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We return to an arbitrary compact complex manifold $X$ with a Hermitian form $\omega $. In (10.14.1), the negative part of $T$ is measured by a closed Kähler form. If the closed perturbation is replaced by the Hermitian form $\omega $, the supremum can be infinite even for the zero Bott–Chern class $\{ 0\} $ and $V=X$. Unbounded Monge–Ampère masses of smooth potentials, such as those on the Hopf threefold, already produce this failure.
Proposition 10.15.1.FCA1263B1141 Suppose there are smooth real functions $v_ j$ on $X$, for positive integers $j$, such that
\[ \omega +\mathrm{dd}^{\mathrm{c}}v_ j>0, \qquad \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}v_ j)^ n\longrightarrow +\infty . \]
Then for every $\varepsilon >0$,
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10.15.2
\begin{equation} \sup _{\substack {T\in \{ 0\} \ \mathrm{smooth,\ closed}\\ T\geq -\varepsilon \omega }} \int _ X(T+\varepsilon \omega )^ n=+\infty . \end{equation}
This occurs for the Gauduchon form on the Hopf threefold in theorem 23.1.1.
Proof.
Fix $\varepsilon >0$ and take $T_ j=\varepsilon \mathrm{dd}^{\mathrm{c}}v_ j$. Each $T_ j$ is smooth and closed and represents the zero Bott–Chern class. Moreover,
\[ T_ j+\varepsilon \omega =\varepsilon (\omega +\mathrm{dd}^{\mathrm{c}}v_ j)>0, \]
and its top mass $\varepsilon ^ n\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}v_ j)^ n$ tends to $+\infty $. The functions $v_ j$ on the Hopf threefold are provided by theorem 23.1.1.
$\square$
The currents $T_ j$ are smooth, so requiring analytic singularities with smooth remainder does not exclude them. Moreover, the supremum is infinite for each fixed $\varepsilon $, so letting $\varepsilon $ tend to zero afterwards cannot produce a finite limit. By contrast, if $\omega $ is Kähler and $T=\mathrm{dd}^{\mathrm{c}}v$ is smooth, Stokes' theorem gives
\[ \int _ X(T+\varepsilon \omega )^ n =\varepsilon ^ n\int _ X\omega ^ n\longrightarrow 0. \]
On the other hand, the zero class is nef, its non-nef locus is empty, and its only positive representative is the zero current. Indeed, a positive representative $\mathrm{dd}^{\mathrm{c}}u\geq 0$ would make $u$ a global plurisubharmonic function on a compact manifold, hence constant. Thus (10.15.2) measures the nonlinear mass of the Hermitian error family rather than positivity of the closed class.
A numerical theory of restricted volumes in a Hermitian background must therefore either control the approximating family or use a different invariant. Two qualitative loci avoid this supremum. The non-Hermitian locus of a big background is defined through its strict currents, whose exterior derivative is prescribed by the background; Dang's null locus tests failure of bigness after restriction to subvarieties. The singularities allowed in the first definition, and the equality of the two loci for nef and big backgrounds, are recalled in section 24.4.
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