Lemma 5.8.2.3C70A4ABB14A Let $\Omega \subset \mathbb {C}^ n$ be open, $n\ge 1$, let $u\colon \Omega \to [-\infty ,\infty )$ be a locally integrable function, let $a\in \mathbb {C}^ n$ be a unit vector, and let $\chi \in C^\infty _ c(\Omega )$ be nonnegative. Write the points of $\mathbb {C}^ n$ as $v+ta$ with $v\in (\mathbb {C}a)^\perp $ and $t\in \mathbb {C}$, and put $K_ v=\{ t\in \mathbb {C}:v+ta\in \operatorname{Supp}\chi \} $. Assume that for almost every $v\in (\mathbb {C}a)^\perp $ there is an open set $G_ v\subset \{ t\in \mathbb {C}:v+ta\in \Omega \} $ containing $K_ v$ such that the restriction of $t\mapsto u(v+ta)$ to each connected component of $G_ v$ is subharmonic, possibly identically $-\infty $. Then
\[ \Bigl \langle \sum _{j,k}a_ j\bar a_ ku_{j\bar k},\chi \Bigr \rangle \geq 0. \]
Consequently, if this hypothesis holds for every unit vector $a$ and every nonnegative $\chi \in C^\infty _ c(\Omega )$, then $\mathrm{dd}^{\mathrm{c}}u\geq 0$. This is the case, in particular, if the restriction of $u$ to each connected component of each affine complex-line section of $\Omega $ is subharmonic.
Proof.
The map $(v,t)\mapsto v+ta$ is a unitary isomorphism of $(\mathbb {C}a)^\perp \times \mathbb {C}$ onto $\mathbb {C}^ n$, so it preserves Lebesgue measure, and by the chain rule $\partial _ t\partial _{\bar t}[\chi (v+ta)]=\sum _{j,k}a_ j\bar a_ k\chi _{j\bar k}(v+ta)$. We extend $u$ and $\chi $ by zero outside $\Omega $. Since $u$ is locally integrable and the derivatives of $\chi $ are bounded and supported in $\operatorname{Supp}\chi $, Fubini's theorem gives
\[ \Bigl \langle \sum _{j,k}a_ j\bar a_ ku_{j\bar k},\chi \Bigr \rangle =\int _{(\mathbb {C}a)^\perp }I(v)\, \mathrm{d}V_{2n-2}(v), \]
where
\[ I(v)=\int _{\mathbb {C}}u(v+ta)\, \partial _ t\partial _{\bar t}[\chi (v+ta)]\, \mathrm{d}V_2(t). \]
Applied to $|u|\mathbf1_{\operatorname{Supp}\chi }$, Fubini's theorem also shows that for almost every $v$ the function $t\mapsto u(v+ta)$ is integrable on $K_ v$. It therefore suffices to show that $I(v)\geq 0$ for every $v$ for which this integrability holds and an open set $G_ v$ as in the hypothesis exists.
Fix such a $v$. The compact set $K_ v$ is covered by the disjoint open connected components of $G_ v$, so it meets only finitely many of them, say $U_1,\ldots ,U_ N$, and each set $K_ v\cap U_ i$ is compact, being the complement in $K_ v$ of the open set $K_ v\cap \bigcup _{l\neq i}U_ l$. Hence the function $\chi _ i(t)=\chi (v+ta)$, $t\in U_ i$, is a nonnegative test function on $U_ i$, and
\[ I(v)=\sum _{i=1}^ N\int _{U_ i}u(v+ta)\, \partial _ t\partial _{\bar t}\chi _ i(t)\, \mathrm{d}V_2(t). \]
If $\chi _ i\equiv 0$, the $i$-th term vanishes. Otherwise $\chi _ i>0$ on a nonempty open subset of $U_ i$, contained in $K_ v$, on which $t\mapsto u(v+ta)$ is integrable and hence finite almost everywhere. Thus the subharmonic function $w_ i(t)=u(v+ta)$ on the domain $U_ i$ is not identically $-\infty $, and it is locally integrable by proposition 3.4.8. Since $\partial _ t\partial _{\bar t}=\frac14\Delta $, the $i$-th term equals $\frac14\langle \Delta w_ i,\chi _ i\rangle $, which is nonnegative by theorem 3.4.12(1). Hence $I(v)\geq 0$.
If the hypothesis holds for every unit vector $a$ and every $\chi $, then $\sum _{j,k}a_ j\bar a_ ku_{j\bar k}$ is a positive distribution for every unit vector $a$, and by homogeneity for every $a\in \mathbb {C}^ n$. Since $\mathrm{dd}^{\mathrm{c}}u$ is a real $(1,1)$-current, the matrix criterion for positivity in section 5.4 shows that $\mathrm{dd}^{\mathrm{c}}u\geq 0$. For the last assertion, take $G_ v=\{ t\in \mathbb {C}:v+ta\in \Omega \} $ for every $v$; its connected components are the connected components of the affine complex-line sections of $\Omega $ in the direction $a$.
$\square$
Pome
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