PomeTheorem 5.4.3.2408CCC1B5A0 Let $T$ be a closed positive $(1,1)$-current on a complex manifold of positive dimension. Every point has a coordinate neighborhood $U$ on which $T=\mathrm{dd}^{\mathrm{c}}u$ for a real function $u\in L^1_{\mathrm{loc}}(U)$. Since $\Delta u=4\sum _ ju_{j\bar j}\geq 0$ by the matrix criterion, the class of $u$ has a canonical subharmonic representative in the coordinates of $U$ (theorem 3.4.12), which we call a canonical local potential of $T$. Two canonical local potentials $u$ and $\widetilde u$ of $T$ on the same coordinate neighborhood differ by a smooth pluriharmonic function $h$, that is, a real smooth function with $\mathrm{dd}^{\mathrm{c}}h=0$: we have $\widetilde u=u+h$ at every point.
Theorem 5.4.32408CCC1B5A0
Proof. Fix a point and a coordinate chart around it, identified with an open set $\Omega \subset \mathbb C^ n$, and write $T$ as above with trace $\mu =\sum _ jt_{j\bar j}$, a positive Radon measure on $\Omega $. Choose a test function $\chi $ on $\Omega $ with $0\leq \chi \leq 1$ that equals one on a neighborhood $W$ of the point (lemma 2.5.1), and let $E_{2n}$ be the fundamental solution of proposition 3.1.13. By proposition 3.4.11, the function $u_0=4E_{2n}*(\chi \mu )$ is locally integrable and subharmonic, and $\Delta u_0=4\chi \mu $; in particular $\Delta u_0=4\mu $ on $W$. The current $R=T-\mathrm{dd}^{\mathrm{c}}u_0$ on $W$ is real and closed, and its coefficients $r_{j\bar k}=t_{j\bar k}-(u_0)_{j\bar k}$ have trace $\sum _ lr_{l\bar l}=\mu -\Delta u_0/4=0$, by (5.2.9). Using both relations (5.4.1) for $R$, we find
\[ \tfrac 14\Delta r_{j\bar k} =\sum _ l\partial _{z_ l}\partial _{\bar z_ l}r_{j\bar k} =\sum _ l\partial _{\bar z_ k}\partial _{z_ l}r_{j\bar l} =\partial _{z_ j}\partial _{\bar z_ k}\sum _ l r_{l\bar l}=0. \]
By Weyl's lemma (theorem 3.1.11) every coefficient $r_{j\bar k}$ is smooth, so $R$ is a smooth real closed $(1,1)$-form on $W$. By lemma 5.4.2, $R=\mathrm{dd}^{\mathrm{c}}v$ on a coordinate neighborhood $U\subset W$ of the point, with $v$ smooth and real. Then $u=u_0+v$ satisfies $\mathrm{dd}^{\mathrm{c}}u=T$ on $U$. Moreover $\Delta v=4\sum _ lr_{l\bar l}=0$, so $v$ is harmonic. Since the sum of a subharmonic function and a real harmonic function is subharmonic (see the discussion following theorem 3.4.5), $u$ is a locally integrable subharmonic function. By the uniqueness in theorem 3.4.12, it is the canonical subharmonic representative of its class.
Now let $u$ and $\widetilde u$ be canonical local potentials of $T$ on the same coordinate neighborhood $U$. Their classes in $L^1_{\mathrm{loc}}(U)$ differ by a real distribution with vanishing complex Hessian, hence with vanishing Laplacian. By Weyl's lemma, this difference is a smooth real harmonic function $h$, and $\mathrm{dd}^{\mathrm{c}}h=0$. Hence, as above, $u+h$ is a locally integrable subharmonic function, and it agrees almost everywhere with $\widetilde u$. By the uniqueness in theorem 3.4.12, $\widetilde u=u+h$ at every point of $U$, including the points where $u=-\infty $. $\square$
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