Seminar: Josias Reppekus (Hörsaal 2, BZ.08.04)
I will continue presenting the results on
Convexity of Complements of Limit Sets for Holomorphic Foliations on Surfaces
by Bertrand Deroin, Christophe Dupont and Victor Kleptsyn, that can be found here: arxiv.org/abs/2202.01007
Consider a holomorphic foliation F of a compact Kähler surface with only hyperbolic singularities and no foliation cycles. In the last talk, I explained the construction of a metric on the normal bundle of F that is positive along F.
In this talk, we show that if the limit set of F is thin (e.g. has Lebesgue measure 0), then we can modify the metric to be positive in all directions near the limit set. Finally, we adapt a construction by Brunella to obtain a strictly plurisubharmonic exhaustion function for the complement near the limit set, using the density of the constructed metric and the negative logarithm of the transversal distance to the limit set.