Example 5.6.65A6B397A8B30

Example 5.6.6.5A6B397A8B30 On $\mathbb {C}^2$, the current $T=[\{ z_1=0\} ]$ has the canonical local potential $\log |z_1|^2$, by theorem 5.5.6 and proposition 5.5.3. For $F(w)=(0,w)$, the composition $\log |z_1|^2\circ F$ is identically $-\infty $, and so is the composition of every canonical local potential, since these differ from $\log |z_1|^2$ by smooth functions; thus proposition 5.6.5 does not define $F^*T$. For $G(w)=(w,0)$, the composition is $\log |w|^2$, and $G^*T=[0]$ is the Dirac mass at the origin. Thus the current of integration over a line can be restricted to a transverse line, but not to the line itself.