Pome9E3723511E88For a smooth Hermitian equation, differentiating the scalar identifies the supporting probability with the adjoint probability of the linearized operator.
Theorem 11.34.9.7E60599DA339 Let $\theta =\omega $ be a Hermitian form, let $\varphi =0$, and let $F$ be a smooth real function. Let $u$ be the smooth solution of (11.33.1), unique up to an additive constant by corollary 14.24.17, and let $\mu _ u$ be its Gauduchon adjoint probability from proposition 11.23.2; for $n=1$, this is $\mu _ u=(\omega +\mathrm{dd}^{\mathrm{c}}u)/\int _ X\omega $. Then
Proof. Suppose first that $n\geq 2$. By uniqueness of the constant, the normalization scalar of the smooth equation in theorem 14.8.1 is $B_0$, and that theorem gives, for every smooth real $G$,
Let $\nu \in \mathcal A_0(F)$. Testing the supporting inequality at $F+tG$ for both signs of $t$ and letting $t\to 0$ shows that $G$ has the same integral against $\nu $ and $\mu _ u$. Since smooth functions are uniformly dense in $C(X)$, it follows that $\nu =\mu _ u$, and the assertion follows from nonemptiness of $\mathcal A_0(F)$. In dimension one, Stokes' theorem gives $c_0(F)=\int _ X\omega /\int _ X\mathrm{e}^ F \mathrm{d}V$; differentiating its logarithm in the same way yields the stated measure, since $\omega +\mathrm{dd}^{\mathrm{c}}u=c_0(F)\mathrm{e}^ F\mathrm{d}V$. $\square$
Consequently, in the Hermitian case the supporting probabilities at a continuous $F$ can be approached through smooth data. If smooth $F_ j$ converge uniformly to $F$, then by proposition 11.34.3 every weak cluster point of the Gauduchon adjoint probabilities of the solutions for $F_ j$ belongs to $\mathcal A_0(F)$, and is therefore absolutely continuous. If $\mathcal A_0(F)$ is a singleton, which holds in particular for every $F\in \mathscr U_0$ and under the continuous positive Hessian hypothesis of theorem 11.34.11, these probabilities converge to its unique member, independently of the approximation. We do not know whether $\mathcal A_0(F)$ is a singleton for every continuous $F$, even in this Hermitian setting; the same question is open for the singular references considered above.
For $n\geq 2$, the Hermitian zero-reference case admits a quantitative strengthening: corollary 12.9.3 bounds the densities of all supporting probabilities uniformly in $L^ q$, for $1<q<n/(n-1)$, when $\| F\| _\infty $ is bounded, and also for $q=n/(n-1)$ when $\omega $ has positive lower mass $v_-(\omega )>0$ in the sense of definition 13.14.14. The proof uses the Gauduchon-current formulation of section 12.1, and we do not claim such a bound for an arbitrary singular reference.
A weak linearized operator requires more than a supporting probability. In the same Hermitian zero-reference setting, theorem 12.8.2 represents each supporting probability by a determinant-weighted positive $\mathrm{dd}^{\mathrm{c}}$-closed current. Identifying that current with a scalar multiple of a weak power $(\omega +\mathrm{dd}^{\mathrm{c}}u)^{n-1}$ would require a well-defined product and its compatibility with approximation.
A similar representation holds for a functional defined by subsolutions rather than by an equation in a prescribed singularity type. Let $0\leq f\in L^1(\mathrm{d}V)$ be nonzero with $f^{1/n}\in \mathcal D_\omega ^0$, the space of densities with uniformly vanishing Gauduchon tails from 15.32.7. Then proposition 12.10.9 describes the supporting probabilities of $F\mapsto \log b_\omega (\mathrm{e}^ Ff)$, where $b_\omega (\cdot )$ is the density threshold of theorem 15.28.4.
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