[geometry-ml:01719] 東大数理・複素解析幾何セミナー (4/15)

[Yoshihiko Matsumoto]

皆様

来週の東大数理・複素解析幾何セミナーのお知らせです。

《日時》4月15日（月）10:30〜12:00
《場所》東大数理（駒場キャンパス）126教室
《講演者》Nikolay Shcherbina氏（University of Wuppertal）
《タイトル》On defining functions for unbounded pseudoconvex domains

《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 $\varphi \colon U \to \mathbb{R}$ 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.

再来週以降の予定については、こちらをご覧ください。
http://seminar.ms.u-tokyo.ac.jp/geocomp/future.html

複素解析幾何セミナー担当
平地健吾、高山茂晴、松本佳彦