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Holomorphic dynamical systems on manifolds. (Sistemas dinamicos holomorfos en superficies.) (Spanish) Zbl 0855.58049
Aiming at the description of global properties of solutions of holomorphic dynamical systems on compact surfaces, this paper follows ideas of Il’yashenko’s seminar [Yu. S. Il’yashenko, Sel. Math. Sov. 5, 141-199 (1986; Zbl 0585.34003); translation from Tr. Semin. Im. I. G. Petrovskogo 4, 83-136 (1978; Zbl 0418.34007)] devoted to the study of the ergodic and topological properties of polynomial differential equations in two complex variables. By the authors’ specification, the contribution of the present paper is to incorporate to Il’yashenko’s ideas the intrinsic point of view of modern algebraic geometry. So, the authors prove that the ergodic and topological properties of polynomial differential equations on the projective complex tangent space to a line described by Il’yashenko hold also for the holomorphic differential equation tangent to an ample curve in a projective surface.
In the first chapter, the relation between ordinary differential equations and holomorphic foliations by curves is given: the solutions of an holomorphic ordinary differential equation define a foliation by curves and, on the other hand, for a foliation by curves, a class of differential equations the solutions of which form the given foliation is defined. By using this local correspondence, an analytical global description of the foliations by curves on a connected complex manifold \(M\) is given. Namely, such a foliation (possibly with singularities) is described as a vector bundle morphism \(\alpha: L\to TM\) which is not identically zero on any connected component of \(M\), \(L\) being a linear complex bundle over \(M\). By this description, the concepts of the family of foliation by curves and of universal family follow in a natural manner. These notions are concretized in the case of foliations by curves in the projective spaces \(\mathbb{C} P^n\) which can be described by homogeneous polynomial fields on \(\mathbb{C}^{n+1}\). In the last part of this chapter, the linearization of Poincaré’s theorem is proved by using a homological method.
The second chapter presents the dynamical concepts: the transverse dynamics of a foliation by curves on a complex manifold \(M\), without singularities, and holonomy groups.
Il’yashenko’s results related to the density, ergodicity and rigidity of the action of the group of germs of conformal transformations on \(\mathbb{C}\) are presented in chapters III and IV. The authors give here a new proof of the ergodicity theorem by using Koebe’s theorem. Il’yashenko’s rigidity theorem for foliations by curves tangent to a line in \(\mathbb{C} P^2\) is proved in chapter V. Remarkable is an example of a foliation with non-abelian holonomy.
The extension of Il’yashenko’s results to the case of foliations tangent to ample curves in projective surfaces are given in the last chapter of the book as an application of the ideas from the previous chapters.
The authors were preoccupied by a clear and rigorous exposition of the results. We consider the work as an excellent reference to the problem of holomorphic dynamical systems on projective surfaces.

37F99 Dynamical systems over complex numbers
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
34M99 Ordinary differential equations in the complex domain
34D20 Stability of solutions to ordinary differential equations
37A99 Ergodic theory
54C70 Entropy in general topology