Now let $s=2$ and $t=1$, so that $X$ is a threefold. A totally positive unit has norm $\sigma _1(\varepsilon )\sigma _2(\varepsilon )|\sigma _3(\varepsilon )|^2=1$, so the form $y_1y_2\, \mathrm{dd}^{\mathrm{c}}|z|^2$ is $\Gamma $-invariant. The Hermitian form
\[ \omega =\theta +\beta ,\qquad \beta =y_1y_2\, \mathrm{dd}^{\mathrm{c}}|z|^2, \]
descends to $X$. Write $\mathrm dV_0=(\mathrm i\, \mathrm dw_1\wedge \mathrm d\overline w_1)\wedge (\mathrm i\, \mathrm dw_2\wedge \mathrm d\overline w_2)\wedge (\mathrm i\, \mathrm dz\wedge \mathrm d\overline z)$ on the universal cover, and $\mathrm M_\omega (u)=\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3$.
A unit that fixes $y_1$ satisfies $\sigma _1(\varepsilon )=1$, hence $\varepsilon =1$ because $\sigma _1$ is injective. Hence the map $X\to T_ U=\mathbb R^2/\Lambda _ U$, $[w,z]\mapsto (\log y_1,\log y_2)$, has as fibers the real tori $(\mathbb R^2\times \mathbb C)/(\sigma _{\mathbb R},\sigma _{\mathbb C})(\mathcal O_ K)$, and for every $\Lambda _ U$-periodic function $g$ of $\varsigma =(\log y_1,\log y_2)$ there is a constant $c_ X>0$, independent of $g$, with
We shall use an elementary property of lattices in the plane: every lattice $\Lambda \subset \mathbb R^2$ contains vectors in each of the four open quadrants. Indeed, let $\ell >0$ be the length of a shortest nonzero vector of $\Lambda $, and let $0<\delta <\ell $. By Minkowski's theorem, the symmetric convex set $\{ k:|k_1-k_2|<\delta ,\ |k_1+k_2|<N\} $ contains a nonzero lattice point once $N$ is large. Such a point satisfies $(k_1+k_2)^2=2|k|^2-(k_1-k_2)^2\geq 2\ell ^2-\delta ^2$, hence $4k_1k_2=(k_1+k_2)^2-(k_1-k_2)^2>0$; replacing $k$ by $-k$ if necessary gives a point in the open first quadrant, and the line $\mathbb R(1,-1)$ gives the other two quadrants in the same way. If $\lambda $ lies in the open first quadrant and $\mu \in \Lambda $ is independent of $\lambda $, then $N\lambda +\mu $ also lies there for large $N$, so the first quadrant contains two independent lattice vectors.
Proof.
(1) Since $\mathrm{dd}^{\mathrm{c}}|z|^2$ has constant coefficients and $y_ j$ is pluriharmonic,
\[ \mathrm{dd}^{\mathrm{c}}\beta =\mathrm{dd}^{\mathrm{c}}(y_1y_2)\wedge \mathrm{dd}^{\mathrm{c}}|z|^2, \]
where
\[ \mathrm{dd}^{\mathrm{c}}(y_1y_2)=\frac{\mathrm i}{2\pi }\bigl (\partial y_1\wedge \overline\partial y_2 +\partial y_2\wedge \overline\partial y_1\bigr ) =\frac{\mathrm i}{8\pi }\bigl (\mathrm dw_1\wedge \mathrm d\overline w_2 +\mathrm dw_2\wedge \mathrm d\overline w_1\bigr ). \]
The Hermitian matrix of $\mathrm{dd}^{\mathrm{c}}(y_1y_2)$ in the basis $\mathrm dw_1,\mathrm dw_2$ is a positive multiple of $\left(\begin{smallmatrix}0& 1\\ 1& 0\end{smallmatrix}\right)$, which is indefinite. Accordingly, for $\eta _\pm =\mathrm dw_1\pm \mathrm dw_2$ the $(3,3)$-forms $\mathrm{dd}^{\mathrm{c}}\omega \wedge \mathrm i\eta _\pm \wedge \overline\eta _\pm $ are respectively negative and positive multiples of $\mathrm dV_0$, so $\mathrm{dd}^{\mathrm{c}}\omega =\mathrm{dd}^{\mathrm{c}}\beta $ is neither semipositive nor seminegative. Since $\theta $ is closed and $\beta ^2=0$, we have $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=2\theta \wedge \mathrm{dd}^{\mathrm{c}}\beta $, and this vanishes because every term of $\theta \wedge (\mathrm dw_1\wedge \mathrm d\overline w_2+\mathrm dw_2\wedge \mathrm d\overline w_1)$ repeats a differential.
(2) By theorem 11.1.5 with $n=3$ and $\mathrm{dd}^{\mathrm{c}}(\omega ^2)=0$, $\mathrm M_\omega (u)-\int _ X\omega ^3=3\int _ Xu\, \mathrm{dd}^{\mathrm{c}}u\wedge \mathrm{dd}^{\mathrm{c}}\omega $. The form $\mathrm{dd}^{\mathrm{c}}\omega $ is closed, so Stokes' theorem gives $\int _ Xu\, \mathrm{dd}^{\mathrm{c}}u\wedge \mathrm{dd}^{\mathrm{c}}\omega =-\int _ X\mathrm du\wedge \mathrm d^{\mathrm c}u\wedge \mathrm{dd}^{\mathrm{c}}\omega $. Now $\mathrm du\wedge \mathrm d^{\mathrm c}u=(\mathrm i/2\pi )\, \partial u\wedge \overline\partial u$, and a direct computation in the coordinates $w_1,w_2,z$ gives
\[ \mathrm i\, \partial u\wedge \overline\partial u\wedge \mathrm i\bigl (\mathrm dw_1\wedge \mathrm d\overline w_2+\mathrm dw_2\wedge \mathrm d\overline w_1\bigr ) \wedge \mathrm i\, \mathrm dz\wedge \mathrm d\overline z =-2\operatorname {Re}\Bigl (\frac{\partial u}{\partial w_1}\, \overline{\frac{\partial u}{\partial w_2}}\Bigr )\mathrm dV_0. \]
Collecting the constants $(1/2\pi )(1/8\pi )(1/2\pi )$ gives the formula.
(3) Since $\mathrm{dd}^{\mathrm{c}}u$ has no components in the $z$-direction, $\omega +\mathrm{dd}^{\mathrm{c}}u=(\theta +\mathrm{dd}^{\mathrm{c}}u)+\beta $ splits into blocks, and $u\in \mathrm{PSH}(X,\omega )$ if and only if $u\in \mathrm{PSH}(X,\theta )$. By theorem 10.29.4, $u=f(y)+\log (y_1y_2)$ with $f$ convex, so $g$ is locally Lipschitz, hence Lipschitz by periodicity. In the variables $\varsigma _ j=\log y_ j$, convexity of $f$ is equivalent to positivity of the symmetric matrix
\[ H=I+D^2g-\operatorname {diag}(\partial _1g,\partial _2g), \]
as one checks from $y_ iy_ j\, \partial ^2f/\partial y_ i\partial y_ j=\partial ^2\Phi /\partial \varsigma _ i\partial \varsigma _ j-\delta _{ij}\, \partial \Phi /\partial \varsigma _ i$ for $\Phi (\varsigma )=f(\mathrm e^{\varsigma })=g(\varsigma )-\varsigma _1-\varsigma _2$. This condition is linear in $g$ with constant coefficients, so it is preserved by convolution with a smooth kernel on $\mathbb R^2$; the regularized functions are periodic, and they converge uniformly to $g$ with gradients converging almost everywhere. By the continuity of bounded Monge–Ampère products under uniform convergence and dominated convergence, it suffices to prove all assertions for smooth $g$ with $H\geq 0$.
For such $g$, the function $u$ depends only on $y$, so $\partial u/\partial w_ j=-\frac{\mathrm i}2\partial u/\partial y_ j$ and the integrand in (2) is $\frac14\partial _{y_1}u\, \partial _{y_2}u=\partial _1g\, \partial _2g/(4y_1y_2)$. A direct computation gives $\omega ^3=3\theta ^2\wedge \beta =\frac{3}{64\pi ^3}\, \mathrm dV_0/(y_1y_2)$. By (10.29.9) and (2),
\[ \mathrm M_\omega (u)=\frac{\int _ X\omega ^3}{|T_ U|}\int _{T_ U} \bigl (1+\partial _1g\, \partial _2g\bigr )\, \mathrm d\varsigma =\frac{\int _ X\omega ^3}{|T_ U|}\int _{T_ U}PQ\, \mathrm d\varsigma , \]
where the second equality uses $\int _{T_ U}\partial _ jg\, \mathrm d\varsigma =0$.
The diagonal entry $H_{11}=P-\partial _1P\geq 0$ says that $\mathrm e^{-\varsigma _1}P$ is nonincreasing in $\varsigma _1$, that is,
62F4D45FFF73
10.29.11
\begin{equation} P(\varsigma +\tau e_1)\leq \mathrm e^{\tau }P(\varsigma )\qquad (\tau \geq 0). \end{equation}
If $P(\varsigma )<0$, the right side tends to $-\infty $ as $\tau \to \infty $, which is impossible for the bounded function $P$; hence $P\geq 0$, and likewise $Q\geq 0$. Thus $\partial _ jg\leq 1$. The lattice $\Lambda _ U$ contains two independent vectors $\lambda ,\lambda '$ with positive coordinates, and every point of $T_ U$ has the form $\varsigma +\tau \lambda +\tau '\lambda '$ with $\tau ,\tau '\in [0,1)$. Along $\tau \mapsto \varsigma +\tau \lambda $, the derivative of $g$ is at most $|\lambda |_1=\lambda _1+\lambda _2$, and its integral over $[0,1]$ vanishes by periodicity, so $|g(\varsigma +\tau \lambda )-g(\varsigma )|\leq |\lambda |_1$. Hence $\operatorname {osc}g\leq L$, with $L=|\lambda |_1+|\lambda '|_1$ depending only on $\Lambda _ U$.
Averaging (10.29.11) in the form $P(\varsigma -\tau e_1)\geq \mathrm e^{-\tau }P(\varsigma )$ over $\tau \in [0,1]$ gives
\[ (1-\mathrm e^{-1})P(\varsigma )\leq \int _0^1P(\varsigma -\tau e_1)\, \mathrm d\tau =1-g(\varsigma )+g(\varsigma -e_1)\leq 1+L. \]
On the other hand, averaging $P(\varsigma )\geq \mathrm e^{-\tau }P(\varsigma +\tau e_1)$ over $\tau \in [0,R]$ with $R=\max (2L,1)$ gives
\[ P(\varsigma )\geq \frac{\mathrm e^{-R}}R\int _0^ RP(\varsigma +\tau e_1)\, \mathrm d\tau =\frac{\mathrm e^{-R}}R\bigl (R-g(\varsigma +Re_1)+g(\varsigma )\bigr ) \geq \frac{\mathrm e^{-R}}2. \]
The same bounds hold for $Q$. Therefore the average of $PQ$ over $T_ U$ lies between $\mathrm e^{-2R}/4$ and $\bigl ((1+L)/(1-\mathrm e^{-1})\bigr )^2$, which proves (3).
(4) Let $k$ belong to the dual lattice $\Lambda _ U^*=\{ k\in \mathbb R^2:\langle k,\lambda \rangle \in \mathbb Z\text{ for all }\lambda \in \Lambda _ U\} $, and put $g_\epsilon (\varsigma )=\epsilon \cos (2\pi \langle k,\varsigma \rangle )$. For small $\epsilon >0$, the matrix $H$ of $g_\epsilon $ is positive definite, so $u_\epsilon =g_\epsilon (\varsigma )$ is a smooth function with $\omega +\mathrm{dd}^{\mathrm{c}}u_\epsilon >0$. By (3),
\[ \frac{\mathrm M_\omega (u_\epsilon )}{\int _ X\omega ^3} =1+\frac1{|T_ U|}\int _{T_ U}\partial _1g_\epsilon \, \partial _2g_\epsilon \, \mathrm d\varsigma =1+2\pi ^2\epsilon ^2k_1k_2. \]
Since the lattice $\Lambda _ U^*$ has vectors in every open quadrant, we may choose $k$ with $k_1k_2>0$, respectively $k_1k_2<0$, which gives the two inequalities.
$\square$
For a general smooth potential, averaging over the fibers separates the invariant part from the fluctuations. For $u\in \mathrm{PSH}(X,\omega )$ let $Au$ be the average of $u$ over the fibers of $X\to T_ U$, that is, over the translations $(w,z)\mapsto (w+r,z+\zeta )$ with $(r,\zeta )\in (\mathbb R^2\times \mathbb C)/(\sigma _{\mathbb R},\sigma _{\mathbb C})(\mathcal O_ K)$. These translations preserve $\omega $ on the universal cover. They do not descend to $X$ individually, but conjugation by a unit $\varepsilon $ carries the translation by $(r,\zeta )$ to the translation by $(\sigma _1(\varepsilon )r_1,\sigma _2(\varepsilon )r_2,\sigma _3(\varepsilon )\zeta )$, and this automorphism of the translation torus preserves its Haar measure because $\varepsilon $ has norm one. Hence $Au$ is a well-defined $\omega $-plurisubharmonic function depending only on $y$, and $c_ U^{-1}\int _ X\omega ^3\leq \mathrm M_\omega (Au)\leq c_ U\int _ X\omega ^3$ by proposition 10.29.10. For smooth $u$, the integral of a $y$-derivative of $u-Au$ over a fiber vanishes, and so does the integral of an $x$-derivative. The cross terms between $Au$ and $u-Au$ in proposition 10.29.10(2) therefore integrate to zero, and
Since the extrema of the masses of bounded potentials equal the extrema over smooth strictly $\omega $-plurisubharmonic potentials (proposition 13.14.15, theorem 12.15.4), $v_+(\omega )<+\infty $ if and only if the fluctuation term in (10.29.12) is bounded above on smooth $\omega $-plurisubharmonic functions; likewise, $v_-(\omega )>0$ if and only if $\mathrm M_\omega (Au)$ plus the fluctuation term is bounded below by a positive constant on such functions.
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