EDIT: Related to "Derivative of a logarithm and Dirac delta function", "2-dimensional delta function (complex plane)" and "How to define a delta function on complex plane?".
In a physics paper by Agaganic (arXiv:0905.3415), the author defines a connection of a certain circle bundle over a space as
$$A_{RR} = \sum_{j}q_{j}\frac{d\phi_j}{\phi_j}$$
where $q_{j}$'s are scalars (real numbers), and $\phi_{j}$'s are in fact coordinates parametrizing the ambient space $\mathbb{C}^{N+d}$ in which one embeds a Calabi-Yau 4-fold $X_{4}$ which is the total space of a circle bundle (with connection $A_{RR}$) over a base space which is $X_{3} \times \mathbb{R}$ (here $X_{3}$ is a Calabi-Yau 3-fold).
The specific question I have is far more elementary than the setting of the physical problem. It is claimed that
$$F_{RR} = dA_{RR} = \sum_{j}q_j \omega_j$$
where $\omega_{j} = \delta^2(\phi_j)d\phi_j \wedge d\overline{\phi}_j$ is a 2-form with $\delta$-function support corresponding to $\phi_j = 0$.
What is the reason for the appearance of this delta-function 2-form? I believe it is related to this post: How to define a delta function on complex plane?, but I think we're saying the following is true for the $\delta^{2}(\phi_j)$ function introduced in this paper.
$$\delta^{2}(z) \stackrel{?}{=} -\overline{\partial}\left(\frac{1}{2\pi i z}\right)$$
