Proposition 12.18.2.89AE8B534538 Let $X=E^ n$, where $E=\mathbb C/(\mathbb Z+\mathrm i\mathbb Z)$ carries its flat area form of total area one, and write $\omega _ i$ for the pullback of this form from the $i$th factor. Set
\[ h=\sum _{i=1}^ n\omega _ i,\qquad \mathrm{d}V=\omega _1\wedge \cdots \wedge \omega _ n,\qquad V=n!,\qquad \alpha =\frac{n-1}{n}. \]
For $F=0$, the unique maximizing current is $T_*=h^{n-1}/V$, with $k_{T_*}=V^{-1/n}$ and $\mathfrak D_0(T_*)=V^{-1/n}$. There are smooth strictly positive closed forms $S_\varepsilon \in \mathscr G_ h$, $0<\varepsilon <\min \{ 1/4,\mathrm e^{-1}\} $, such that
\[ \| k_{S_\varepsilon }\| _{L^ q}=V^{-1/n},\qquad \mathfrak D_0(S_\varepsilon )\longrightarrow V^{-1/n}, \]
while for every fixed finite $p>q$,
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\begin{equation} \int _ X k_{S_\varepsilon }^{p}\, \mathrm{d}V \geq C_{n,p}[\log (1/\varepsilon )]^{-\alpha p} \varepsilon ^{2-2\alpha p}\longrightarrow +\infty . \end{equation}
Their $L^\infty $ norms also diverge. Each $S_\varepsilon $ is the $(n-1)$st power of a Kähler form in the fixed class $V^{-1/(n-1)}[h]$, and has the same Bott–Chern, de Rham and Aeppli classes as $T_*$. No $S_\varepsilon $ attains the maximum of $\mathfrak D_0$.
Proof.
Put $H_ i=\bigwedge _{j\ne i}\omega _ j$. Expanding the wedge powers gives $h^{n-1}=(n-1)!\sum _ iH_ i$ and $h\wedge H_ i=\mathrm{d}V$. Thus $T_*=(1/n)\sum _ iH_ i$ is normalized, and the determinant formula gives $k_{T_*}=V^{-1/n}$. The smooth equation for $F=0$ is solved by the zero potential, since $h^ n=V\, \mathrm{d}V$; theorem 12.7.3 therefore gives the maximal value and the uniqueness of the maximizer.
To construct $S_\varepsilon $, choose a nonnegative smooth function $\beta $ compactly supported in the unit disk in $\mathbb C$, of integral one, with $\beta \geq \beta _0>0$ on a disk of radius $r_0\in (0,1)$. On an embedded disk in the first elliptic factor put $b_\varepsilon (z)=\varepsilon ^{-2}\beta (z/\varepsilon )$, extended by zero; it has integral one. For $0<t<1$, set
\[ \begin{gathered} f_{\varepsilon ,t}=1-t+tb_\varepsilon ,\qquad a=V^{-1/(n-1)},\\ \gamma _{\varepsilon ,t}=a(f_{\varepsilon ,t}\omega _1+\omega _2+\cdots +\omega _ n), \qquad T_{\varepsilon ,t}=\gamma _{\varepsilon ,t}^{n-1}. \end{gathered} \]
The form $\gamma _{\varepsilon ,t}$ is strictly positive, and it is closed because $f_{\varepsilon ,t}$ depends only on the first factor, which has real dimension two. Direct expansion gives
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\begin{equation} T_{\varepsilon ,t}=\frac1n\left(H_1+f_{\varepsilon ,t}\sum _{i=2}^ nH_ i\right), \qquad k_{T_{\varepsilon ,t}}=V^{-1/n}f_{\varepsilon ,t}^{\alpha }. \end{equation}
Since $\int _ Xf_{\varepsilon ,t}\, \mathrm{d}V=1$, the trace integral is one, so $T_{\varepsilon ,t}\in \mathscr G_ h$, and $\| k_{T_{\varepsilon ,t}}\| _{L^ q}=V^{-1/n}$ because $\alpha q=1$.
The mean-zero function $f_{\varepsilon ,t}-1$ has a smooth real potential $\psi $ on $E$ with $\mathrm{dd}^{\mathrm{c}}\psi =(f_{\varepsilon ,t}-1)\omega _1$; for example, the mean-zero Fourier series solving $\Delta \psi =4\pi (f_{\varepsilon ,t}-1)$ in the area-one coordinates. Consequently $\gamma _{\varepsilon ,t}-ah=\mathrm{dd}^{\mathrm{c}}(a\psi )$. Since $T_*=(ah)^{n-1}$, the closed form $R=\sum _{j=0}^{n-2}\gamma _{\varepsilon ,t}^{n-2-j}\wedge (ah)^ j$ satisfies
\[ T_{\varepsilon ,t}-T_*=\mathrm{dd}^{\mathrm{c}}(a\psi R). \]
This proves all the assertions about classes, including the case $n=2$, where $R=1$.
Jensen's inequality and $f_{\varepsilon ,t}\geq 1-t$ give
\[ V^{-1/n}(1-t)^\alpha \leq \mathfrak D_0(T_{\varepsilon ,t}) =V^{-1/n}\int _ Xf_{\varepsilon ,t}^{\alpha }\, \mathrm{d}V <V^{-1/n}, \]
where the inequality is strict because $f_{\varepsilon ,t}$ is nonconstant and $0<\alpha <1$. On the product of the disk of radius $r_0\varepsilon $ with the other factors, we have $f_{\varepsilon ,t}\geq t\beta _0\varepsilon ^{-2}$, and this set has volume $\pi r_0^2\varepsilon ^2$. Hence
\[ \begin{split} \int _ Xk_{T_{\varepsilon ,t}}^ p\, \mathrm{d}V & \geq V^{-p/n}\pi r_0^2\beta _0^{\alpha p}t^{\alpha p} \varepsilon ^{2-2\alpha p},\\ \| k_{T_{\varepsilon ,t}}\| _\infty & \geq V^{-1/n}\beta _0^\alpha t^\alpha \varepsilon ^{-2\alpha }. \end{split} \]
Now take $t=1/\log (1/\varepsilon )$, which lies in $(0,1)$ by the restriction on $\varepsilon $, and $S_\varepsilon =T_{\varepsilon ,t}$. Then $\mathfrak D_0(S_\varepsilon )$ converges to its maximum, while both lower bounds diverge for $p>q$ because $\alpha p>1$. All constants are independent of $\varepsilon $.
$\square$
Pome
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