5.36.4 Smooth remainders after resolution70049B9396E7

For arguments on a modification, the bounded remainder in an analytic presentation is sometimes insufficient. The following version of regularization gives a smooth remainder after resolution.

This classical theorem is applied to the closed current $\mathrm{dd}^{\mathrm{c}}u$ with the smooth lower bound $-\alpha $; see section 25.3 for its source. The lower bound need not be closed, and $X$ need not be Kähler. What matters here is the smooth remainder $b$ after resolution; an analytic type, whose remainder is merely bounded, would not suffice. The sequence $(\widetilde u_ j)$ need not be the multiplier-ideal sequence of theorem 5.36.1, and we do not claim that the conclusions of the two theorems hold for one and the same sequence. If $\theta +\mathrm{dd}^{\mathrm{c}}u\geq \varepsilon \omega $ for some $\varepsilon >0$, applying the theorem with $\alpha =\theta -\varepsilon \omega $ and discarding the terms with $\delta _ j\geq \varepsilon $ gives approximants with $\theta +\mathrm{dd}^{\mathrm{c}}\widetilde u_ j\geq (\varepsilon -\delta _ j)\omega >0$; thus strict positivity is preserved.