Theorem 12.19.1.38D9206B1A2F There is a connected compact complex threefold $X$ such that, for every Hermitian form $h$ on $X$, there are smooth real functions $u_ j$ satisfying
\[ h+\mathrm{dd}^{\mathrm{c}}u_ j>0,\qquad \sup _ Xu_ j=0,\qquad \int _ X(h+\mathrm{dd}^{\mathrm{c}}u_ j)^3\longrightarrow 0. \]
In particular, $v_-(h)=0$ for every Hermitian form $h$ on this manifold.
of theorem 12.19.1.
Fix a Hermitian form $h$ on $X$, and choose $A>0$ with $h\leq A\omega $. Let $\eta _ j$ be the positive root of $\Gamma _ j$, so that $\eta _ j^2=\Gamma _ j$. These metrics are Gauduchon, and
\[ P_ j=\int _ Xh\wedge \eta _ j^2\leq AC_2,\qquad R_ j=\int _ X\eta _ j^3\longrightarrow +\infty . \]
Set $\mu _ j=\eta _ j^3/R_ j$ and solve the smooth equation in the background $h$:
\[ h_ j=h+\mathrm{dd}^{\mathrm{c}}u_ j>0,\qquad h_ j^3=c_ h(\mu _ j)\mu _ j, \qquad \sup _ Xu_ j=0. \]
Since $D_{\mu _ j}(\eta _ j)=R_ j^{2/3}$, using $\eta _ j$ as a test in theorem 12.2.3 gives
\[ 0<\int _ Xh_ j^3=c_ h(\mu _ j)=\mathcal V_{\mu _ j}(h) \leq \frac{P_ j^3}{R_ j^2}\longrightarrow 0. \]
The potentials $u_ j$ are smooth, hence bounded, so $v_-(h)=0$.
$\square$
Pome
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