Theorem 12.18.13.BDB7ADF7845B On $X=(\mathbb C/(\mathbb Z+i\mathbb Z))^ n$, $n\geq 2$, there are positive metrics $\omega _\varepsilon $ in a fixed Kähler class, $0<\varepsilon \leq 1$, with fixed complete current and determinant-measure families and fixed endpoint (12.18.9), while their least relative eigenvalue equals $\varepsilon $. They have uniformly bounded smooth coefficients and global potentials, and
\[ v_-(\omega _\varepsilon )=v_+(\omega _\varepsilon )=n!>0. \]
The manifold has a global holomorphic volume form and $c_1=0$. Every displayed metric is already balanced and pluriclosed.
Proof.
Write $z_ j=x_ j+iy_ j$, $\beta _ j=(i/2)dz_ j\wedge d\bar z_ j=dx_ j\wedge dy_ j$, $\beta =\sum _ j\beta _ j$, $V=n!$, and $\mathrm dV=\beta ^ n/V$. Set
\[ u_\varepsilon =\frac{1-\varepsilon }{4\pi }\cos (4\pi x_1), \qquad a_\varepsilon =1-(1-\varepsilon )\cos (4\pi x_1). \]
For a function of $x_1$, $\mathrm{dd}^{\mathrm{c}}u=(u''/(4\pi ))\beta _1$. Hence
\[ \omega _\varepsilon =\beta +\mathrm{dd}^{\mathrm{c}}u_\varepsilon =a_\varepsilon \beta _1+\sum _{j=2}^ n\beta _ j. \]
These are globally periodic smooth positive closed forms with $\varepsilon \leq a_\varepsilon \leq 2-\varepsilon $ and $\omega _\varepsilon \leq 2\beta $. Their minimum relative eigenvalue is $\varepsilon $. They converge smoothly to a semipositive form losing rank on $\{ x_1\in \tfrac 12\mathbb Z\} $. All fixed-order smooth norms of the coefficients and potentials are uniformly bounded. Theorem 12.18.8 proves equality of the entire current data and the exact norm at every $\varepsilon $.
Since $\omega _\varepsilon $ is closed, theorem 11.1.7 gives mass $V$ for every bounded $\omega _\varepsilon $-plurisubharmonic potential. The explicit formula also gives $\int _ X\omega _\varepsilon ^ n=V$, since $a_\varepsilon $ has mean one. Thus both extrema over bounded potentials equal $V$. The global form $dz_1\wedge \cdots \wedge dz_ n$ trivializes the canonical bundle; closed positive metrics are balanced and pluriclosed.
$\square$
Pome
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