PomeDECDD98C3557Corollary 16.28.48.DECDD98C3557 Let $\tau >0$, $\Psi \in \mathcal L_\tau \cap C^2(\mathbb C^ n)$ and $a\in K$. If $F_{\tau ,K,\Psi |_ K}(a)=\Psi (a)$, but $F_{\delta ,K,\Psi |_ K}(a)>\Psi (a)$ for some $\delta >\tau $, then
Proof. The complex Hessian of $\Psi $ is semipositive. If its top power did not vanish at $a$, it would be positive definite at $a$, and by continuity $\mathrm{dd}^{\mathrm{c}}\Psi \geq c\omega _0$ on a neighborhood of $a$ for some $c>0$. Theorem 16.28.46 would then give regular contact at $a$ for every $\delta \geq \tau $, a contradiction. $\square$
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