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These notes are a slightly enlarged version of my habilitation thesis, where our research interest and main results in the past few years are summarized. Most of the discussion revolves around complex ordinary differential equations and their underling foliations, singularity theory and dynamical systems. Compared to the original text, a section containing some background material on holomorphic foliations was added. Also some new results obtained in the past three years that are in line with the one presented in the habilitation were included.

In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of the latter foliation exerts influence on several problems that apparently have a purely local nature. In the course of the discussion, a few recent results and open problems in the area will be reviewed as well.

Let $\mathcal{F}$ be a foliation defined on a complex projective manifold $M$ of dimension $n$ and admitting a holomorphic vector field $X$ tangent to it along some non-empty Zariski-open set. In this paper we prove that if $X$ has sufficiently many integral curves that are given by meromorphic functions defined on $\mathbb{C}$ then the restriction of $\mathcal{F}$ to any invariant complex $2$-dimensional analytic set admits a first integral of Liouvillean type. In particular, on $\mathbb{C}^3$, every rational vector fields whose solutions are meromorphic functions defined on $\mathbb{C}$ admits a non-empty invariant analytic set of dimension $2$ where the restriction of the vector field yields a Liouvillean integrable foliation.

We provide examples of vector fields on $(\mathbb{C}^3, 0)$ admitting a formal first integral but no holomorphic first integral. These examples are related to a question raised by D. Cerveau and motivated by the celebrated theorems of Malgrange and Mattei-Moussu.

Starting from some remarkable singularities of holomorphic vector fields, we construct (open) complex surfaces over which the singularities in question are realized by complete vector fields. Our constructions lead to manifolds and vector fields beyond the algebraic setting and provide examples of complete vector fields with some new dynamical phenomena., Comment: to appear in Moscow Mathematical Journal

We classify degenerate singular points of $\C^2$-actions on complex surfaces.

This paper is devoted to the resolution of singularities of holomorphic vector fields and of one-dimensional holomorphic foliations in dimension 3 and it has two main objectives. First, from the general perspective of one-dimensional foliations, we build upon the work of Cano-Roche-Spivakovsky and essentially complete it. As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan-Panazzolo but proved by means of rather different methods. The second objective of this paper consists of looking at a special class of singularities of foliations containing, in particular, all singularities of complete holomorphic vector fields on complex manifolds of dimension 3. We then prove that for this class of holomorphic foliations, there holds a much sharper resolution theorem. This second result was the initial motivation of this paper and it relies on the combination of the previous resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations., Comment: To appear in Russian Mathematical Surveys

We introduce a method to estimate the size of the domain of definition of the solutions of a meromorphic vector field on a neighborhood of its pole divisor. The corresponding techniques are, in a certain sense, quantitative versions of some well-known phenomena related to the presence of metrics with positive curvature. Several applications of these ideas are provided including a type of "confinement theorem" for solutions of complete polynomial vector fields on $\mathbb{C}^n$ and obstructions for certain (germs of) vector fields to be realized by a global holomorphic vector field on a compact K\"ahler manifold. As a complement a new approach to certain classical equations is proposed and detailed in the case of Halphen equations., Comment: This is a revised version of arXiv:1111.2365

These are slightly informal lecture notes intended for graduate students about the standard local theory of holomorphic foliations and vector fields. Though the material presented here is well-known some of the proofs differs slightly from the classical arguments., Comment: 115 pages, 9 figures