Example 10.17.9C15ADAE71322

Example 10.17.9.C15ADAE71322 Let $\mu :Y\to \mathbb P^2$ be the blowup of a point, with exceptional curve $E$, let $\omega _{\mathrm{FS}}$ be the Fubini–Study form on $\mathbb P^2$ with total volume one, and put $\alpha _ Y=\mu ^*\omega _{\mathrm{FS}}$ and $v=0$. The decomposition (10.17.1) holds on $Z=Y$ with no divisor, the residual class $\{ \alpha _ Y\} $ is nef and big, and

Nevertheless, there is no bounded $q\in \mathrm{PSH}(Y,\alpha _ Y)$ for which $\alpha _ Y+\mathrm{dd}^{\mathrm{c}}q\geq \delta \kappa $ with $\delta >0$ and $\kappa $ Kähler. Indeed, since $q$ is bounded, its restriction to $E$ is defined, and the disk criterion of lemma 5.43.2 gives

because $\alpha _ Y|_ E=0$. Integration over the compact curve $E$ contradicts Stokes' theorem. Thus positive trace mass gives the one-sided comparison of proposition 10.17.7, but a Kähler current in the class may need additional singularities.