Abstract
Let F be a holomorphic foliation by Riemann surfaces on a compact Kähler surface X. Assume that it is generic in the sense that all the singularities are hyperbolic, and the foliation admits no directed positive closed -current. Then there exists a unique (up to a multiplicative constant) positive -closed -current directed by F. This is a very strong ergodic property of F showing that all leaves of F have the same asymptotic behavior. Our proof uses an extension of the theory of densities to a class of non--closed currents. This is independent of foliation theory and represents a new tool in pluripotential theory. A complete description of the cone of directed positive -closed -currents is also given when F admits directed positive closed currents.
Citation
Tien-Cuong Dinh. Viêt-Anh Nguyên. Nessim Sibony. "Unique ergodicity for foliations on compact Kähler surfaces." Duke Math. J. 171 (13) 2627 - 2698, 15 September 2022. https://doi.org/10.1215/00127094-2022-0044
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