Lemma 5.46.43B857D475432

Proof. On a small coordinate ball, take plurisubharmonic local potentials $a,b$ with $T=\mathrm{dd}^{\mathrm{c}}a$ and $S=\mathrm{dd}^{\mathrm{c}}b$, by Theorem 5.4.3. Put $h=a-b\in L^1_{\mathrm{loc}}$. The trace identity gives $\mathrm{dd}^{\mathrm{c}}h\wedge \eta ^{n-1}=0$ in the sense of distributions. Dividing by the smooth positive volume form $\eta ^ n$ gives a scalar second-order operator with smooth coefficients whose principal matrix is a positive multiple of the inverse of the Hermitian matrix of $\eta $; it is uniformly elliptic on every smaller ball. Local elliptic regularity therefore makes $h$ smooth. Thus $T-S=\mathrm{dd}^{\mathrm{c}}h$ is smooth and carried by a Lebesgue null set, so its coefficient measures vanish. This proves equality locally, and hence globally. $\square$