Dynamics of Singular Holomorphic Foliations on the Complex Projective Plane

This manuscript is a revised version of my Master’s thesis which was originally written in 1992 and was presented to the Mathematics Department of University of Tehran. My initial goal was to give, in a language accessible to non-experts, highlights of the 1978 influential paper of Il’yashenko on singular holomorphic foliations on CP [I3], providing short, self-contained proofs. Parts of the exposition in chapters 1 and 3 were greatly influenced by the beautiful work of Gómez-Mont and Ortiz-Bobadilla [GO] in Spanish, which contains more material, different from what we discuss here. It must be noted that much progress has been made in this area since 1992, especially in local theory (see for instance the collection [I6] and the references cited there). However, Hilbert’s 16th Problem and the Minimal Set Problem are still unsolved. There is a well-known connection between holomorphic foliations in dimension 2 and dynamics of iterations of holomorphic maps in dimension 1, but many believe that this connection has not been fully exploited. It seems that some experts in each area keep an eye on progress in the other, but so far there have been rather few examples of a fruitful interaction. The conference on Laminations and Foliations held in May 1998 at Stony Brook was a successful attempt to bring both groups together. As a result, many people in dynamics expressed their interest in learning