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):
- $dd^{c}\log|z_{1}|\wedge dd^{c}\log|z_{2}|$ and
- $(dd^{c}log\|z\|)^{2}$.
My feeling is that both of them will be dirac-delta at the origin but unable to show it precisely.
Using by Poincare-Lelong formula [see https://www.scielo.cl/pdf/proy/v32n1/art01.pdf] we get $dd^c\log |z_1|=[z_1=0]$ and $dd^c \log |z_2|=[z_2=0]$. From $\text{supp}(T_1 \wedge T_2)\subset \text{supp}(T_1)\cap \text{supp}(T_2)$ implies that $\text{supp}(dd^c\log |z_1|\wedge dd^c\log |z_2|)=\{z=0\}$. It means that $dd^c\log |z_1|\wedge dd^c\log |z_2|=C\delta_0$. In otherside we get $[z_k=0]=\frac{1}{2i}dz_k\wedge d\bar{z}_k,k\in\{1,2\}$ and $$\langle dd^c\log |z_1| \wedge dd^c\log |z_2|, \phi \rangle=\langle dx_1 \wedge dx_2 \wedge dy_1 \wedge dy_2, \phi \rangle=\phi(0)$$ and $C=1$. Last answer is $dd^c\log |z_1| \wedge dd^c\log |z_2|=\delta_0(z)$.