Lemma 5.24.7F3A890BB2575

Proof. First suppose that $u$ is plurisubharmonic. Near a point of $Z$, choose a plurisubharmonic function $\psi \leq 0$ with $\psi =-\infty $ on $Z$. For $t>0$, extend $u+t\psi $ by $-\infty $ on $Z$. The local upper bound for $u$ makes this extension upper semicontinuous. It is plurisubharmonic off $Z$, and its submean inequality on complex lines is automatic at points of $Z$, where its value is $-\infty $. It is therefore plurisubharmonic on the whole neighborhood. As $t\downarrow 0$, these functions increase, with a common local upper bound. By the local upper-envelope theorem, the regularization of their limit is a plurisubharmonic function that agrees with $u$ almost everywhere off $Z$, hence everywhere off $Z$ by lemma 5.24.3. By upper semicontinuity and the submean inequality, its value at a point of $Z$ is the upper limit of $u$ off $Z$, so it is the upper-limit extension of $u$.

For a smooth background, fix a positive constant form $\kappa $ on a coordinate chart and let $\epsilon >0$. On a sufficiently small ball, freeze $\theta $ to a constant form $\theta '$ with $-\epsilon \kappa \leq \theta -\theta '\leq \epsilon \kappa $, and choose a quadratic potential $q$ for $\theta '+\epsilon \kappa $. Applying the preceding argument to the plurisubharmonic function $u+q$ shows that the upper-limit extension satisfies $\theta +\mathrm{dd}^{\mathrm{c}}u\geq -2\epsilon \kappa $ on the ball. Since the extension is defined by upper limits, it does not depend on the ball, and letting $\epsilon \downarrow 0$ gives $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$. Finally, analytic subsets are locally pluripolar: locally, a proper analytic subset is contained in the $-\infty $ locus of the logarithm of the sum of the squared moduli of local defining functions. $\square$