Complex Analysis and Complexes of Differential Operators

Abstract

Introduction. Complex analysis and the theory of complexes of differential operators are two closely related subjects. First we notice that the study of the Dolbeault complex is essential to understand holomorphic functions of several complex variables: this complex is a particular example of a complex of differential operators and thus many results in complex analysis could be considered as particular instances of the more general theory of differential complexes. Our knowledge of the Dolbeault complex is a powerful source of intuition to forecast the behaviour of general differential complexes. On the other hand, complex analysis and the Dolbeault complex playa very peculiar role because they are also an essential tool for the study of differential operators. In my lectures, I will try to give an idea of the close relationship of the two fields. The arguments that I will discuss will be the following: 1) the theory of Ehrenpreis-Malgrange for differential operators with constant coefficients. 2) Differential equations with constant coefficients in the class of real-analytic functions (analytic convexity). 3) The theory of convexity and the theorem of Cartan Thullen for general operators. 4) Boundary complexes and boundary values of pluriharmonic functions. 5) The Lemma of Poincaré for complexes of differential operators with smooth (non constant) coefficients.