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Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure? Every paper or article I looked at, talk only about singular laminations on $\mathbb{CP}^2$. I was wondering why. If you know something about it or you can give some reference, it would be nice.

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It is conjectured that $\mathbb {CP}^2$ contains no embedded compact laminated set (without singularities) apart the smooth algebraic curves.

This is a strong form of the "Minimal Exceptional" conjecture, stating that for a singular holomorphic foliation of $\mathbb{CP}^2$, every leaf accumulates in the singular set.

Nice references about this subject are the following surveys:

  • É. Ghys - Laminations par surfaces de Riemann. Dynamique et géométrie complexes - Panor. Synthèses, 8, 1999.

  • S. Zakeri - Dynamics of singular holomorphic foliations on the complex projective plane. Laminations and foliations in dynamics, geometry and topology - Contemp. Math., 269, Amer. Math. Soc., 2001.

  • J. E. Fornaess and N. Sibony - Riemann surface laminations with singularities. - J. Geom. Anal. 18 (2008), no. 2, 400–442.

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  • $\begingroup$ Thanks! Can you please give the name of the articles. I guess you are talking about Riemann surface laminations by Ghys and Dynamics of \mathbb{P}^2 : examples by Fornaess and Sibony. I couldn't find the article by Zakeri. $\endgroup$ Jun 25, 2016 at 16:14
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    $\begingroup$ Dear @DivakaranDivakaran , I added more details about the references. I hope it helps. $\endgroup$ Jun 25, 2016 at 18:11

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