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11.38 Entropy bounds531A2D2A66CF

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For a Hermitian background and the zero reference, a lower bound for the $L^ q$ norm of the density in the equation and an upper bound coming from a Gauduchon trace place $B_0$ between two logarithmic integrals. Dualizing gives two-sided entropy estimates, which identify the set where $H_0$ is finite. Throughout this section, $\mathrm{d}V$ is normalized by $\mathrm{d}V(X)=1$.

Proposition 11.38.1.170451FB65CB Let $\theta =\omega $ be a Hermitian form, and let $\varphi =0$. There are real constants $A$ and, for every $q>1$, $a_ q$ such that every Borel probability measure $\nu $ satisfies

\[ a_ q+\frac1q\operatorname {Ent}_{\mathrm{d}V}(\nu ) \leq H_0(\nu ) \leq A+n\operatorname {Ent}_{\mathrm{d}V}(\nu ). \]

The inequalities include the value $+\infty $. Thus $H_0(\nu )<\infty $ if and only if $\nu $ has finite entropy, and a family of probability measures has uniformly bounded values of $H_0$ if and only if its entropies are uniformly bounded.

Proof. In dimension one, $\omega $ is a Kähler form, and proposition 11.37.3, which rests on (11.37.2), gives $H_0=\log \int _ X\omega +\operatorname {Ent}_{\mathrm{d}V}$, so the assertions follow. Suppose that $n\geq 2$.

For $G\in C(X)$, apply the lower bound for $\| F\| _ q$ in theorem 14.19.19 to a bounded solution with density $F=\mathrm{e}^{B_0(G)+G}$. It gives a constant $c_ q>0$, independent of $G$, such that

\[ B_0(G)\geq \log c_ q-\frac1q\log \int _ X\mathrm{e}^{qG}\, \mathrm{d}V. \]

For the upper bound, fix a smooth Gauduchon metric $\gamma $, rescaled so that $\int _ X\omega \wedge \gamma ^{n-1}=1$, and write

\[ k_\gamma =\left(\frac{\gamma ^ n}{\mathrm{d}V}\right)^{(n-1)/n}, \qquad m=\min _ X k_\gamma >0. \]

For smooth $G$, the trace bound (11.5.6) gives

\[ \mathrm{e}^{B_0(G)/n}\int _ X\mathrm{e}^{G/n}k_\gamma \, \mathrm{d}V\leq 1. \]

Uniform approximation and continuity of $B_0$ extend this inequality to continuous $G$. Consequently

\[ B_0(G)\leq -n\log m-n\log \int _ X\mathrm{e}^{G/n}\, \mathrm{d}V. \]

Taking the defining suprema in the two scalar bounds and applying lemma 11.37.1 with $F=qG$ and $F=G/n$, respectively, proves both inequalities for every probability measure, including infinite values, with $a_ q=\log c_ q$ and $A=-n\log m$. The finite-domain and bounded-family assertions follow by taking $q=2$. $\square$

11.38.2 The mass bounds as entropy differences7BF91F477A54

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.

11.38.4 Entropy for a prescribed ordinary type562884633BF8

Return to the standing setting: $\theta $ is semipositive and big, and $\varphi $ is a model reference, with $\varepsilon \in (0,1)$ as in (16.28.67). The part $(1-\varepsilon )\theta $ of the background left over by the smaller-background envelope gives a uniform lower bound for the density norm, while comparison in a larger Hermitian background gives the upper bound. Together they extend the entropy estimates to the prescribed type.

Corollary 11.38.5.A13D502ACAAC For every $q>1$ there is a finite real $a_ q$, and there is a finite real $A$, such that

\[ a_ q+\frac1q\operatorname {Ent}_{\mathrm{d}V}(\nu ) \leq H_\varphi (\nu ) \leq A+n\operatorname {Ent}_{\mathrm{d}V}(\nu ) \]

for every Borel probability measure $\nu $, including infinite values. Thus $H_\varphi (\nu )$ is finite exactly when $\nu $ has finite entropy, and uniformly bounded dual values are equivalent to uniformly bounded entropies. For fixed $\theta $ and $\varepsilon $, the constants can be chosen independently of the model reference $\varphi $.

Proof. For $G\in C(X)$, let $u\sim \varphi $ solve the equation with density $c\mathrm{e}^ G$, where $c=\mathrm{e}^{B_\varphi (G)}$. The lower bound $b_ q>0$ of lemma 15.12.2 for the $L^ q$ norm of the density gives $c\| \mathrm{e}^ G\| _ q\geq b_ q$, hence

\[ B_\varphi (G)\geq \log b_ q-\frac1q \log \int _ X\mathrm{e}^{qG}\, \mathrm{d}V. \]

For the upper bound, choose a Hermitian form $\Omega \geq \theta $, and let $B_{\Omega ,0}$ and $H_{\Omega ,0}$ denote its normalization scalar and dual functional for the zero reference. Let $v$ be a bounded solution for $G$ in this background; being bounded, it has a smaller-background envelope $P_{\delta \Omega }(v)\geq \inf _ Xv$ for every $0<\delta <1$. Moreover $u\in \mathrm{PSH}(X,\Omega )$, $u\preceq v$, and positivity gives

\[ \mathrm{MA}^{\mathrm{ext}}_\Omega (u) \geq \mathrm{MA}^{\mathrm{ext}}_\theta (u) =c\mathrm{e}^ G\, \mathrm{d}V. \]

If $c>\mathrm{e}^{B_{\Omega ,0}(G)}$, the singular strict domination principle, applied in the background $\Omega $ to $v$ and $u+t$ for every real $t$, would give $v\geq u+t$ for all $t$, which is impossible. Hence $B_\varphi \leq B_{\Omega ,0}$, and therefore $H_\varphi \leq H_{\Omega ,0}$; proposition 11.38.1 now gives the upper bound.

Taking the defining supremum in the lower scalar estimate and applying lemma 11.37.1 with $F=qG$ gives the lower bound for every probability measure, including infinite values, with $a_ q=\log b_ q$. The finite-domain and bounded-family assertions follow by taking $q=2$. Finally, lemma 15.12.2 is uniform in the reference for fixed $\theta $ and $\varepsilon $, and the upper bound uses only $\Omega $, which proves the last assertion. $\square$

This gives compactness when the reference varies. Let the model references $\varphi _ j$ satisfy the smaller-background condition with the same $\varepsilon $, let $G_ j\in L^\infty (\mathrm{d}V)$ satisfy $\| G_ j\| _\infty \leq M$ for a fixed constant $M$, and choose $\nu _ j\in \mathcal A_{\varphi _ j}(G_ j)$. With $\Omega $ as in the proof,

\[ H_{\varphi _ j}(\nu _ j) =B_{\varphi _ j}(G_ j)+\int _ XG_ j\, \mathrm{d}\nu _ j \leq B_{\Omega ,0}(0)+2M. \]

The uniform lower estimate of corollary 11.38.5 bounds their entropies, and the Hermitian upper estimate then places all $\nu _ j$ in a single finite sublevel set of $H_{\Omega ,0}$. Hence theorem 11.35.4, applied in the background $\Omega $, gives weak compactness and uniform integrability of their densities, and every weak limit has finite entropy. Identifying such a limit as a supporting probability for a limiting reference requires further information.

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