8B86B2B0A308
When every positive potential perturbation of $\omega $ is Gauduchon, the adjoint probability is the normalized Monge–Ampère measure. Mass invariance and the continuity of bounded products then extend this description to bounded potentials.
Proposition 11.25.1.3E5CF25D6E23 Assume that $\omega _ v$ is Gauduchon for every $v\in \mathcal H_\omega $. For every $u\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)$, the measure
\[ \mu _ u:=M^{-1}(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n \]
is a probability measure, which agrees with the preceding definition when $u\in \mathcal H_\omega $. If $u_ j,u\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)$ are uniformly bounded and $u_ j\to u$ in capacity, then $\mu _{u_ j}\rightharpoonup \mu _ u$.
Proof.
By proposition 11.23.4, theorem 11.1.7, the positive bounded product $(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n$ has mass $M$, and for $u\in \mathcal H_\omega $ its normalization is the smooth adjoint measure by proposition 11.23.4. The continuity of bounded products under convergence in capacity, in the sense of section 5.29, gives weak convergence of the products, and division by $M$ preserves it.
$\square$
The proposition applies in particular to Gauduchon surfaces. For a general Hermitian background, passing from smooth $u_ j\in \mathcal H_\omega $ to a limit $u\in \mathrm{PSH}(X,\omega )$ requires control of the multipliers $\rho _{u_ j}$ as well as of the potentials. The supporting probabilities of section 11.32 recover $\mu _ u$ in the smooth case. If a bounded solution has a continuous positive complex Hessian, theorem 11.34.11 gives a stationary supporting probability and approximation independence under uniform convergence of the logarithmic densities. We do not know whether, in general, the limit is independent of the approximating sequence, or whether a weak linearization satisfies an adjoint identity.
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