Lemma 5.38.47A2D4C65C667

Proof. The local $\mathrm{dd}^{\mathrm{c}}$-Poincaré lemma gives real distributional potentials $v_ i$ on the sets of a cover of $Z$ by coordinate neighborhoods, with $T=\mathrm{dd}^{\mathrm{c}}v_ i$; for positive $T$ these potentials are plurisubharmonic. See [Chapter III, Section 1.18 and Proposition 1.19, DemCADG]. On overlaps $\mathrm{dd}^{\mathrm{c}}(v_ i-v_ j)=0$. Taking the Euclidean trace shows that $v_ i-v_ j$ is a harmonic distribution, so it is smooth by elliptic regularity.

Choose a smooth locally finite partition of unity $(\chi _ i)$ subordinate to the cover and put $v=\sum _ i\chi _ i v_ i$, each term being extended by zero as a distribution. On the $j$th neighborhood,

is smooth. Thus $\beta =T-\mathrm{dd}^{\mathrm{c}}v=-\mathrm{dd}^{\mathrm{c}}(v-v_ j)$ is smooth on each neighborhood, hence on $Z$, and it is real and closed. For positive $T$, the same identity expresses $v$ locally as a plurisubharmonic function plus a smooth function. Finally, if $T=\beta +\mathrm{dd}^{\mathrm{c}}v=\beta '+\mathrm{dd}^{\mathrm{c}}v'$ are two such decompositions, then $\mathrm{dd}^{\mathrm{c}}(v-v')=\beta '-\beta $ is smooth; taking a smooth Hermitian trace and applying elliptic regularity shows that $v-v'$ is smooth. $\square$