Structural stability of singular holomorphic foliations having a meromorphic first integral

Foliations in Algebraic Surfaces Having a Rational First Integral

Given a foliation F in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case S = P2 some new counter-examples to the classic formulation of the Poincaré problem are presented. If S is a rational surface and F has singularities of type (1, 1) or (1,−1) we prove that the general solution is a non-singular curve.

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 ex...

A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations

Existence and Stability of Foliations by J–Holomorphic Spheres

We study the existence and stability of holomorphic foliations in dimension greater than 4 under perturbations of the underlying almost complex structure. An example is given to show that, unlike in dimension 4, J–holomorphic foliations are not stable under large perturbations of almost complex structure.

Stability of Holomorphic Foliations with Split Tangent Sheaf

We show that the set of singular holomorphic foliations on projective spaces with split tangent sheaf and good singular set is open in the space of holomorphic foliations. We also give a cohomological criterion for the rigidity of holomorphic foliations induced by group actions and prove the existence of rigid codimension one foliations of degree n − 1 on P for every n ≥ 3.