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Seminar on Geometric Complex Analysis

Seminar information archive ~06/27Next seminarFuture seminars 06/28~

Date, time & place Monday 10:30 - 12:00 128Room #128 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Kengo Hirachi, Shigeharu Takayama

2013/04/15

10:30-12:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Nikolay Shcherbina (University of Wuppertal)
On defining functions for unbounded pseudoconvex domains (ENGLISH)
[ Abstract ]
We show that every strictly pseudoconvex domain Ω with smooth boundary in a complex manifold M admits a global defining function, i.e. a smooth plurisubharmonic function φ:UR defined on an open neighbourhood UM of Ω such that Ω={φ<0}, dφ0 on bΩ and φ is strictly plurisubharmonic near bΩ. We then introduce the notion of the kernel K(Ω) of an arbitrary domain ΩM as the set of all points where every smooth and bounded from above plurisubharmonic function on Ω fails to be strictly plurisubharmonic. If Ω is not relatively compact in M, then in general K(Ω) is nonempty, even in the case when M is Stein. It is shown that every strictly pseudoconvex domain ΩM with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of K(Ω). 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.