Let $K$ be number field and $\mathcal{O}_K$ its ring of integers. In Arakelov theory the idea is to enrich an arithmetic scheme $X$ over $\mathcal{O}_K$ "at infinity", that is to add data at the archemedian places. Now in higher dimensions, an Arakelov divisor of $X$ is given by the datum of a cycle of $X$ and the added data of a Green current $g_D$ (or rather one for each archemedian place of $K$).
My understanding of why currents show up is roughly this: if $M$ is a manifold, and $C,D$ closed submanifolds of complementary dimension, then $$C\cdot D=\int_C \delta_D$$ where $\delta_D$ is the "integrations along D"-current. Or rather this is true if $\delta_D$ were smooth along $C$. This doesn't have to be true, and so we would like to change $\delta_D$. What sort of class is smooth, well for instance the elements $A^{p-1,p-1}(X)\subset D^{p-1,p-1}(X)$. So we change $\delta_D$, namely we add $dd^{\mathrm{c}} g$ for all well-choose $g$ such that $$dd^{\mathrm{c}}g+\delta_D\in A^{p-1,p-1}(X),$$ and in particular smooth.
The existence of such Green currents can therefore, if I understand correctly, be understood as some sort of "analytic moving lemma". My problem now is that it seems that the "smoothing out"-data of a Green current seems very auxiliary, similar to a principal divisor in algebraic geometry used to have cycles meet transversely.
I'm aware that this isn't the case, and I'm sure there is a good reason why, I'm just not aware of it. So why isn't a Green current auxiliary data?
