Theorem 10.20.1C4F23D785797

Proof. Fix a Kähler form $\omega $ and put $f=\mu /\omega ^ n$. Since $X$ is compact and $\mu $ is smooth, $f\in L^ p(X,\omega ^ n)$ for every finite $p>1$. The classical estimate for this equation, discussed in section 25.21, shows that $\varphi $ has minimal singularities: there is a constant $C>0$ such that

The upper bound follows from the normalization.

Let $T=\theta +\mathrm{dd}^{\mathrm{c}}\rho $ be a Kähler current with analytic singularities as in theorem 10.13.2. Its potential $\rho $ has analytic singularities with smooth remainder, and for some constant $\varepsilon >0$,

Thus $\rho $ is smooth on $\Omega =\operatorname {Amp}(\alpha )$. After subtracting a constant we may assume $\rho \leq 0$, hence $\rho \leq V_\theta $, and (10.20.3) gives

In particular $\varphi $ is locally bounded on $\Omega $, and by locality its non-pluripolar equation there is the equation for bounded potentials. The density $f$ belongs to $L^ p$ on $X$ and is smooth and strictly positive on $\Omega $. Hence all hypotheses of corollary 15.18.50 hold for the strict barrier $\rho $, so $\varphi $ is smooth and $\theta +\mathrm{dd}^{\mathrm{c}}\varphi >0$ on $\Omega $. Since $\theta $ is closed, this form is Kähler. $\square$