Corollary 11.6.2.F83E15A1F529 For a smooth real $(1,1)$-form $\theta $, set
\[ \kappa _\omega (\theta )= \inf _{\gamma \ \mathrm{Gauduchon}} \frac{\int _ X\theta \wedge \gamma ^{n-1}}{\int _ X\omega \wedge \gamma ^{n-1}}. \]
This is a finite real number, and it is the largest real $a$ for which there is a quasi-plurisubharmonic function $u$ satisfying
\[ \theta +\mathrm{dd}^{\mathrm{c}}u\geq a\omega . \]
In particular, this largest value is attained, and $\theta $ is big, in the sense of section 5.40, precisely when $\kappa _\omega (\theta )>0$.
Proof.
Since $\theta $ is smooth, $-C\omega \leq \theta \leq C\omega $ for some $C$, so the infimum lies in $[-C,C]$. If $\theta +\mathrm{dd}^{\mathrm{c}}u\geq a\omega $, the trace identity applied to the background $\theta -a\omega $ shows that $a$ is at most each of the ratios in the definition, hence $a\leq \kappa _\omega (\theta )$. Conversely, for $a=\kappa _\omega (\theta )$, all Gauduchon degrees of $\theta -a\omega $ are nonnegative, and theorem 11.6.1 applied to this smooth form gives the required $u$. The last assertion follows, since bigness means the existence of a $\theta $-plurisubharmonic $u$ with $\theta +\mathrm{dd}^{\mathrm{c}}u\geq \delta \omega $ for some $\delta >0$.
$\square$
Pome
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