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For a Gauduchon metric $\gamma $, Stokes' theorem shows that the pairing $\int _ X\theta \wedge \gamma ^{n-1}$ is unchanged by a smooth $\mathrm{dd}^{\mathrm{c}}$ change of $\theta $; this pairing will be the numerator of our quotient. For the denominator, put
\[ D_\mu (\gamma )=\int _ X \left(\frac{\mu }{\gamma ^ n}\right)^{1/n}\gamma ^ n. \]
This number is positive. Like the numerator, it is multiplied by $r^{n-1}$ when $\gamma $ is multiplied by a positive constant $r$, so the quotient below depends only on the ray of $\gamma $.
Definition 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 $.
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.
$\square$
The smooth equation identifies both the value of the quotient and every minimizing metric. Let $\mathscr P$ be the set of backgrounds $\theta $ admitting a smooth real function $u$ with $\theta +\mathrm{dd}^{\mathrm{c}}u>0$. Choosing such a Hermitian representative of $\theta $ and writing $\mu $ as a smooth positive density relative to its volume form, we may apply theorem 14.2.1. It gives, for each $\theta \in \mathscr P$, a unique positive form $\Omega _\theta $, which we call the solution form, and a unique constant $c_\theta (\mu )>0$ such that
\[ \Omega _\theta =\theta +\mathrm{dd}^{\mathrm{c}}u_\theta , \qquad \Omega _\theta ^ n=c_\theta (\mu )\mu . \]
The potential $u_\theta $ is unique after any fixed additive normalization.
Theorem 12.2.3.2DB664A6EFDF For every $\theta \in \mathscr P$,
\[ \lambda _\mu (\theta )=c_\theta (\mu )^{1/n}, \qquad \mathcal V_\mu (\theta )=c_\theta (\mu ). \]
The infimum in definition 12.2.1 is attained. When $n\geq 2$, its minimizing Gauduchon metrics are precisely the constant multiples of the Gauduchon metric conformal to $\Omega _\theta $.
Proof.
For any Hermitian metric $\gamma $, the arithmetic–geometric mean inequality for the eigenvalues of $\Omega _\theta $ relative to $\gamma $ gives the pointwise inequality
\[ \Omega _\theta \wedge \gamma ^{n-1} \geq \left(\frac{\Omega _\theta ^ n}{\gamma ^ n}\right)^{1/n}\gamma ^ n =c_\theta (\mu )^{1/n} \left(\frac{\mu }{\gamma ^ n}\right)^{1/n}\gamma ^ n. \]
If $\gamma $ is Gauduchon, integration and Stokes' theorem replace $\Omega _\theta $ in the numerator by $\theta $. Every test quotient is therefore at least $c_\theta (\mu )^{1/n}$.
For $n\geq 2$, take the Gauduchon metric $\gamma =\mathrm{e}^ g\Omega _\theta $ conformal to $\Omega _\theta $, given by theorem 11.4.2, where $g$ is a smooth real function. Since all eigenvalues of $\Omega _\theta $ relative to $\gamma $ are equal at every point, equality holds pointwise and hence in the quotient. This proves the formula and attainment. Conversely, if a Gauduchon metric attains the infimum, the nonnegative smooth difference of the two sides of the pointwise inequality has zero integral. It therefore vanishes everywhere, and the equality case of the arithmetic–geometric mean inequality makes $\gamma $ conformal to $\Omega _\theta $. Since the Gauduchon metric in a conformal class is unique up to a constant factor, this proves the last assertion.
For $n=1$, we integrate the equation and use $\int _ X\mathrm{dd}^{\mathrm{c}}u_\theta =0$. This gives $c_\theta (\mu )=\int _ X\theta $, which equals every quotient in the definition.
$\square$
For a Hermitian background $\omega $, a smooth positive probability volume $\mathrm{d}V$ and a nonzero nonnegative density $f\in L^1(\mathrm{d}V)$, the analogous quotient, with $\mu $ replaced by $f\, \mathrm{d}V$, computes the subsolution threshold $b_\omega (f)$ of section 15.28; see theorem 15.29.1. Attainment in this extremal problem, and the regularity of the extremal tests, are studied from section 12.7 on.
Corollary 12.2.4.A486E36565B6 If $\theta $ is big, then $\lambda _\mu (\theta )>0$.
Proof.
By the definition of bigness, there are a Hermitian form $h$ and $\delta >0$ such that $\theta -\delta h$ is pseudoeffective. Monotonicity, homogeneity and the theorem, applied to $h\in \mathscr P$, give $\lambda _\mu (\theta )\geq \delta \lambda _\mu (h)>0$.
$\square$
The converse fails for nef nonclosed backgrounds: theorem 12.14.6 gives, on a projective manifold, a nef background of positive volume that is not big.
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