Theorem 23.1.76C1374FAD9C60

Proof. Put $m=b-V_\Omega >0$. For necessity, let $\sigma $ be a shrinking defect for $(\Omega ,b)$. Proposition 13.15.20, applied with $h=\Omega $, gives a positive closed limit $K$ of the squares of the intermediate forms with $\sigma =3\Omega \wedge K$, and $\alpha \wedge K=0$ as in the proof of lemma 23.1.72. The local coefficient and Fourier argument of that proof gives $K=\pi ^*\tau $ for a positive measure $\tau $ on $\mathbb P^2$. Consequently

and $\mu $ has mass $m$.

Conversely, fix such a $\mu $ and set $\tau =\mu /(3A_\Omega )$. By theorem 23.1.74, choose a sequence $f_ j=t_ ju_ j$ realizing the shrinking defect $\mathcal L(6\tau )$ for $\omega $. The positive forms $H_ j=\mathrm{dd}^{\mathrm{c}}f_ j+t_ j\omega $ have trace mass $O(t_ j)$, and $H_ j^2\rightharpoonup \pi ^*\tau $ by lemma 23.1.72. Choose $C\geq 1$ with $\omega \leq C\Omega $, and put

The trace mass of $G_ j$ is $O(t_ j)$, and $G_ j^2-H_ j^2$ has trace mass $O(t_ j^2)$. Thus $G_ j^2\rightharpoonup \pi ^*\tau $ and $3\Omega \wedge G_ j^2\rightharpoonup \mathcal L_\Omega \mu $.

Unlike $\omega $, the form $\Omega $ need not be Gauduchon, so the mass polynomial may have a linear term. Define

These estimates follow by expanding $\mathrm{dd}^{\mathrm{c}}f_ j=G_ j-Ct_ j\Omega $ and using the preceding trace bounds. Stokes' formula removes the cubic integral, so

For large $j$, $q_ j>0$. The positive root

gives $M_\Omega (\lambda _ jf_ j)=b$. The normalized function $U_ j=u_ j/C$ is $\Omega $-plurisubharmonic, and $w_ j=\lambda _ j f_ j=r_ jU_ j$ with $r_ j=\lambda _ jCt_ j\to 0$.

To identify the full weak limit, first note that expanding $G_ j=\mathrm{dd}^{\mathrm{c}}f_ j+Ct_ j\Omega $ gives

Hence $G_ j^3\to 0$ in total variation. The positive expansion

converges weakly to $\Omega ^3+\mathcal L_\Omega \mu $, since the second and fourth terms tend to zero in total variation. Given prescribed bounds $\varepsilon _ j$, pass to a sufficiently late subsequence with $r_ j<\varepsilon _ j$ at every index. This preserves the normalization, the exact mass and the weak limit.

The fiber probabilities $h_ p^\Omega $ have smooth densities relative to $h_ p$ that are bounded above and below by positive constants, uniformly in $p$. Thus $\mathcal L_\Omega \mu $ and $\mathcal L\mu $ are bounded by fixed positive multiples of each other, and the assertions about support, point atoms and pluripolar sets follow from theorem 23.1.74. The global functions with fiber poles also remain available: if $\Omega \geq c\omega $, then $c\psi \in \mathrm{PSH}(X,\Omega )$.

Finally, for smooth $u\in \mathrm{PSH}(X,\omega )$ and $c>0$ with $\Omega \geq c\omega $,

Since $v_+(\omega )=+\infty $ by corollary 23.1.40 and the supremum may be computed over smooth potentials (proposition 13.14.15), it follows that $v_+(\Omega )=+\infty $. $\square$