(Motivational Discussion on singularities) We denote

$$f_{S}=1-z_{1}-z_{2}=1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}$$

and write the 3-current $S$ as

$$S=\delta(f_{S})\frac{\sqrt{-1}}{4\pi^{2}}df_{S}\wedge d\bar{f}_{S}\wedge d\mu,$$

which defines a generalised function $\delta(f_{S})$ satisfying the measures identities:

$$\int\frac{\sqrt{-1}}{4\pi^{2}}d\eta_{p}\wedge d\bar{\eta}_{q}\wedge\delta(f_{S})\wedge df_{S}\wedge d\bar{f}_{S}\wedge d\mu=\int_{S}d\eta_{p}\wedge d\bar{\eta}_{q},$$

where the notation $\int_{S}d\eta_{p}\wedge d\bar{\eta}_{q}$ is the shorthand for the complex measure $f\mapsto\int_{S}fd\eta_{p}\wedge d\bar{\eta}_{q}$, and similarly for the LHS. Now $df_{S}=-2\pi\sqrt{-1}(z_{1}d\eta_{1}+z_{2}d\eta_{2})$, so

$$\begin{cases}-\int_{S}|z_{2}|^{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{1}\wedge d\bar{\eta}_{1},\\ -\int_{S}|z_{1}|^{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{2}\wedge d\bar{\eta}_{2},\\ \int_{S}\bar{z}_{1}z_{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{1}\wedge d\bar{\eta}_{2},\\ \int_{S}\bar{z}_{2}z_{1}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{2}\wedge d\bar{\eta}_{1}.\\ \end{cases}$$

The distributional equation (4.6) is written in components as

$$-\frac{1}{4\pi}\left(\frac{\partial^{2}w^{p\bar{q}}}{\partial\mu\partial\mu}+4\frac{\partial^{2}v}{\partial\eta_{p}\partial\bar{\eta}_{q}}\right)=\delta(f_{S})z_{p}\bar{z}_{q}.$$

Multiplying these equations by $a^{p\bar{q}}$ and summing up, we obtain

(4.7)

$$-\frac{1}{4\pi}\Delta_{a}v=\delta(f_{S})a^{p\bar{q}}z_{p}\bar{z}_{q},$$

or equivalently the measure equality

$$(\Delta_{a}v)d\text{Vol}_{a}=-\int_{S}\pi\sqrt{-1}A^{1/2}a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}=-\int_{S}2\pi A^{1/2}d\mathcal{A},$$

where $d\mathcal{A}=\frac{\sqrt{-1}}{2}a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}$ is the natural area form on $S$. A natural guess for $w^{p\bar{q}}$ is then

(4.8)

$$-\frac{1}{4\pi}\Delta_{a}w^{p\bar{q}}=A^{-1}\delta(f_{S})z_{p}\bar{z}_{q},$$

or equivalently

$$(\Delta_{a}w^{p\bar{q}})d\text{Vol}_{a}=-\int_{S}2\pi\frac{A^{-1/2}z_{p}\bar{z}_{q}}{a^{i\bar{j}}z_{i}\bar{z}_{j}}d\mathcal{A},$$

where summation convention is used. The singularity around $S$ to leading order looks like (cf. Section 4.4 below)

$$v\sim\frac{A^{1/2}}{2R},\quad w^{p\bar{q}}\sim\frac{A^{-1/2}z_{p}\bar{z}_{q}}{2Ra^{i\bar{j}}z_{i}\bar{z}_{j}},\quad R\sim(\frac{|f_{S}|^{2}}{4\pi^{2}a^{i\bar{j}}z_{i}\bar{z}_{j}}+A\mu^{2})^{1/2},$$

which is compatible with the singularity in the distributional equation (4.6).