PomeD2CCD76890C8Proposition 11.34.13.D2CCD76890C8 Let $a>0$, let $b$ be smooth and positive on $\mathbb P^1$, and let $A$ be continuous and positive there, with $\int _{\mathbb P^1}Aw=a$. For
one has
Proof. The Green solution of $\mathrm{dd}^{\mathrm{c}}u=(A-a)w$ on $\mathbb P^1$ has continuous complex Hessian and is bounded. Its pullback yields $\sigma =Aw+b\kappa >0$, and $\sigma ^{k+1}=(k+1)Ab^ kw\wedge \kappa ^ k$. Thus scalar uniqueness gives $c_0(F)=1$, and theorem 11.34.11 gives a unique support. For every smooth $h$ depending on all variables, the block decomposition
shows that $(A/a)w\otimes \nu _\kappa $ is stationary: the first term integrates to zero by Stokes on $\mathbb P^1$, and the second by stationarity on $Y$ and Fubini. This candidate has a bounded density with respect to $\mathrm dV$. The one-dimensional annihilator of lemma 11.24.1 and probability normalization therefore identify it with the supporting probability. $\square$
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