Lemma 5.24.998E72E23E884

Proof. In coordinates centered at $x$, let $q$ be a quadratic polynomial with $\mathrm{dd}^{\mathrm{c}}q=\theta (x)$, and let $\varepsilon >0$. On a sufficiently small ball about $x$, we have $\theta \leq \theta (x)+\varepsilon \mathrm{dd}^{\mathrm{c}}|z|^2$, so $u+q+\varepsilon |z|^2$ is plurisubharmonic, and $a+q+\varepsilon |z|^2$ touches it from above at $x$. By the submean inequality on complex lines through $x$, a $C^2$ function touching a plurisubharmonic function from above at $x$ has nonnegative complex Hessian at $x$. Hence $\theta (x)+\mathrm{dd}^{\mathrm{c}}a(x)+\varepsilon \mathrm{dd}^{\mathrm{c}}|z|^2\geq 0$, and letting $\varepsilon \to 0$ proves the lemma. $\square$