For a general Hermitian background, a uniform bound on the moments of normalized potentials with respect to the probabilities of one smooth maximizing sequence can replace the regularity of the maximizing current. Together with the envelope condition in a smaller background, it shows that the cluster limits of the solutions of the exponential equations are exact threshold solutions.
For a nonzero $0\leq f\in L^1(\mathrm{d}V)$ with $b_\omega (f)>0$, let $\mathcal S_ f$ be the class of threshold subsolutions with supremum zero and $\Phi _ f$ its greatest member, from proposition 15.28.6. As in proposition 12.10.20, $u_\lambda $ denotes the greatest solution of the exponential equation $\mathrm{MA}^{\mathrm{ext}}_\omega (u)=\mathrm{e}^{\lambda u}f\, \mathrm{d}V$, and $v_\lambda =u_\lambda -\sup _ Xu_\lambda $, $c_\lambda =\mathrm{e}^{\lambda \sup _ Xu_\lambda }$.
Theorem 12.10.43.A528C756F776 Let $0\leq f\in L^1(\mathrm{d}V)$ be nonzero with $b=b_\omega (f)>0$, and put $p=f^{1/n}$ and $R=\rho _\omega (p)=b^{-1/n}$. Suppose that there are smooth strictly positive $T_ j\in \mathscr G_\omega $, with associated probability measures $\nu _ j$, such that
\[ d_ j=D_ p(T_ j)\longrightarrow R,\qquad \mathrm{d}\nu _ j=d_ j^{-1}p k_{T_ j}\, \mathrm{d}V, \]
and a constant $C$ satisfying
797F5974140E
12.10.44
\begin{equation} \int _ X(-v)\, \mathrm{d}\nu _ j\leq C \quad \text{for all }j\text{ and all }v\in \mathrm{PSH}(X,\omega ) \text{ with }\sup _ Xv=0. \end{equation}
Then, for every $\lambda >0$,
\[ b\leq c_\lambda \leq b\mathrm{e}^{C\lambda },\qquad \Phi _ f-C\leq v_\lambda \leq 0. \]
Suppose in addition that $P_{\delta \omega }(\Phi _ f)$ is not identically $-\infty $ for some $0<\delta <1$. Whenever $\lambda _ k\downarrow 0$ and $v_{\lambda _ k}\to v$ in $L^1(X)$, the convergence holds in capacity and
\[ \mathrm{MA}^{\mathrm{ext}}_\omega (v)=bf\, \mathrm{d}V, \qquad \Phi _ f-C\leq v\leq \Phi _ f. \]
In particular an exact threshold solution exists.
Proof.
Since $\mathrm{MA}^{\mathrm{ext}}_\omega (v_\lambda )=c_\lambda \mathrm{e}^{\lambda v_\lambda }f\, \mathrm{d}V$, the trace inequality of lemma 13.14.12 and Stokes' theorem give, for each $j$ and $\lambda >0$,
\[ 1\geq c_\lambda ^{1/n}d_ j \int _ X\mathrm{e}^{\lambda v_\lambda /n}\, \mathrm{d}\nu _ j \geq c_\lambda ^{1/n}d_ j\mathrm{e}^{-C\lambda /n}. \]
The second inequality follows from Jensen's inequality and (12.10.44). Letting $j\to \infty $ and using $R=b^{-1/n}$, we obtain the upper bound for $c_\lambda $. The lower bound and the comparison with every normalized threshold subsolution were proved in corollary 15.35.3; applying that comparison to $\Phi _ f$, we obtain
\[ v_\lambda \geq \Phi _ f-\lambda ^{-1}\log (c_\lambda /b) \geq \Phi _ f-C. \]
Now let $v_{\lambda _ k}\to v$ in $L^1$. By proposition 15.28.6, the limit $v$ lies in $\mathcal S_ f$, hence below $\Phi _ f$. The lower bound passes to the limit, so $\Phi _ f-C\leq v\leq \Phi _ f$; in particular $v\sim \Phi _ f$ in the sense of definition 16.7.1. For all $k$ with $\lambda _ k\leq 1$,
\[ \mathrm{MA}^{\mathrm{ext}}_\omega (v_{\lambda _ k}) =c_{\lambda _ k}\mathrm{e}^{\lambda _ k v_{\lambda _ k}}f\, \mathrm{d}V \leq b\mathrm{e}^{C}f\, \mathrm{d}V. \]
The functions $|\mathrm{e}^{v_{\lambda _ k}}-\mathrm{e}^ v|$ are uniformly bounded and tend to zero in $\mathrm{d}V$-measure, so their integrals against the fixed $L^1$ density $f$ tend to zero. Hence the error term in (15.9.2) tends to zero, as required by proposition 15.9.3 with $\varphi =\Phi _ f$. The additional hypothesis $v\preceq \Phi _ f$ of that proposition holds since $v\leq \Phi _ f$, and the condition on $P_{\delta \omega }(\Phi _ f)$ is assumed. That proposition therefore gives convergence in capacity.
The continuity of damped measures under convergence in capacity, theorem 13.15.9, now identifies the equation without a mass-conservation argument. Write $g_ k=c_{\lambda _ k}\mathrm{e}^{\lambda _ k v_{\lambda _ k}}f$. We know $g_ k\to bf$ in $L^1$ and $0\leq g_ k\leq b\mathrm{e}^ Cf$ for large $k$. Hence
\[ \mathrm{e}^{v_{\lambda _ k}}\mathrm{MA}^{\mathrm{ext}}_\omega (v_{\lambda _ k}) =\mathrm{e}^{v_{\lambda _ k}}g_ k\, \mathrm{d}V \longrightarrow \mathrm{e}^ v bf\, \mathrm{d}V \]
in total variation, while theorem 13.15.9 gives the weak limit $\mathrm{e}^ v\mathrm{MA}^{\mathrm{ext}}_\omega (v)$. Dividing by $\mathrm{e}^ v$ off the pole set of $v$, which neither measure charges, gives the equation and the finiteness of the mass. Finally, compactness of normalized $\omega $-plurisubharmonic functions provides an $L^1$ cluster limit along any sequence $\lambda _ k\downarrow 0$, which proves existence.
$\square$
The moment condition holds if the densities of the $\nu _ j$ are uniformly bounded in $L^ q(\mathrm{d}V)$ for some $q>1$, by Hölder's inequality and the uniform bounds for the moments of normalized $\omega $-plurisubharmonic functions. More generally, a uniform entropy bound suffices. Write $a_ j=\mathrm{d}\nu _ j/\mathrm{d}V$ and suppose $\int _ Xa_ j\log a_ j\, \mathrm{d}V\leq H$ for all $j$. (By proposition 11.38.1, a uniform entropy bound is equivalent to a uniform bound for the dual entropy $H_0(\nu _ j)$ with zero reference defined there.) Choose $\alpha >0$ and $A$ from lemma 5.26.2, so that $\int _ X\mathrm{e}^{-\alpha v}\, \mathrm{d}V\leq A$ for every $v\in \mathrm{PSH}(X,\omega )$ with $\sup _ Xv=0$. The elementary entropy inequality, applied first to the truncations $v_ T=\max (v,-T)$ at levels $T>0$, gives
\[ \alpha \int _ X(-v_ T)a_ j\, \mathrm{d}V \leq \int _ Xa_ j\log a_ j\, \mathrm{d}V +\log \int _ X\mathrm{e}^{-\alpha v_ T}\, \mathrm{d}V \leq H+\log A. \]
To prove this inequality, apply $t\log t-t+1\geq 0$ to $t=a_ j/q$, integrate against $q\, \mathrm{d}V$, where $q=\mathrm{e}^{-\alpha v_ T}/\int _ X\mathrm{e}^{-\alpha v_ T}\, \mathrm{d}V$ is a probability density, and rearrange. Monotone convergence as $T\to \infty $ proves (12.10.44) with $C=(H+\log A)/\alpha $.
The same entropy bound also gives attainment of the dual supremum along this sequence. Indeed, $t\log t\geq -\mathrm{e}^{-1}$ and, for $K>1$,
\[ \int _{\{ a_ j>K\} }a_ j\, \mathrm{d}V \leq \frac{H+\mathrm{e}^{-1}}{\log K}. \]
This proves uniform integrability of the $a_ j$, and since $d_ j\to R>0$, also of the densities $p k_{T_ j}=d_ ja_ j$. By proposition 12.10.6, every weak cluster limit $T$ of $(T_ j)$ maximizes $D_ p$, and along a subsequence converging to $T$ the probabilities $\nu _ j$ converge to $R^{-1}p k_ T\, \mathrm{d}V$. For each normalized $v$, the function $-v$ is nonnegative and lower semicontinuous, so the lower semicontinuity of its integral under weak convergence of measures passes the moment bound to this limiting probability. These conclusions give no regularity of $T$.
The envelope condition holds if $\Phi _ f$ is bounded. In general, it and the moment bound are additional hypotheses; the tail condition on $f^{1/n}$ alone does not provide them in this argument. The theorem identifies the equation satisfied by each cluster limit. Uniqueness, which gave convergence of the whole family in proposition 12.10.20, is not available here.
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.