In a closed background the difference between $H_0$ and the entropy is constant. In general it measures how much the total Monge–Ampère mass of bounded potentials varies: its infimum and supremum recover the lower and upper masses $v_-(\omega )$ and $v_+(\omega )$ of definition 13.14.14.
Let $\mathcal P_{\mathrm{ent}}(X)$ denote the set of probability measures of finite entropy relative to the fixed smooth probability volume $\mathrm{d}V$. By proposition 11.38.1, the difference $H_0(\nu )-\operatorname {Ent}_{\mathrm{d}V}(\nu )$ is a finite real number for every $\nu $ in this set.
Proposition 11.38.3.A3720120A48C For every Hermitian background $\omega $,
\[ \begin{split} \inf _{\nu \in \mathcal P_{\mathrm{ent}}(X)} \bigl (H_0(\nu )-\operatorname {Ent}_{\mathrm{d}V}(\nu )\bigr ) & =\log v_-(\omega ),\\ \sup _{\nu \in \mathcal P_{\mathrm{ent}}(X)} \bigl (H_0(\nu )-\operatorname {Ent}_{\mathrm{d}V}(\nu )\bigr ) & =\log v_+(\omega ). \end{split} \]
The same extrema are obtained over smooth strictly positive probability measures. Here $\log 0=-\infty $ and $\log (+\infty )=+\infty $. Thus the difference is bounded above if and only if $X$ has the bounded mass property, and bounded below if and only if $\omega $ has positive lower mass.
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
\[ B_0(G)\geq \log v_-(\omega )-\log Z_ G. \]
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
\[ H_0(\nu _ G)-\operatorname {Ent}_{\mathrm{d}V}(\nu _ G) \geq B_0(G)+\log Z_ G. \]
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
\[ H_0(\sigma _ G)-\operatorname {Ent}_{\mathrm{d}V}(\sigma _ G) \leq B_0(G)+\log Z_ G. \]
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.
$\square$
The difference does not depend on the smooth positive reference volume. For smooth real $a$, replacing $\mathrm{d}V$ by $\mathrm{d}V'=\mathrm{e}^ a\mathrm{d}V$ changes both $H_0(\nu )$ and $\operatorname {Ent}(\nu )$ by $-\int _ Xa\, \mathrm{d}\nu $, by the transformation formula preceding proposition 11.38.1, and finiteness of the entropy is unchanged because $a$ is bounded. Similarly, if $h$ is a smooth real function and $\omega +\mathrm{dd}^{\mathrm{c}}h$ is also Hermitian, transporting bounded solutions by $u\mapsto u-h$ preserves $B_0$, and hence this difference. Finally, the difference is constant on $\mathcal P_{\mathrm{ent}}(X)$ exactly when $v_-(\omega )=v_+(\omega )$, that is, when all bounded $\omega $-plurisubharmonic functions have the same mass.
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