Theorem 5.36.1 (Demailly regularization)3CA98F11CC5B

Theorem 5.36.1 (Demailly regularization).3CA98F11CC5B Let $u$ be quasi-plurisubharmonic on $X$ and let $\gamma $ be a continuous real $(1,1)$-form with $\mathrm{dd}^{\mathrm{c}}u\geq \gamma $. There are quasi-plurisubharmonic functions $u_ m$, defined for all sufficiently large positive integers $m$, and numbers $\epsilon _ m\downarrow 0$ such that

1. $u_ m$ has analytic type $(1/m,\mathcal I(mu))$;

2. $u_ m\geq u$ and $u_ m\to u$ pointwise and in $L^1(X)$;

3. $\mathrm{dd}^{\mathrm{c}}u_ m\geq \gamma -\epsilon _ m\omega $;

4. $\max \{ \nu (u,x)-n/m,0\} \leq \nu (u_ m,x)\leq \nu (u,x)$ for all $x$.

The sequence $(u_ m)$ need not be monotone.