Theorem 12.18.17.6072DDDAB57F On $X=(\mathbb C/(\mathbb Z+\mathrm i\mathbb Z))^ n$, let $\beta _ j=\mathrm dx_ j\wedge \mathrm dy_ j$, $\beta =\sum _ j\beta _ j$, $V=n!$ and $\mathrm dV=\beta ^ n/V$. There are smooth positive closed $A_\varepsilon =\beta +\mathrm{dd}^{\mathrm{c}}w_\varepsilon $, $0<\varepsilon <1$, and $T_\varepsilon =A_\varepsilon ^{n-1}/V \in \mathscr G_\beta $ satisfying
\[ \int _ X k_{T_\varepsilon }^{n/(n-1)}\mathrm dV=V^{-1/(n-1)}. \]
Each $T_\varepsilon $ is an actual dual maximizer for a specified unit $L^ n$ root. The potentials $w_\varepsilon $ have uniformly bounded oscillation and converge uniformly to a continuous canonical $\beta $-psh $w_0$. The currents converge weakly to $T_0=(\beta +\mathrm{dd}^{\mathrm{c}}w_0)^{n-1}/V\in \mathscr G_\beta $, but $\Gamma _0(T_0)=0$. The power densities above are not uniformly integrable. Nevertheless every $w_\varepsilon $, including $w_0$, has bounded-potential Monge–Ampère mass $V$.
Proof.
Put $r=1-\varepsilon $ and
\[ P_ r(x)=\frac{1-r^2}{1-2r\cos (2\pi x)+r^2},\qquad f_\varepsilon =\varepsilon +(1-\varepsilon )P_ r. \]
The positive periodic Poisson kernel has mean one and Fourier series
\[ P_ r=1+2\sum _{m\geq 1}r^ m\cos (2\pi mx). \]
Define
\[ w_\varepsilon =-\frac{2(1-\varepsilon )}{\pi } \sum _{m\geq 1}\frac{r^ m\cos (2\pi mx_1)}{m^2}. \]
For fixed $\varepsilon >0$ this is smooth. Differentiation gives $w_\varepsilon ''=4\pi (f_\varepsilon -1)$, and hence
\[ A_\varepsilon =f_\varepsilon (x_1)\beta _1+ \sum _{j=2}^ n\beta _ j=\beta +\mathrm{dd}^{\mathrm{c}}w_\varepsilon >0. \]
It is closed, and $\int A_\varepsilon ^ n=V$. Theorem 12.18.8 and the smooth-root formula give
\[ k_{T_\varepsilon }=V^{-1/n}f_\varepsilon ^{(n-1)/n}, \qquad k_{T_\varepsilon }^{n/(n-1)} =V^{-1/(n-1)}f_\varepsilon . \]
Thus the displayed power integral attains the upper bound over all currents, not a bound for a selected family. More precisely, $p_\varepsilon =f_\varepsilon ^{1/n}$ has $L^ n$ norm one, and $\int p_\varepsilon k_{T_\varepsilon }\mathrm dV=V^{-1/n}$, so $T_\varepsilon $ attains the original entire-family dual supremum for this varying root. No common fixed density is asserted.
The Fourier series bounds $|w_\varepsilon |\leq \pi /3$ and implies uniform convergence to
\[ w_0=-\frac2\pi \sum _{m\geq 1}\frac{\cos (2\pi mx_1)}{m^2}. \]
This limit is continuous and canonically $\beta $-psh by distributional positivity of $A_0=\beta +\mathrm{dd}^{\mathrm{c}}w_0$. The probability measures $P_ r(x_1)dx_1$ converge weakly to the periodic Dirac mass at $x_1=0$, as follows from their Fourier coefficients $r^{|m|}$. Thus
\[ A_0=\delta _{x_1=0}\beta _1+\sum _{j=2}^ n\beta _ j, \]
where the Dirac measure is tensored with Lebesgue measure in the remaining real coordinates. Its bounded local products are genuine Bedford–Taylor products. Locally the first-coordinate current $\beta _1+\mathrm{dd}^{\mathrm{c}}w_0$ is pulled back from one complex coordinate, so its powers of degree at least two vanish in the bounded-product recursion. The remaining factors are smooth and closed. Multilinear expansion therefore contains at most one first-coordinate factor; the same expansion for $A_\varepsilon $ passes directly to the distributional limit. Consequently $T_\varepsilon \to T_0=A_0^{n-1}/V$ weakly. This current is positive, closed and normalized by $\int \beta \wedge T_0=1$.
In its positive coefficient matrix the absolutely continuous part is proportional to the first diagonal matrix below; the singular part on $x_1=0$ is proportional to the second:
\[ \operatorname {diag}(1,0,\ldots ,0),\qquad \operatorname {diag}(0,1,\ldots ,1). \]
Their ranks are respectively one and $n-1$, both less than $n$. With a measure dominating these mutually singular parts, formula (12.18.7) therefore gives $\Gamma _0(T_0)=0$. This is not the Monge–Ampère measure of $w_0$, which has total mass $V$ by theorem 11.1.7.
Away from $x_1=0$, $f_\varepsilon \to 0$ pointwise; its integral against $\mathrm dV$ remains one. Thus the positive power densities are not uniformly integrable: almost-everywhere convergence to zero and uniform integrability would force their fixed positive integrals to vanish. In particular $T\mapsto \int k_ T^{n/(n-1)}\mathrm dV$ is not continuous under these weak current limits, even with uniformly convergent continuous potentials. The intrinsic measures themselves satisfy $\Gamma _0(T_\varepsilon )\to 0$ in total mass; this follows from their uniform $L^{n/(n-1)}$ bound by splitting off arbitrarily small neighborhoods of $x_1=0$. The discontinuity concerns their power integrals, not that last measure limit.
$\square$
The maximizing root varies with $\varepsilon $. This construction concerns the complete dual profile and the critical power integral; it does not give a sequence of exact maximizers for one fixed smooth density.
Pome
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