II Currents and complex structures

Abstract

In this chapter we introduce two new ideas, one coming from differential geometry (currents) and the other coming from analysis (complex analytic manifolds and their associated complex structures), which we will use frequently throughout the rest of this book. We start by defining currents, which for differential forms play the role that distributions play for functions, and then consider the regularisation problem for currents defined on a C^∞ differential manifold. Solving this problem, which is easy in $\mathbb{R}^{n}$ by means of convolution, obliges us to introduce kernels with similar properties to the convolution kernel. We will also study the Kronecker index of two currents, which generalises the pairing of a current and a differential form. This index enables us to prove a fairly general Stokes’ formula which will be used in Chapters III and IV.