Corollary 18.26.20C441ECFEBCE3

Proof. By theorem 11.1.7, every bounded $\omega $-plurisubharmonic potential has mass $A$. Thus $v_+(\omega )=A$ for the upper mass of definition 13.14.14, and $X$ has the bounded mass property by proposition 13.14.15.

Fix $v\sim tH$. Its pole set is $S$, and it is locally bounded off $S$. In a coordinate neighborhood of $S$, choose a smooth local correction $q$ such that $a=v+q$ is plurisubharmonic. Since $\chi =1$ there, $a-th$ is bounded near $S$. By theorem 5.27.18, the full product of $h$ is the weak limit of the measures $(\mathrm{dd}^{\mathrm{c}}h_ L)^ n$, and the support and mass assertions of lemma 18.26.15 show that this limit is supported on $S$ with total mass $M$. The invariance of residual measures in lemma 5.27.19 therefore gives

Set $v_ k=\max (v,-k)$. After each fixed local smooth correction, these functions decrease to a plurisubharmonic function whose unbounded locus lies in $S$. The theorem on full local products and the expansion (5.28.1) identify the weak limit of $(\omega +\mathrm{dd}^{\mathrm{c}}v_ k)^ n$ with a global positive measure $T_ v$ of total mass $A$; the local expressions agree on overlaps because they are limits of the same bounded global measures. By corollary 5.34.3, every closed positive power $(\mathrm{dd}^{\mathrm{c}}a)^ j_{\mathrm{full}}$ with $j<n$ has coefficient measures vanishing on $S$, and smooth forms also give $S$ zero mass. Hence only the top-degree power contributes on $S$, and $T_ v(S)=Mt^ n$. Off $S$, the truncations eventually agree with $v$ on each compact set, so there $T_ v$ equals the non-pluripolar measure of $v$. Removing the mass on $S$ proves the formula for every representative $v\sim tH$.

Both singularity volumes of $tH$ thus equal $A-Mt^ n$. The current inequality $\omega +\mathrm{dd}^{\mathrm{c}}(tH)\geq (1-Ct)\omega $ shows that this volume is positive, uniformly for $0<t\leq t_0<1/C$. By theorem 13.19.5, ordinary projection preserves both volumes and gives the normalized models. Their order and distances follow as in corollary 18.26.18, with the zero endpoint defined separately.

The support and Lelong-number argument in the proof of proposition 18.26.17 uses that the background is smooth and that its polar restriction is closed, but not that the background is closed. It therefore gives vanishing first polar currents on all modifications and trivial multiplier ideals at all weights in the present setting as well. The strict volume deficit shows that every model with positive parameter is not an ideal model. Choosing $t_0$ small enough puts the interval in $\mathcal M_\delta (\omega )$. Finally, on a surface the only condition is $\mathrm{dd}^{\mathrm{c}}\omega =0$, which is the Gauduchon condition, and Gauduchon forms exist by theorem 11.4.2. $\square$