Proposition 12.2.2.4AF37F3C8091 For a pseudoeffective $\theta $, the number $\lambda _\mu (\theta )$ is finite and nonnegative. It is unchanged by a smooth $\mathrm{dd}^{\mathrm{c}}$ change of $\theta $, positively homogeneous of degree one, and concave on the cone of pseudoeffective forms. If $\beta $ is a smooth real $(1,1)$-form and $\beta -\theta $ is pseudoeffective, then
\[ \lambda _\mu (\beta )\geq \lambda _\mu (\theta ). \]
Finally, $\lambda _\mu $ is upper semicontinuous under smooth convergence within this cone.
Proof.
Choose $u\in \mathrm{PSH}(X,\theta )$. For every Gauduchon metric $\gamma $, Stokes' theorem for currents and the Gauduchon condition give
\[ \int _ X\theta \wedge \gamma ^{n-1} =\int _ X(\theta +\mathrm{dd}^{\mathrm{c}}u)\wedge \gamma ^{n-1}\geq 0. \]
Thus every quotient is nonnegative, and testing one fixed Gauduchon metric gives a finite upper bound. The same identity shows that each quotient is unchanged by a smooth $\mathrm{dd}^{\mathrm{c}}$ change of the background, and, applied to $\beta -\theta $, it proves the monotonicity.
Each quotient is a linear function of the background, so their infimum is concave and positively homogeneous. Each individual quotient is also continuous in the smooth topology. Given a smoothly converging sequence, we bound the infimum by any one quotient, take the upper limit, and finally take the infimum over the test metric. This proves upper semicontinuity.
$\square$
Pome
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