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I am new to the theory of currents and pursuing this note-"https://webusers.imj-prg.fr/~tien-cuong.dinh/Cours2005/Master/cours.pdf". My concept of product of positive currents is limited to that note and based on that I am stuck in computing the following two $(2,2)$ currents(distributions) on $\mathbb{C}^2$ (Exercise 6.3.5 and 6.3.6):

  1. $dd^{c}\log|z_{1}|\wedge dd^{c}\log|z_{2}|$ and
  2. $(dd^{c}log\|z\|)^{2}$.

My feeling is that both of them will be dirac-delta at the origin but unable to show it precisely.

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    Your first statement holds for $u\in\mathcal{C}^2$, otherwise (like in the case of $u=\log|f|$ where f can have zeroes in the domain) $\frac{{\partial}^2 u}{\partial z_{k}\partial\bar{z_{l}}}(z)$ is a distribution and product of distributions is not defined. So, I don't see how "det" function is coming. – 2017-02-26
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    How $(dd^{c}\log\|z\|)^{2}$ is the standard volume form, is it not $(dd^{c}\|z\|)^{2} ?$. – 2017-02-26

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