24.3 Positive Gauduchon volume and bigness6CDBB1FBABB2

Fix a smooth strictly positive probability form $\mu $ on $X$. For a pseudoeffective smooth real $(1,1)$-form $\theta $, the Gauduchon variational volume $\mathcal V_\mu (\theta )=\lambda _\mu (\theta )^ n$ of definition 12.2.1 is an infimum over Gauduchon metrics, whereas bigness asks for one potential $u$ making $\theta +\mathrm{dd}^{\mathrm{c}}u$ strictly positive. Bigness implies $\mathcal V_\mu (\theta )>0$ (corollary 12.2.4). The converse fails for nef nonclosed backgrounds: for every $n\geq 3$, theorem 12.14.6 constructs a smooth semipositive background $\theta $ on $\mathbb P^{n-1}\times \mathbb P^1$ with $\mathcal V_\mu (\theta )=n$ that is not big: no current $\theta +\mathrm{dd}^{\mathrm{c}}u$ is bounded below by a positive multiple of a Hermitian form. In this example a radial potential compensates the determinant, while a closed test current carried by a degenerate fiber detects the missing strict positivity.

Positivity of $\mathcal V_\mu $ does not depend on the choice of $\mu $ (proposition 12.15.1), but the infimum over all such $\mu $ is a different invariant: by theorem 12.15.4 it is the lower mass, and it vanishes for every Hermitian form on the threefold of theorem 12.19.1, although Hermitian forms are big. In dimension one, positive volume and bigness are equivalent, since the volume is $\int _ X\theta $. For a closed nef class on a compact Kähler manifold, $\mathcal V_\mu $ is the top self-intersection, and the equivalence is the nef-and-big criterion recalled in [SGPT]. On a compact Kähler manifold, theorem 12.12.2 also gives bigness when a closed nef core of positive volume is supplemented by a semipositive form. Whether positive volume implies bigness for general nef nonclosed backgrounds on surfaces is open. In higher dimension, any such criterion requires geometric hypotheses that exclude the radial obstruction.

24.3.1 Collapse of normalized Gauduchon testsC56273368CE1

The root volume and the bigness threshold divide the same pairing by different quantities. Positive volume can fail to imply bigness only if these denominators separate along tests approaching the threshold, and the following calculation isolates this degeneration. For the radial background of theorem 12.14.6, the normalized $(n-1)$-st powers of a family of balanced test metrics (definition 11.4.1) converge weakly to the integration current of the degenerate fiber (proposition 12.14.13). The volume-minimizing family approaches the bigness threshold only in dimension three, by proposition 12.14.10.

Fix a Hermitian form $h$ on $X$, and let $\theta $ be a smooth pseudoeffective real $(1,1)$-form. For a Gauduchon metric $\gamma $, abbreviate the pairings by

The threshold $\kappa _ h(\theta )$ of corollary 11.6.2 is the infimum of the quotient $A_\theta /A_ h$ over Gauduchon metrics, and it is positive exactly when $\theta $ is big. The root volume $\lambda _\mu (\theta )$ is the infimum of $A_\theta /D_\mu $, with the determinant denominator $D_\mu (\gamma )$ of definition 12.2.1.

The quotient $D_\mu /A_ h$ is always bounded above. Indeed, the smooth equation for $h$, or (24.3.3) with $\theta =h$, gives

What is missing for the converse direction is a lower bound for $D_\mu $ along suitable sequences minimizing the threshold. Compactness of normalized potentials does not give compactness of these Gauduchon metrics.

In dimension one this normalization issue does not arise: the root volume is $\int _ X\theta $, and $\kappa _ h(\theta )=\int _ X\theta /\int _ Xh$. Positive volume and bigness are therefore equivalent by the same threshold criterion.

One way to rule out this collapse is to deform the test metrics. Solving a Gauduchon equation replaces the volume density of a test metric while keeping the Aeppli class of $\gamma ^{n-1}$. When $\theta $ is closed, this leaves the numerator $A_\theta $ unchanged. For nonclosed $\theta $, integration by parts leaves derivatives of the background, so such a deformation must also control the change of the pairing; this is the additional estimate needed to use the smooth equation against collapse.