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| 開催情報 | 月曜日 10:30～12:00 数理科学研究科棟(駒場) 128号室 |
|---|---|
| 担当者 | 平地 健吾, 高山 茂晴 |
2013年04月15日(月)
10:30-12:00 数理科学研究科棟(駒場) 126号室
Nikolay Shcherbina 氏 (University of Wuppertal)
On defining functions for unbounded pseudoconvex domains (ENGLISH)
Nikolay Shcherbina 氏 (University of Wuppertal)
On defining functions for unbounded pseudoconvex domains (ENGLISH)
[ 講演概要 ]
We show that every strictly pseudoconvex domain $\Omega$ with smooth boundary in a complex manifold $M$ admits a global defining function, i.e. a smooth plurisubharmonic function $\varphi \colon U \to \mathbf{R}$ defined on an open neighbourhood $U \subset M$ of $\Omega$ such that $\Omega =\{ \varphi < 0 \}$, $d\varphi \not= 0$ on $b\Omega$ and $\varphi$ is strictly plurisubharmonic near $b\Omega$. We then introduce the notion of the kernel $K(\Omega)$ of an arbitrary domain $\Omega \subset M$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $\Omega$ fails to be strictly plurisubharmonic. If $\Omega$ is not relatively compact in $M$, then in general $K(\Omega)$ is nonempty, even in the case when $M$ is Stein. It is shown that every strictly pseudoconvex domain $\Omega \subset M$ with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of $K(\Omega)$. We then investigate properties of the kernel. Among other results we prove 1-pseudoconcavity of the kernel, we show that in general the kernel does not possess any analytic structure, and we investigate Liouville type properties of the kernel.
We show that every strictly pseudoconvex domain $\Omega$ with smooth boundary in a complex manifold $M$ admits a global defining function, i.e. a smooth plurisubharmonic function $\varphi \colon U \to \mathbf{R}$ defined on an open neighbourhood $U \subset M$ of $\Omega$ such that $\Omega =\{ \varphi < 0 \}$, $d\varphi \not= 0$ on $b\Omega$ and $\varphi$ is strictly plurisubharmonic near $b\Omega$. We then introduce the notion of the kernel $K(\Omega)$ of an arbitrary domain $\Omega \subset M$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $\Omega$ fails to be strictly plurisubharmonic. If $\Omega$ is not relatively compact in $M$, then in general $K(\Omega)$ is nonempty, even in the case when $M$ is Stein. It is shown that every strictly pseudoconvex domain $\Omega \subset M$ with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of $K(\Omega)$. We then investigate properties of the kernel. Among other results we prove 1-pseudoconcavity of the kernel, we show that in general the kernel does not possess any analytic structure, and we investigate Liouville type properties of the kernel.
