Unique ergodicity for foliations in P2 with an invariant curve

• 成果类型: Article
• 署名作者: Dinh, Tien-Cuong; Sibony, Nessim
• 署名单位: National University of Singapore; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0744-2
• 发表日期: 2018
• 页码: 1-38
• 关键词: harmonic currents; equation; MAPS; equidistribution; laminations; extension; points

摘要

Consider a foliation in the projective plane admitting a projective line as the unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a unique positive dd(c)-closed (1, 1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A unique ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. The result uses an extension of our theory of densities for currents. Foliations on compact Kahler surface with one or more invariant curves are also considered.