Proposition 11.38.3A3720120A48C

Proof. Write $Z_ G=\int _ X\mathrm{e}^ G\, \mathrm{d}V$. A bounded solution for $G\in C(X)$ has mass $\mathrm{e}^{B_0(G)}Z_ G$. If $v_-(\omega )>0$, the lower mass bound gives

Taking the defining supremum for $H_0$ and using lemma 11.37.1 gives $H_0-\operatorname {Ent}_{\mathrm{d}V}\geq \log v_-(\omega )$ on $\mathcal P_{\mathrm{ent}}(X)$. In the same way, if $v_+(\omega )<\infty $, the upper mass bound gives $H_0-\operatorname {Ent}_{\mathrm{d}V}\leq \log v_+(\omega )$. These bounds hold trivially when $v_-(\omega )=0$ or $v_+(\omega )=+\infty $.

To prove sharpness, first take a smooth real $G$ and its Gibbs probability $\nu _ G=Z_ G^{-1}\mathrm{e}^ G\, \mathrm{d}V$, which has finite entropy. Testing the supremum defining $H_0$ with $G$ gives

In the other direction, let $\sigma _ G$ be the smooth strictly positive Gauduchon adjoint probability of theorem 11.34.9. It lies in $\mathcal A_0(G)$, so its dual value is $B_0(G)+\int _ XG\, \mathrm{d}\sigma _ G$, and lemma 11.37.1 yields

As $G$ runs over smooth real functions, the masses $\mathrm{e}^{B_0(G)}Z_ G$ run over the masses $\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n$ of all smooth $u$ with $\omega +\mathrm{dd}^{\mathrm{c}}u>0$, since such a $u$ solves the equation for $G=\log ((\omega +\mathrm{dd}^{\mathrm{c}}u)^ n/\mathrm{d}V)$. By theorem 5.32.1, the mass of every bounded $\omega $-plurisubharmonic function is a limit of such masses. Hence the infimum and supremum of $\mathrm{e}^{B_0(G)}Z_ G$ over smooth $G$ are $v_-(\omega )$ and $v_+(\omega )$. Since $\nu _ G$ and $\sigma _ G$ are smooth and strictly positive, the last two inequalities prove sharpness on this smaller domain, including the cases $v_-(\omega )=0$ and $v_+(\omega )=+\infty $. The two equivalences follow. $\square$