Pome7E60599DA339Theorem 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 $\omega +\mathrm{dd}^{\mathrm{c}}u=c_0(F)\mathrm{e}^ F\mathrm{d}V$. $\square$
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