Currents in complex geometry

Question

I would like to know some motivations for the study of currents in complex geometry, notably what theorems one can prove with them and why they are useful.

Answer 1

Here's one motivation for the study of currents (in complex geometry). In Kähler geometry, the study of canonical metrics is of particular interest. For example, Kähler--Einstein metrics have been the subject of great interest for decades. A Kähler metric (more precisely, Kähler form) is a smooth $(1,1)$-form $\omega$ which is given by $\omega(u,v) = g(Ju,v)$, where $g$ is the associated Riemannian metric, and $J$ is the underlying complex structure. A Kähler-Einstein metric is a Kähler metric whose Ricci curvature is a constant multiple of the metric, i.e., a Kähler metric $\omega$ satisfying $$\text{Ric}(\omega) = \lambda \omega$$ for some constant $\lambda \in \mathbb{R}$.

A standard approach to studying/finding (Kähler-)Einstein metrics is to use the (Kähler-)Ricci flow $$\frac{\partial \omega_t}{\partial t} = - \text{Ric}(\omega_t).$$

If the flow encounters singularities, or we wish to consider mildly singular metrics which naturally arise as candidates for "canonical metrics", then one deals with singular analogs of $(1,1)$-forms, namely, currents.