PomeExample 5.19.6.AAD0C8775B34 As an important example of 5.19.5, given $T=\theta +\mathrm{dd}^{\mathrm{c}}\varphi \in \mathcal{Z}_+(X)$ and $x\in X$, we define
Again, as in 5.16.3, this differs from the definitions in some literature by a factor of $2$. But given our normalization
\[ \mathrm{dd}^{\mathrm{c}}=\frac{\mathrm{i}}{2\pi }\partial \overline{\partial }, \]
(5.19.7) seems to be the most natural choice.
The key example to keep in mind is the following:
\[ \nu ([0],0)=1, \]
where $[0]$ is the current of integration at $0\in \mathbb {P}^1$. In fact, as a simple application of the Green's second identity, one can verify that
\[ \frac{\mathrm{i}}{2\pi }\partial \overline{\partial }\log |z|^2=\delta _0, \]
where the right-hand side is the Dirac delta distribution at $0\in \mathbb {C}$.
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