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We now turn to the metric geometry of solutions of Monge–Ampère equations in big classes. In this section $X$ is a compact Kähler manifold of dimension $n\geq 1$, and $\alpha \in H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$ is a big class with a smooth closed representative $\theta $. Its ample locus is $\operatorname {Amp}(\alpha )=X\setminus E_{\mathrm{nK}}(\alpha )$. The solution of the Monge–Ampère equation with a smooth positive density has minimal singularities, and a Kähler current with analytic singularities provides a barrier that is smooth on the ample locus. The interior estimate of corollary 15.18.50 then gives smoothness there.
Let $\operatorname {vol}(\alpha )$ be the volume of $\alpha $, the non-pluripolar mass of a current with minimal singularities in $\alpha $, and let $\mathcal E(X,\theta )$ be the class of potentials $\varphi \in \mathrm{PSH}(X,\theta )$ of full mass, that is, with $\int _ X\langle (\theta +\mathrm{dd}^{\mathrm{c}}\varphi )^ n\rangle =\operatorname {vol}(\alpha )$.
Theorem 10.20.1.C4F23D785797 Let $\mu $ be a smooth positive volume form with $\mu (X)=\operatorname {vol}(\alpha )$. The normalized full-mass solution $\varphi \in \mathcal E(X,\theta )$ of
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10.20.2
\begin{equation} \bigl \langle (\theta +\mathrm{dd}^{\mathrm{c}}\varphi )^ n\bigr \rangle =\mu , \qquad \sup _ X\varphi =0, \end{equation}
is smooth on $\operatorname {Amp}(\alpha )$, and the restriction of $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ to this locus is a Kähler form.
The existence and uniqueness of $\varphi $ are part of the classical theory of Monge–Ampère equations in big classes, discussed in section 25.21; only the regularity assertion needs proof.
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
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10.20.3
\begin{equation} V_\theta -C\leq \varphi \leq V_\theta . \end{equation}
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 $\varepsilon >0$,
\[ T=\theta +\mathrm{dd}^{\mathrm{c}}\rho \geq \varepsilon \omega , \qquad \{ \rho =-\infty \} =E_{\mathrm{nK}}(\alpha ). \]
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
\[ \rho -C\leq \varphi \leq 0. \]
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.
$\square$
The proof uses that the density is smooth and positive on the entire regular locus of the barrier. The theorem says nothing about the distance from the ample locus to its complement: the metric $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ may reach the omitted set along paths of finite length. To determine that length, one must compare the normal and tangential directions of the metric. The following sections first examine zeros of the density; singular radial densities then separate boundedness of the potential from completeness of the metric.
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