A short introduction to several complex variables

This text is a short two hour introduction into the theory of several complex variables. These lectures were given in May 1996 at Caltech to the class Ma 108 substituting for someone else. As prerequisite, the topic requires some familiarity with complex analysis in one dimension. The higher dimensional generalization of complex analysis in one variables is an important part of mathematics. Our short visit of this theory will allow us to repeat some facts of the theory in one complex variable. A motivating example, where multi-dimensional complex analysis can occur in mathematical research, is the iteration of multi-dimensional analytic maps like for example the Hénon map family C 2 → C 2 T : z w → z 2 + c − aw z , where a, c are complex parameters. If a = 0, one obtains, by restriction to the first coordinate, the iteration of a one dimensional map z → z 2 + c. A theorem in dynamical system theory assures that the fixed point (z c , w c) near the center is stable: if we start iterating with an initial condition near that point, we stay near that point. This stability does no more hold in C 2. The theory to understand the iteration of the Hénon map in C 2 heavily needs the theory of complex variables in several dimensions. The aim of this two hour introduction is 1. to show that part of complex analysis in several variables can be obtained from the one-dimensional theory essentially by replacing indices with multi-indices. Examples of results which extend are Cauchy's theorem, the Taylor expansion, the open mapping theorem or the maximum theorem. 2. to show in two examples that there are new features in several dimensions. For example, a multi-dimensional Riemann mapping theorem or a multi-dimensional Piccard theorem does no 1