Seminar: Unique Ergodicity for foliations on compact K\"ahler surfaces
Time:
Venue/Location: C102, VIASM
Speaker: Nguyễn Việt Anh (ĐH Lille)
Content:Let $\mathcal F$ be a holomorphic foliation by Riemann surfaces on a compact K\"ahler surface $X.$
Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed $(1,1)$-current.
Then there exists a unique (up to a multiplicative constant) positive $\ddc$-closed $(1,1)$-current directed by $\mathcal F.$
This is a very strong ergodic property of $\mathcal F$ showing that {\it all} leaves of $\mathcal F$ have the same asymptotic behaviour.
Our proof uses an extension of the theory of densities to a class of non-$\ddc$-closed currents.
This is independent of foliation theory and is a new tool in pluripotential theory.
A complete description of the cone of directed positive $\ddc$-closed $(1,1)$-currents is also given when $\mathcal F$ admits directed positive closed currents.