Proposition 15.20.1.1750C7E75315 Let $\omega $ be a Hermitian form on $X$, let $\mathrm{d}V$ be a smooth positive volume form, and let $0\leq f\in L^ p(\mathrm{d}V)$, $p>1$, be nonzero. Let $c>0$ be the unique scalar for the bounded equation $\mathrm{MA}_\omega (u)=cf\, \mathrm{d}V$ (theorem 14.19.2). For $\lambda >0$ denote by $u_\lambda $ the unique bounded solution of
\[ \mathrm{MA}_\omega (u_\lambda )=\mathrm{e}^{\lambda u_\lambda }f\, \mathrm{d}V, \qquad a_\lambda =u_\lambda -\sup _ Xu_\lambda . \]
The family $\{ a_\lambda :0<\lambda \leq 1\} $ is relatively compact in $C(X)$. Every cluster limit as $\lambda \downarrow 0$ is a normalized bounded solution of $\mathrm{MA}_\omega (a)=cf\, \mathrm{d}V$.
Suppose a group $G$ of biholomorphisms of $X$ preserves $\omega $ and the measure $f\, \mathrm{d}V$. Then every $a_\lambda $ and every such cluster limit is $G$-invariant. In particular, the zero-exponent equation has a $G$-invariant normalized solution.
Proof.
The barriers of proposition 14.19.34 give a uniform oscillation bound for $u_\lambda $ and uniform convergence $\mathrm{e}^{\lambda u_\lambda }\to c$. Thus the equation densities are uniformly bounded in $L^ p$ for $0<\lambda \leq 1$ and tend to $cf$ in $L^ p$ as $\lambda \downarrow 0$. Apply theorem 14.19.21 to obtain relative compactness of $a_\lambda $ in $C(X)$ and the asserted equation for every cluster limit.
For each $g\in G$, pullback of the positive-exponent equation shows that $u_\lambda \circ g$ solves the same equation. Its uniqueness gives $u_\lambda \circ g=u_\lambda $. Surjectivity of $g$ preserves the supremum, so $a_\lambda \circ g=a_\lambda $. Uniform convergence preserves these equalities for every $g$. Compactness gives at least one cluster limit as $\lambda \downarrow 0$, yielding the invariant solution.
$\square$
Pome
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