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arXiv · 1811.07450

Unique Ergodicity for foliations on compact K\"ahler surfaces

Tien-Cuong Dinh·Viet-Anh Nguyen·Nessim Sibony
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Abstract

Let \Fc 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 \Fc. This is a very strong ergodic property of \Fc. Our proof uses an extension of the theory of densities to a class of non-\ddc-closed currents. A complete description of the cone of directed positive \ddc-closed (1,1)-currents is also given when \Fc admits directed positive closed currents.

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Tien-Cuong Dinh, Viet-Anh Nguyen, Nessim Sibony. 2018-11-19. Unique Ergodicity for foliations on compact K\"ahler surfaces. https://arxiv.org/abs/1811.07450

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