Theorem 12.18.15.17E4315391CC On the fixed torus above, fix $0<\tau <1/2$. There is a positive nonclosed pluriclosed metric $h_\tau $ and a family $\Omega _\varepsilon =h_\tau +\mathrm{dd}^{\mathrm{c}}u_\varepsilon >0$, $0<\varepsilon \leq 1/2$, with identical actual current sets and intrinsic determinant-measure families. For any fixed smooth positive probability volume their entire-family endpoint is
\[ \sup _{T\in \mathscr G_{\Omega _\varepsilon }} \int _ X|p|\, \Gamma _0(T) =V_\tau ^{-1/n}\| p\| _{L^ n(\mathrm dV)},\qquad V_\tau =n!(1-\tau ^2/2). \]
Both bounded-potential masses equal $V_\tau $. All fixed-order smooth coefficient and potential norms are uniformly bounded, while
\[ \min _ X\lambda _{\min }(\Omega _\varepsilon \mid \beta )=\varepsilon , \qquad d\Omega _\varepsilon =dh_\tau \ne 0. \]
The flat $\beta $ is a fixed balanced metric on this same manifold with $c_1=0$; each $\Omega _\varepsilon $ is pluriclosed.
Proof.
With the torus coordinates above set $s=\sin (2\pi x_1)$ and
\[ \xi =\frac i2(dz_1\wedge d\bar z_2+dz_2\wedge d\bar z_1) =dx_1\wedge dy_2+dx_2\wedge dy_1, \qquad h_\tau =\beta +\tau s\xi . \]
The first relative Hermitian block is $\left(\begin{smallmatrix}1& \tau s\\ \tau s& 1\end{smallmatrix}\right)$, so positivity follows from $|\tau s|<1$. Moreover
\[ dh_\tau =2\pi \tau \cos (2\pi x_1) dx_1\wedge dx_2\wedge dy_1\ne 0. \]
Because $\mathrm{dd}^{\mathrm{c}}s$ is proportional to $dz_1\wedge d\bar z_1$, $\mathrm{dd}^{\mathrm{c}}(s\xi )=0$. The first two-coordinate metric is thus pluriclosed, and its square is top degree on that complex surface. Expanding with the remaining closed curve factors proves
\[ \mathrm{dd}^{\mathrm{c}}(h_\tau ^ j)=0\quad (1\leq j\leq n-1),\qquad \int _ X h_\tau ^ n=n!\int _0^1(1-\tau ^2\sin ^2(2\pi x_1))dx_1 =V_\tau . \]
Use the same $u_\varepsilon $ and $a_\varepsilon $ as in Theorem 12.18.13, now with $\varepsilon \leq 1/2$. Then
\[ \Omega _\varepsilon =a_\varepsilon \beta _1+ \sum _{j\geq 2}\beta _ j+\tau s\xi , \qquad a_\varepsilon =\varepsilon +2(1-\varepsilon )s^2. \]
The first block of $\Omega _\varepsilon -\varepsilon \beta $ has diagonal entries $2(1-\varepsilon )s^2$ and $1-\varepsilon $, and determinant
\[ \bigl [2(1-\varepsilon )^2-\tau ^2\bigr ]s^2\geq 0. \]
It is positive semidefinite; the remaining diagonal entries are $1-\varepsilon $. Hence $\Omega _\varepsilon \geq \varepsilon \beta $, with equality in the least eigenvalue at $s=0$. Also $\Omega _\varepsilon \leq 3\beta $, all fixed-order smooth norms are uniformly bounded, and the smooth semipositive limit loses rank at $s=0$. Since $d\mathrm{dd}^{\mathrm{c}}u_\varepsilon =0$, the exterior derivative is the displayed fixed nonzero $dh_\tau $.
The lower-power expansion in the proof of theorem 12.18.8 applies to $\Omega _\varepsilon =h_\tau +\mathrm{dd}^{\mathrm{c}}u_\varepsilon $; it gives $\mathrm{dd}^{\mathrm{c}}(\Omega _\varepsilon ^ j)=0$ for $j\leq n-1$, in particular pluriclosedness. Its volume is $V_\tau $, either by integrating the displayed determinant or by theorem 11.1.7. That theorem also gives mass $V_\tau $ for every bounded $\Omega _\varepsilon $-plurisubharmonic potential, so both mass extrema equal $V_\tau $. Theorem 12.18.8 applied to $h_\tau $ gives equality of the entire current and measure sets and their exact endpoint for every $p\in L^ n$. The balanced trace is fixed as well:
\[ \int _ X\Omega _\varepsilon \wedge \beta ^{n-1}=n!. \]
The off-diagonal term has zero trace and the mean of $a_\varepsilon $ equals one. The global holomorphic volume form gives $c_1=0$. The manifold itself is already Kähler, so these nonclosed metrics show that the complete determinant profile and positive bounded-potential mass extrema can stay fixed while the least relative eigenvalue tends to zero. The coexistence problem of 24.4.22 remains open.
$\square$
Pome
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