Pome

Proposition 10.15.1FCA1263B1141

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 $\varepsilon >0$,

B03D3D606FEB 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. $\square$

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