERSITY, FUKUOKA 812-8581, JApAN-rnail address: oeffiath.
$E$$\mathrm{j}$$\mathrm{k}\mathrm{y}\mathrm{u}\mathrm{s}\mathrm{h}\mathrm{u}-\mathrm{u}.\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}$