PomeTheorem 5.32.1.240BA0465940 For every $u\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)$, there are smooth functions $u_ j\downarrow u$ such that
\[ \omega +\mathrm{dd}^{\mathrm{c}}u_ j>0. \]
The functions $u_ j$ are uniformly bounded.
5.32 Regularization in a fixed Hermitian background26C061D9E9B2
26C061D9E9B2
For $\theta =\omega $, rescaling the approximants absorbs the curvature error into the positive background. Subtracting a constant first makes the approximants negative, so that this rescaling preserves their decreasing order.
Proof. Subtracting a constant, we may assume $u\leq -2$. Apply proposition 5.31.2 with $\theta =\omega $, obtaining smooth functions $v_ j\downarrow u$ and positive numbers $\varepsilon _ j\downarrow 0$ with $(1+\varepsilon _ j)\omega +\mathrm{dd}^{\mathrm{c}}v_ j\geq 0$. The compact sets $\{ v_ j\geq -1\} $ decrease to the empty set, since $u\leq -2$. After discarding finitely many terms, we may therefore assume $v_ j<0$ everywhere. Put
\[ u_ j=\frac{v_ j}{1+2\varepsilon _ j}. \]
Then
\[ \omega +\mathrm{dd}^{\mathrm{c}}u_ j \geq \frac{\varepsilon _ j}{1+2\varepsilon _ j}\omega >0. \]
Since $v_ j$ is negative and decreases, while the positive denominator decreases to one, $u_ j$ also decreases, and it converges to $u$. Moreover, $u\leq v_ j\leq u_ j\leq u_1$, which gives uniform boundedness. Adding back the subtracted constant proves the theorem. $\square$
This regularization provides the approximation used for products and mass in section 13.14. For an arbitrary smooth form $\theta $, proposition 5.31.2 only gives approximants whose curvature error tends to zero.
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