5.36.3 Regularization of closed currentsD4C469F61056

We also use the regularization of closed currents in the form recalled in 5.18 Analytic singularities. Let $T=\eta +\mathrm{dd}^{\mathrm{c}}v$ be a closed real $(1,1)$-current, where $\eta $ is a smooth closed form and $v$ is quasi-plurisubharmonic, and set $\nu (T,x)=\nu (v,x)$. Suppose that $T\geq \gamma $ for a smooth real $(1,1)$-form $\gamma $. Smooth regularization gives smooth representatives $T_ j=\eta +\mathrm{dd}^{\mathrm{c}}v_ j$ with

where $C>0$ is a fixed constant and the continuous functions $\lambda _ j$ decrease pointwise to $\nu (T,\cdot )$. If all Lelong numbers of $T$ vanish, Dini's theorem shows that $\lambda _ j\to 0$ uniformly, so the loss of positivity becomes uniformly small. Analytic regularization instead gives representatives $S_ j$ with analytic singularities and

Thus analytic regularization preserves strict positivity up to a small loss and creates no positive Lelong number at a point where the Lelong number of $T$ vanishes.