Minimal kernels and compact analytic objects in complex surfaces (Samuele Mongodi - Politecnico di Milano)

The minimal kernel of a weakly complete space was defined by
Slodkowski and Tomassini to be the set of points where all the psh exhaustions
fail to be strictly psh; together with Slodkowski and Tomassini, we studied
the minimal kernel of complex surfaces with a real-analytic psh exhaustion,
obtaining a classification result for such surfaces: they are either proper
over a Stein space or foliated in Levi-flat hypersurfaces whose Levi-foliation
has dense leaves. We called the latter "Grauert-type" surfaces and we studied
their geometry to some detail. I would like to present these results and
discuss the relation between the minimal kernel and the presence of imbedded
complex curves with compact closure; if time permits, I will also outline some
more recent results where no real-analyticity assumption is made.