Proposition 5.5.337EEAF275A32

Proof. Local integrability. Near a point where $f\neq 0$, the function $\log |f|^2$ is smooth. Let $a$ be a zero of $f$. After a translation and a complex-linear change of coordinates, which preserve local integrability, we may assume that $a=0$ and that $f(0,w)$ does not vanish identically in the last variable $w$ (lemma 5.5.1). Then $f(0,\cdot )$ has a zero of some order $m\geq 1$ at $0$, and by theorem 5.5.2, $f=gW$ on a polydisc $P'\times \{ |w|<\rho \} $, where $g$ is a holomorphic unit and $W(z',w)=\prod _{l=1}^ m(w-\lambda _ l(z'))$ with all roots in $\{ |w|<\rho _1\} $ for some $\rho _1<\rho $. Choose $r\in (\rho _1,\rho )$ and replace $P'$ by a smaller concentric polydisc with compact closure in $P'$; since $g$ is continuous and nowhere zero, $\log |g|^2$ is then bounded on $P'\times \{ |w|<r\} $. For $|\lambda |\leq \rho _1$ we have

with $C$ independent of $\lambda $: the positive part of $\log |w-\lambda |$ is at most $\max \{ \log (r+\rho _1),0\} $, and after the translation $\zeta =w-\lambda $ the integral of its negative part is at most $\int _{|\zeta |<1}(-\log |\zeta |)\, \mathrm dV_2(\zeta )$. Since $|\log |W(z',w)||\leq \sum _ l|\log |w-\lambda _ l(z')||$, it follows that $\int _{|w|<r}|\log |W(z',w)||\, \mathrm dV_2(w)\leq mC$ for every $z'\in P'$. (The estimate holds for each fiber separately, so we need not choose the roots as measurable functions of $z'$.) By Tonelli's theorem $\log |W|^2$, and hence $\log |f|^2$, is integrable on $P'\times \{ |w|<r\} $, a neighborhood of $a=0$.

The slice formula. For $n=1$, example 3.4.15 gives $\Delta \log |f|^2=4\pi \sum _ am(a)\delta _ a$ locally, that is, $(\log |f|^2)_{z\bar z}=\pi \sum _ am(a)\delta _ a$; this is (5.5.4) with $\mathbb C^0$ a point. In general, write $P=P'\times D$ with $D$ a disc in the variable $z_ j$. By lemma 5.5.1, the set $N_0$ of points $z'\in P'$ over which $f$ vanishes identically on $D$ is null. By Fubini's theorem,

the inner integral being an integrable function of $z'$. For $z'\notin N_0$, the one-variable case applied to $f(z',\cdot )$ on $D$ shows that the inner integral equals $\pi \sum _ am(z',a)\varphi (z',a)$. This proves (5.5.4); the integrand on the right is measurable and integrable, being equal almost everywhere to the inner integral.

Positivity. By (5.5.4), $(\log |f|^2)_{j\bar j}$ is nonnegative on nonnegative test functions supported in polydiscs contained in $\Omega $, hence on all nonnegative test functions, which are finite sums of such functions by lemma 2.5.2. This holds in every system of complex linear coordinates, since $f$ remains holomorphic after a linear change of coordinates. Applying this after a unitary change of coordinates, as in the matrix criterion of section 5.4, we see that $\sum _{j,k}(\log |f|^2)_{j\bar k}\xi _ j\bar\xi _ k$ is a positive measure for every $\xi \in \mathbb C^ n$. By the matrix criterion, $\mathrm{dd}^{\mathrm{c}}\log |f|^2\geq 0$, and by theorem 5.3.7 its coefficients are measures.

Null sets. Let $\nu =(\log |f|^2)_{j\bar j}$ on $P=P'\times D$, and let $\eta \geq 0$ be a test function on $D$. By (5.5.4), for every nonnegative test function $\psi $ on $P'$,

Both sides define positive Radon measures on $P'$ in $\psi $, the function $h_\eta $ being locally integrable by the identity itself. By the uniqueness in theorem 2.5.6, the image measure $A\mapsto \int _{\pi _ j^{-1}(A)}\eta \, \mathrm d\nu $ of $\eta \nu $ under $\pi _ j$ is therefore $h_\eta \, \mathrm dV_{2n-2}$. Every null set is contained in a Borel null set, by outer regularity of Lebesgue measure, so we may assume that $N$ is a Borel set. Then $\int _{\pi _ j^{-1}(N)}\eta \, \mathrm d\nu =\int _ Nh_\eta \, \mathrm dV_{2n-2}=0$. Letting $\eta $ increase to $1$ on $D$ along a sequence, we obtain $\nu (\pi _ j^{-1}(N))=0$ by monotone convergence.

The canonical representative. The trace of $\mathrm{dd}^{\mathrm{c}}\log |f|^2$ is positive, so by theorem 3.4.12 the class of $\log |f|^2$ has a canonical subharmonic representative $v$, given by the limits of ball averages in (3.4.13). At a point where $f\neq 0$, the function $\log |f|^2$ is continuous, so $v$ agrees with it there. At a zero $a$, there is $C$ with $|f(z)|\leq C|z-a|$ near $a$, since $f$ is smooth and $f(a)=0$; hence the average of $\log |f|^2$ over $B(a,r)$ is at most $\log C^2+2\log r$, which tends to $-\infty $ as $r\to 0$. Thus $v(a)=-\infty =\log |f(a)|^2$, and $v=\log |f|^2$ everywhere. $\square$