This book is devoted to the analytic theory of ordinary differential equations with complex time. Methods for the investigation of local and global properties of solutions are deeply examined so that the current state of typical problems like the 16th Hilbert problem (the number of limit cycles of polynomial planar vector fields) and the Riemann-Hilbert problem (i.e. the 21st Hilbert problem on the existence of a linear system with prescribed monodromy group and position of all singularities) can be presented. The first two chapters are devoted to an analysis of singular points of holomorphic vector fields by holomorphic normal forms. A local analysis of singularities is based on the notion of algebraic and analytic solvability. Chapter 3 deals with the local and global theory of linear systems. This chapter also contains positive and negative results on the solvability of the Riemann-Hilbert problem. Chapter 4 is concerned with analytic classification of resonant singularities. The main working tool here is an almost complex structure and quasiconformal maps. In the last chapter, the global theory of polynomial differential equations on the real and complex plane is investigated. Tools for the study of limit cycles (the 16th Hilbert problem) near poly-cycles or those which bifurcate from non-isolated periodic orbits (Abelian integrals) are presented. It is also shown how generic properties of complex foliations differ from real ones.
The book is carefully written and important notions are motivated and explained in detail. All sections end with exercises and problems often lying at the frontier of current research. A good knowledge of various parts of analysis (e.g. complex analysis in several variables) and topology is required. The book is aimed at senior graduate students in differential equations. Professionals can also find here an initiation into the present-day level of research and interesting applications of algebraic geometry to differential equations.