12.1 A Gauduchon variational volume881C26425D7A

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Let $X$ be a connected compact complex manifold of dimension $n\geq 1$, let $\theta $ be a smooth real $(1,1)$-form, which we call the background, and let $\mu $ be a smooth strictly positive probability form on $X$. If $\theta +\mathrm{dd}^{\mathrm{c}}v>0$ for some smooth real function $v$, there is exactly one constant $c>0$ for which the Monge–Ampère equation $(\theta +\mathrm{dd}^{\mathrm{c}}u)^ n=c\mu $ has a smooth solution $u$ with $\theta +\mathrm{dd}^{\mathrm{c}}u>0$. When $\theta $ is closed, Stokes' theorem gives $c=\int _ X\theta ^ n$, independently of $\mu $. For a nonclosed background the total mass $\int _ X(\theta +\mathrm{dd}^{\mathrm{c}}u)^ n$ can depend on the potential, and the constant $c$ is the one number that the equation attaches to the pair $(\theta ,\mu )$. This chapter studies it as a function of the background.

The starting point is a dual formula. Integrating the arithmetic–geometric mean inequality against the $(n-1)$st power of a Gauduchon metric $\gamma $, that is, of a Hermitian form with $\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$, shows that $c^{1/n}$ is the infimum of a quotient whose numerator $\int _ X\theta \wedge \gamma ^{n-1}$ is linear in $\theta $ and unchanged by smooth $\mathrm{dd}^{\mathrm{c}}$ changes of $\theta $. Taking this infimum for every pseudoeffective background defines the root of the Gauduchon variational volume. The root volume is concave. On the positive cone, a minimizing Gauduchon metric computes its derivative, and the Hessian of its negative logarithm defines a Riemannian metric on background classes in each finite-dimensional family. Invariant forms on compact quotients of complex Lie groups and warped products with a curve show that the unit-volume hypersurface of this metric may be complete or may acquire a boundary at finite distance.

For $n\geq 2$, the same quotient computes the subsolution threshold $b_\omega (f)$ for a Hermitian background $\omega $ and an integrable density $f$. Allowing positive $\mathrm{dd}^{\mathrm{c}}$-closed currents as tests makes this dual problem compact. For continuous densities the maximizing currents represent exactly the supporting probabilities of the normalization functional; for integrable densities, a uniform tail condition passes the determinant integral to weak limits and again gives a maximizing current. A maximizing test that is a smooth strictly positive form gives more: equality in the trace inequality determines the solution current and forces an exact smooth equation. The exact equation persists when the maximizing form degenerates along a proper analytic set, whereas a threshold solution with a point pole can admit no smooth maximizing test at all.

The last part of the chapter concerns the boundary of the positive cone and the dependence on $\mu $. At a nef background the variational volume is the limit of the smooth equation constants of nearby positive backgrounds. A projective radial model gives positive variational volume without bigness, even at the finite maximal time of a Chern–Ricci flow. Varying $\mu $ recovers the upper and lower Monge–Ampère masses. A threefold with zero lower mass shows that positivity for each smooth density need not be uniform over densities, whereas on an elliptic threefold a nonnegative sum of curvature squares gives a bounded cubic energy identity and hence a positive lower bound for every potential, even when the manifold admits no pluriclosed metric. Finally, nef degeneration gives a further contrast with closed Kähler classes: intersection polynomials are replaced by fractional, logarithmic or oscillating decay, and a single blowup can change its order.

Throughout the chapter, a smooth real $(1,1)$-form $\theta $ is called pseudoeffective if $\mathrm{PSH}(X,\theta )\ne \varnothing $, that is, if $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$ for some quasi-plurisubharmonic $u$; by theorem 11.6.1 this holds exactly when $\int _ X\theta \wedge \gamma ^{n-1}\geq 0$ for every Gauduchon metric $\gamma $. Bigness is defined in section 5.40, and $\theta $ is nef if, for a Hermitian form $h$ and every $\epsilon >0$, there is a smooth real $f_\epsilon $ with $\theta +\mathrm{dd}^{\mathrm{c}}f_\epsilon \geq -\epsilon h$; this is the condition of subsection 20.4.16 for the affine class of $\theta $ in subsection 10.30.1. None of these notions requires $\theta $ to be closed, and each is unchanged by a smooth $\mathrm{dd}^{\mathrm{c}}$ change of $\theta $.