PomeDefinition 12.2.1.0C4AF7E2CEBB For a pseudoeffective smooth real $(1,1)$-form $\theta $ on $X$, define
\[ \lambda _\mu (\theta )= \inf _{\gamma \text{ Gauduchon}} \frac{\displaystyle \int _ X\theta \wedge \gamma ^{n-1}}{D_\mu (\gamma )}, \qquad \mathcal V_\mu (\theta )=\lambda _\mu (\theta )^ n. \]
We call $\mathcal V_\mu (\theta )$ the Gauduchon variational volume of $\theta $ with respect to $\mu $, and $\lambda _\mu (\theta )$ its root volume. In dimension one every Hermitian metric is Gauduchon, and every quotient equals $\int _ X\theta $.
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