Theorem 5.5.6 (Poincaré–Lelong)8D4BA339507E

Proof. If $X$ has dimension zero, then $f$ has no zeros and both sides vanish. Let $X$ have dimension $n\geq 1$. The set of points near which $f$ vanishes identically is open, and it is closed by the identity theorem, applied in connected coordinate neighborhoods; since $f$ does not vanish identically on any component of $X$, this set is empty. Hence proposition 5.5.3 applies in every connected chart: the function $\log |f|^2$ is locally integrable, and $T=\mathrm{dd}^{\mathrm{c}}\log |f|^2$, given by $T(\varphi )=\int _ X\log |f|^2\, \mathrm{dd}^{\mathrm{c}}\varphi $ independently of coordinates, is a positive current with measure coefficients. It is closed because $\mathrm d\mathrm{dd}^{\mathrm{c}}=\mathrm d\mathrm d\mathrm d^{\mathrm c}=0$. We show that $T$ gives zero mass to $S$ and agrees with $m[V_{\mathrm{reg}}]$ outside $S$.

Step 1: $T$ gives $S$ zero mass. Let $x_0\in S$, and choose coordinates and a polydisc $P$ centered at $x_0$ as in lemma 5.5.5. For each $j$, the last assertion of proposition 5.5.3 shows that the diagonal coefficient $(\log |f|^2)_{j\bar j}$ vanishes on $S\cap P\subset \pi _ j^{-1}(N_ j)$. By the computation of the trace measure in section 5.4, the sum of these diagonal coefficients is a positive multiple of the trace measure of $T$ in these coordinates, which therefore vanishes on $S\cap P$; by theorem 5.3.7 so do all coefficient measures of $T$ in these coordinates. By (5.3.6), the vanishing of the coefficient measures on a Borel set does not depend on the chart. Every compact subset of $S\cap U$, for a chart $U$, is covered by finitely many such polydiscs $P$, so by inner regularity all coefficient measures of $T$ vanish on $S\cap U$.

Step 2: $T=m[V_{\mathrm{reg}}]$ on $X\setminus S$. Since $T$ is defined without reference to coordinates, we may compute it in any holomorphic coordinates. Away from $V$ the function $\log |f|^2$ is pluriharmonic, so $T=0$ there. Near a point $p\in V_{\mathrm{reg}}$ write $f=g\, a^ m$ as above, and use holomorphic coordinates $(t,x')$ with $t=a$. Then $\log |f|^2=\log |g|^2+m\log |t|^2$, where $\log |g|^2$ is pluriharmonic. In one variable, $\partial _ t\partial _{\bar t}\log |t|^2=\pi \delta _0$ by example 3.4.15. Since $\log |t|^2$ does not depend on $x'$, Fubini's theorem shows that its distributional derivatives involving $x'$ or $\bar x'$ vanish and that $\langle (\log |t|^2)_{t\bar t},\psi \rangle =\pi \int \psi (0,x')\, \mathrm dV_{2n-2}(x')$ for test functions $\psi $; thus $\mathrm{dd}^{\mathrm{c}}\log |t|^2=(\mathrm i/2\pi )(\log |t|^2)_{t\bar t}\, \mathrm dt\wedge \mathrm d\bar t$. Let $\varphi $ be a test form of bidegree $(n-1,n-1)$, and let $\varphi _0=c(t,x')\bigwedge _ l(\frac{\mathrm i}2\mathrm dx'_ l\wedge \mathrm d\bar x'_ l)$ be its component without $\mathrm dt$ and $\mathrm d\bar t$, the only one that contributes. Since $(\mathrm i/2\pi )\, \mathrm dt\wedge \mathrm d\bar t\wedge \varphi _0=\pi ^{-1}c\, \mathrm dV_{2n}$, we obtain

that is, $\mathrm{dd}^{\mathrm{c}}\log |t|^2=[\{ t=0\} ]$ in the sense of example 5.3.9, the hypersurface carrying its complex orientation. Therefore $T=m[V_{\mathrm{reg}}]$ near $p$.

Step 3: extension across $S$. Let $U$ be a chart and $K\subset U$ compact. By Step 2, the trace measure of the positive current $m[V_{\mathrm{reg}}]$ on $U\setminus S$ is the restriction of $\mu _ T$, so its mass on $K\setminus S$ is at most $\mu _ T(K)<\infty $. By the coefficient bound of theorem 5.3.7, applied on $U\setminus S$, the coefficient measures of $m[V_{\mathrm{reg}}]$, a positive current of bidegree $(1,1)$, have total variation at most $C_{n,1}\mu _ T(K)$ on $K\setminus S$. Hence they extend by zero on $S$ to complex Radon measures on $U$. Indeed, each coefficient measure $\sigma $ is a complex combination of the positive and negative variations of $\operatorname{Re}\sigma $ and $\operatorname{Im}\sigma $, which are positive Radon measures on $U\setminus S$ bounded by $|\sigma |$; extended by zero on $S$, they become Borel measures on $U$ that are finite on compact sets, hence Radon by proposition 2.2.2. By Step 2 the coefficient measures of $T$ agree with those of $m[V_{\mathrm{reg}}]$ on $U\setminus S$, and by Step 1 they vanish on $S$. Thus the extensions by zero are the coefficient measures of the current $T$ of order zero; we denote this current by $[\operatorname{div}(f)]$. This proves (5.5.7), and $[\operatorname{div}(f)]=T$ is positive and closed. $\square$