Example 13.5.63D09B5B530AA

Example 13.5.6.3D09B5B530AA On $\mathbb P^1$, the current of integration $[0]$ at the point $0$ satisfies $\langle [0]\rangle =0$. Indeed, near $0$ it has the potential $\log |z|^2$ (theorem 5.5.6), whose pole set is $\{ 0\} $, so $\langle [0]\rangle =\mathbf1_{\{ z\neq 0\} }[0]=0$ there by proposition 13.4.7; away from $0$ the current vanishes. Thus $[0]$ has mass one while $\langle [0]\rangle $ has mass zero: discarding the poles can change the cohomology class, even in degree one.