Example 5.3.91176DC9B7704

Example 5.3.9.1176DC9B7704 Let $Z$ be a complex submanifold of $X$ of dimension $r$, with its complex orientation. Then

defines a closed positive current of bidegree $(n-r,n-r)$ on $X$, the current of integration over $Z$. Since $Z$ is closed, $Z\cap \operatorname{Supp}\varphi $ is compact for every test form $\varphi $, so the integral is defined. In a chart in which $Z=\{ z_{r+1}=\cdots =z_ n=0\} $, the value $[Z](\varphi )$ is a constant times the integral over this coordinate plane of the coefficient of $\mathrm dz_ I\wedge \mathrm d\bar z_ I$ in $\varphi $, where $I=(1,\ldots ,r)$, because the pullbacks of $\mathrm dz_ j$ and $\mathrm d\bar z_ j$ to $Z$ vanish for $j>r$. Hence $[Z]$ is a current, and its coefficients in such a chart are constant multiples of Lebesgue measure on the plane. Since $[Z]$ vanishes on $X\setminus Z$, the charts of this kind and the charts contained in $X\setminus Z$ form an atlas in which $[Z]$ has measure coefficients, and $[Z]$ is a current of order zero. If $\varphi $ is strongly positive, so is its pullback $\varphi |_ Z$ to $Z$, which is therefore a nonnegative multiple of the volume form of $Z$ at each point (proposition 5.3.2); thus $[Z](\varphi )\geq 0$. Finally, since pullback commutes with $\mathrm d$ by (5.2.5), $[Z](\mathrm d\psi )=\int _ Z\mathrm d(\psi |_ Z)=0$ by proposition 5.2.6 applied on $Z$ when $r\geq 1$; when $r=0$ the current $[Z]$ has degree $2n$, and $\mathrm d[Z]=0$ because currents of degree $2n+1$ are zero.