Title of the course: Introduction to Several Complex Variables
Instructor: Prof. Pascal J. Thomas
Institution: Institut de Mathématiques de Toulouse
Dates: 22-28 September 2025
Prerequisites: knowledge of the basics of one-variable holomorphic functions: Cauchy formula, analytic continuation, series expansion, maximum principle; desirable but not necessary: harmonic and subharmonic functions.
Level: Graduate
Abstract: We define analytic (holomorphic) functions of several variables as those locally representable by convergent power series of several variables. The main question we will tackle will be that of the natural domains of existence of analytic functions. First we show that the domains of convergence of power series must be log-convex complete Reinhardt domains. We exhibit the Hartogs phenomenon showing that some domains in $\mathbb C^n$ force extension of all analytic functions defined on them to the same strictly larger domain. We then define pseudoconvexity for $\mathcal C^{^2}$ domains using the Levi form, and show that a (smooth) domain of existence for analytic functions is necessarily pseudoconvex. Time permitting, we will tackle the general definition of holomorphic convexity and pseudoconvexity, and hints about the converse: a pseudoconvex domain admits a holomorphic function with no analytic extension to any larger domain.
Language: English