10.31.21 Exponential and normalized equations6720AA8AAF70

Proof. First suppose that $\mu $ is a probability measure with $A_\theta (\mu )<\infty $, and let $\mu _ j$ be its smooth probability approximations. The theorems for the minimal ordinary type, theorem 15.16.2, theorem 15.16.7, apply to their smooth, nonzero densities.

For the normalized equation, they give $v_ j\sim V$ with $\sup _ Xv_ j=0$ and $\mu _{v_ j}=c_ j\mu _ j$. Since $\mu _ j$ is a probability measure, the mass bounds $\ell _ V\leq \mu _{v_ j}(X)\leq M_ V$ give $\ell _ V\leq c_ j\leq M_ V$. After extraction, $c_ j\to c>0$ and $v_ j\to v$ in $L^1(X)$. The preceding proposition applies, so $v\in \mathcal E_ V$ and $\mu _{v_ j}\rightharpoonup \mu _ v$. The right-hand sides converge weakly to $c\mu $, which proves the equation.

For the positive-exponent equation, take $u_ j\sim V$ satisfying $\mu _{u_ j}=\mathrm{e}^{\lambda u_ j}\mu _ j$, and write $a_ j=\sup _ Xu_ j$ and $v_ j=u_ j-a_ j$. The mass bounds and Jensen's inequality give

and $\sup _ jA_\theta (\mu _ j)<\infty $ by lemma 5.44.12. Thus $a_ j$ is bounded above and below. After extraction, $a_ j\to a$, $v_ j\to v$ in $L^1(X)$, and $\mu _{v_ j}\leq C\mu _ j$. The preceding proposition gives $v\in \mathcal E_ V$, convergence in $L^1(\mu )$ and weak convergence of the Monge–Ampère measures. Set $u=v+a$. By the smoothing lemma, the right-hand sides converge weakly to $\mathrm{e}^{\lambda u}\mu $. This proves the equation, and $u\in \mathcal E_ V$.

The same conclusions hold for every nonzero finite measure whose normalization to a probability measure has finite $A_\theta $, since scaling the measure only changes the positive scalar, or translates the solution of the exponential equation by a constant.

Now let $\mu $ be an arbitrary finite non-pluripolar measure of positive mass. The bounded-carrier decomposition lemma 15.24.4, applied to $\omega $, writes

For $\mu _ k=\min (f,k)\nu $, we have $\mu _ k(X)>0$. By lemma 5.44.10, the normalization of $\mu _ k$ has finite $A_\theta $ for each fixed $k$.

For the normalized equation, the preceding case gives $v_ k\in \mathcal E_ V$ with $\sup _ Xv_ k=0$ and $\mu _{v_ k}=c_ k\mu _ k$, where

After extraction, $c_ k\to c>0$ and $v_ k$ converges in $L^1$ to some $v\not\equiv -\infty $. All the measures $\mu _{v_ k}$ are bounded by the fixed finite non-pluripolar measure $M_ V\mu /\mu _1(X)$. The fixed-dominator theorem theorem 10.31.5 gives $v\in \mathcal E_ V$ and weak convergence to $\mu _ v$. Since $c_ k\mu _ k\to c\mu $ in total variation, $\mu _ v=c\mu $.

For the positive-exponent equation, let $u_ k\in \mathcal E_ V$ solve $\mu _{u_ k}=\mathrm{e}^{\lambda u_ k}\mu _ k$. Since $\mu _ k$ increases with $k$, exponential order (corollary 13.19.3) gives $u_{k+1}\leq u_ k$. Also

The lower bound on the suprema shows that the decreasing limit $u$ is not identically $-\infty $. The same fixed-dominator theorem puts $u$ in $\mathcal E_ V$ and gives weak convergence of the left-hand sides. Dominated convergence with respect to $\nu $, with the integrable bound $\mathrm{e}^{\lambda \sup _ Xu_1}f$, gives convergence of the right-hand sides to $\mathrm{e}^{\lambda u}\mu $ in total variation.

Finally, corollary 13.19.3 with reference $V$, applied in both directions, gives uniqueness of the exponential solution and of the positive scalar. $\square$

Proof. Choose $\rho \in \mathrm{PSH}(X,\theta )$ with $\theta _\rho \geq a\omega $ for some $a>0$. Then $\rho \in \mathrm{PSH}(X,\theta -a\omega )$. Volume addition (proposition 13.17.2), followed by monotonicity in the singularities (theorem 13.17.1), gives

In the addition step the big background is $a\omega $, and the added positive current is $(\theta -a\omega )_\rho $. Bounded mass gives finiteness of the upper volume. $\square$