Although every Hermitian form on the threefold of theorem 12.19.1 has zero lower mass, the metric $\omega $ of (12.19.3) is balanced and has finite upper mass. A direct calculation expresses the loss of mass under a potential as a vertical Dirichlet energy along the elliptic fibers. The same calculation shows that the degeneration cannot be realized by potentials that are the sum of a periodic function of $t$ and a function on the base: their masses stay above $8$. Put $D=\mathrm{d}t\wedge \mathrm d^{\mathrm c}t$, so that $\omega =\alpha +\beta +D$.
Proposition 12.19.18.94417AD14985 On the threefold $X$ of theorem 12.19.1, the Hermitian form $\omega =\alpha +\beta +D$ satisfies
\[ \mathrm{d}(\omega ^2)=0,\qquad \mathrm{dd}^{\mathrm{c}}\omega =2\alpha \wedge \beta \geq 0, \qquad \int _ X\omega ^3=24\pi . \]
For every smooth real function $u$,
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12.19.19
\begin{equation} \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3 =24\pi -6\int _ X\mathrm{d}u\wedge \mathrm d^{\mathrm c}u\wedge \alpha \wedge \beta . \end{equation}
Consequently $v_+(\omega )=24\pi $. In particular, $X$ has the bounded mass property, while $v_-(h)=0$ for every Hermitian form $h$ on $X$.
Proof.
Although $t$ is defined only on the covering space $\Omega $, both $\mathrm{d}t$ and $\mathrm d^{\mathrm c}t$ descend to $X$. The Fubini–Study potentials $\log \| z\| ^2$ and $\log \| w\| ^2$ give $\mathrm{dd}^{\mathrm{c}}t=\beta -\alpha $. The forms $\alpha ,\beta $ are closed, and $\alpha ^2=\beta ^2=0$ since each is pulled back from a curve. As $D^2=0$, we obtain
\[ \mathrm{d}D=-\mathrm{d}t\wedge (\beta -\alpha ),\qquad \mathrm{dd}^{\mathrm{c}}D=-(\beta -\alpha )^2=2\alpha \wedge \beta , \]
\[ \mathrm{d}(\omega ^2) =-2(\alpha +\beta +D)\wedge \mathrm{d}t\wedge (\beta -\alpha )=0. \]
In the last identity, the term involving $D$ contains $\mathrm{d}t$ twice, and $(\alpha +\beta )\wedge (\beta -\alpha )=0$. Thus $\omega $ is balanced, hence Gauduchon, and the formula for $\mathrm{dd}^{\mathrm{c}}\omega $ follows from that for $\mathrm{dd}^{\mathrm{c}}D$.
On a fiber, write $\zeta =a+\mathrm{i}b$, where both real periods are $2\pi $. There $t=2a$ plus a constant, so
\[ D|_ E=\frac1\pi \, \mathrm{d}a\wedge \mathrm{d}b, \qquad \int _ ED=4\pi . \]
Since $\omega ^3=6\alpha \wedge \beta \wedge D$, fiber integration and the normalization of the Fubini–Study forms give $\int _ X\omega ^3=24\pi $.
The identity (12.16.4) for Gauduchon metrics on threefolds, with $\mathrm{dd}^{\mathrm{c}}\omega =2\alpha \wedge \beta $, gives (12.19.19). Its last integral is nonnegative. Thus every smooth strictly $\omega $-plurisubharmonic potential has mass at most $24\pi $, and the constant potential attains this value. Regularization of bounded potentials, as in the proof of theorem 12.15.4, gives the same supremum over bounded potentials, so $v_+(\omega )=24\pi $. The metric independence in proposition 13.14.15 then gives the bounded mass property of $X$. The assertion about the lower mass is theorem 12.19.1.
$\square$
For an admissible smooth $u$, equality in the upper-mass bound holds precisely when $u$ is constant along every elliptic fiber. Indeed, $\alpha \wedge \beta $ is a positive volume form in the base directions, so the nonnegative integral in (12.19.19) vanishes exactly when the derivative of $u$ along the fibers vanishes. The fibers are connected, and smooth local sections then show that $u$ is the pullback of a smooth function on $\mathbb P^1\times \mathbb P^1$. For the potentials with masses tending to zero in theorem 12.19.1, the same integral instead tends to $4\pi $.
A function $f(t)$ descends to $X$ when $f$ is $4\pi $-periodic, since $t\circ g=t+4\pi $. Even after adding an arbitrary potential pulled back from the base, such functions leave a positive mass gap. The sharp gap follows from the bounds on the first two derivatives of $f$ imposed by positivity.
Proposition 12.19.20.3290D6E85075 Let $\pi :X\to B=\mathbb P^1\times \mathbb P^1$ be the elliptic fibration above, and put $L=4\pi $. Consider the smooth potentials
\[ u=f(t)+\pi ^*v,\qquad \omega +\mathrm{dd}^{\mathrm{c}}u>0, \]
where $f:\mathbb R\to \mathbb R$ is smooth and $L$-periodic and $v\in C^\infty (B,\mathbb R)$. The infimum of $\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3$ over these potentials is $8$. Every such mass is strictly greater than $8$, and the infimum is approached already with $v=0$.
Proof.
Write $p=f'$, an $L$-periodic function with mean zero. The chain rule gives
\[ \omega +\mathrm{dd}^{\mathrm{c}}u =(1-p)\alpha +(1+p)\beta +\pi ^*\mathrm{dd}^{\mathrm{c}}v+(1+p')D. \]
Positivity on the vertical line implies $p'>-1$. On the complex plane $\ker \partial t$, the form $D$ vanishes and the projection to the base is an isomorphism. For each value of $t$ modulo $L$, every base point has a point above it with that value. Hence, writing $\alpha _ B,\beta _ B$ for the two Fubini–Study forms on $B$, the form $(1-p(s))\alpha _ B+(1+p(s))\beta _ B+\mathrm{dd}^{\mathrm{c}}v$ is positive on $B$ for every $s$. Integration over the two rulings gives $1-p>0$ and $1+p>0$. Thus $|p|<1$ and $p'>-1$; when $v=0$, these inequalities also suffice for strict positivity.
In a logarithmic fiber coordinate $x+\mathrm{i}y$, both real periods are $2\pi $, $t=2x$ plus a constant, and $D|_ E=\pi ^{-1}\mathrm{d}x\wedge \mathrm{d}y$. Hence $\int _ E F(t)D=\int _0^ L F(s)\, \mathrm{d}s$ for every $L$-periodic function $F$. Only the derivative of $u$ along the fibers contributes to (12.19.19), and it equals $p(t)$ times the derivative of $t$. Fiber integration therefore gives
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\begin{equation} \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^3 =6\left(L-\int _0^ L p(s)^2\, \mathrm{d}s\right). \end{equation}
We now bound $\int _0^ Lp^2$. Suppose first that $p\not\equiv 0$, and put $m_+=\max p>0$, $m_-=-\min p>0$ and $r=m_++m_-<2$. The function $G(s)=(m_+-s)(s+m_-)$ is nonnegative on $[-m_-,m_+]$. Choose an interval $I$ of length at most $L$ that runs from a maximum point of $p$ forward to the next minimum point. Since $p'\geq -1$, we have $G(p)\geq -G(p)p'$. Using that $p$ has mean zero, we obtain
\[ \begin{split} m_+m_-L-\int _0^ L p^2\, \mathrm{d}s & =\int _0^ L G(p)\, \mathrm{d}s \geq -\int _ I G(p)p'\, \mathrm{d}s\\ & =\int _{-m_-}^{m_+}G(s)\, \mathrm{d}s=\frac{r^3}{6}. \end{split} \]
As $m_+m_-\leq r^2/4$, it follows that
\[ \int _0^ L p^2\, \mathrm{d}s \leq \frac{Lr^2}{4}-\frac{r^3}{6}<L-\frac43, \]
where the last inequality holds because the middle expression is strictly increasing in $r\in (0,2]$, as $L>2$. The inequality $\int _0^ Lp^2<L-4/3$ holds trivially when $p=0$. Equation (12.19.21) now gives the strict lower bound $8$.
To prove sharpness, put $a=(L-2)/2$ and define an $L$-periodic function $p_0$ by
\[ p_0(s)= \begin{cases} 1,& 0\leq s<a,\\ 1-(s-a),& a\leq s<a+2,\\ -1,& a+2\leq s<L. \end{cases} \]
It has mean zero, $|p_0|\leq 1$ and $\int _0^ L p_0^2\, \mathrm{d}s=L-4/3$. Its distributional derivative is $-1$ on the descending interval and zero on the two constant intervals, and it has a positive atom where the period closes up. Thus $p_0'\geq -1$ as distributions on the circle. Let $\rho _\varepsilon $ be a nonnegative smooth periodic approximate identity of integral one, and set $p_\varepsilon =(1-\varepsilon )(p_0*\rho _\varepsilon )$ for $0<\varepsilon <1$. Then $p_\varepsilon $ is smooth, has mean zero, and
\[ |p_\varepsilon |\leq 1-\varepsilon , \qquad p_\varepsilon '\geq -1+\varepsilon , \qquad p_\varepsilon \longrightarrow p_0\quad \hbox{in }L^2. \]
Its primitive $f_\varepsilon $ is smooth and periodic, so $u_\varepsilon =f_\varepsilon (t)$ is admissible. By (12.19.21), its mass tends to $8$.
$\square$
Proposition 12.19.22.69BB70EA26D9 The threefold $X$ admits neither a pluriclosed Hermitian metric nor a closed Kähler current.
Proof.
By proposition 12.19.18, the nonzero smooth semipositive $(2,2)$-form $\alpha \wedge \beta =\mathrm{dd}^{\mathrm{c}}(\omega /2)$ is $\mathrm{dd}^{\mathrm{c}}$-exact. If a Hermitian metric $\eta $ satisfied $\mathrm{dd}^{\mathrm{c}}\eta =0$, Stokes' theorem would give
\[ 0<\int _ X\eta \wedge \alpha \wedge \beta =\frac12\int _ X\eta \wedge \mathrm{dd}^{\mathrm{c}}\omega =\frac12\int _ X\omega \wedge \mathrm{dd}^{\mathrm{c}}\eta =0, \]
a contradiction. The first inequality holds because $\eta $ is positive in the vertical direction at every point.
Similarly, if a closed positive current $T$ satisfied $T\geq \varepsilon \omega $ for some $\varepsilon >0$, positivity and Stokes' theorem for currents would give
\[ 0<\varepsilon \int _ X\omega \wedge \alpha \wedge \beta \leq \int _ XT\wedge \alpha \wedge \beta =\frac12\int _ X\omega \wedge \mathrm{dd}^{\mathrm{c}}T=0. \]
Hence there is no closed Kähler current.
$\square$
Since a compact complex manifold carries a closed Kähler current exactly when it belongs to Fujiki's class $\mathcal C$, as recalled in section 24.4, the threefold $X$ lies outside this class. The example shows that a balanced metric does not guarantee positive lower mass, even together with bounded upper mass. The Hopf threefold, in contrast, has positive lower mass and infinite upper mass, so the two properties are logically independent, and the products in corollary 12.19.16 show that both can fail at once. In particular, in a theorem asserting solvability for every nonzero finite non-pluripolar measure, bounded mass alone cannot replace the hypothesis of positive lower mass.
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