### Seminar: Unique Ergodicity for foliations on compact K\"ahler surfaces

Thời gian: 14:00 đến 15:30 Ngày 09/08/2022

Địa điểm: C102, VIASM

Báo cáo viên: Nguyễn Việt Anh (ĐH Lille)

Tóm tắt:

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.