Example 5.5.8DB42A2BDBC5B

Example 5.5.8.DB42A2BDBC5B Let $1\leq r\leq n$ and $f(z)=z_1^{a_1}\cdots z_ r^{a_ r}$ on $\mathbb C^ n$, with positive integers $a_ j$. Then $\log |f|^2=\sum _ ja_ j\log |z_ j|^2$ everywhere, both sides being $-\infty $ on the zero set, and theorem 5.5.6 applied to each $z_ j$ gives $\mathrm{dd}^{\mathrm{c}}\log |f|^2=\sum _{j=1}^ r a_ j[\{ z_ j=0\} ]$ on all of $\mathbb C^ n$, including the intersections of the coordinate hyperplanes. More generally, let $g$ and $h$ be holomorphic and not identically zero on a connected coordinate neighborhood. The function $\log |g|^2-\log |h|^2$, defined outside the null set where $g$ or $h$ vanishes, is locally integrable, and its complex Hessian is $\mathrm{dd}^{\mathrm{c}}(\log |g|^2-\log |h|^2)=[\operatorname{div}(g)]-[\operatorname{div}(h)]$. This current depends only on the quotient $g/h$: if $g',h'$ is another such pair with $gh'=g'h$, the two logarithmic differences agree almost everywhere, hence as distributions. We regard the difference only as an element of $L^1_{\mathrm{loc}}$; in particular, it is assigned no values at the common zeros of $g$ and $h$.