24.4.1 Nef classes, bigness and the Fujiki class806079150C7A

A compact complex manifold belongs to Fujiki's class $\mathcal C$ exactly when it carries a closed Kähler current, that is, a closed positive $(1,1)$-current bounded below by a positive multiple of a Hermitian form. The transcendental Grauert–Riemenschneider conjecture asks whether a closed nef real $(1,1)$-class $\alpha $ with

is big. For such a class, bigness is equivalent to $X\in \mathcal C$; the precise statements and sources are given in section 25.22.

The obstruction can be expressed through Monge–Ampère mass. Under the same hypotheses on $\alpha $, one has

for a Hermitian metric $h$ on $X$, where $v_+(h)$ is the upper mass of definition 13.14.14. There is also a characterization of $\mathcal C$ by bounded mass together with positivity of an upper volume of some closed Bott–Chern class; that volume is different from the Gauduchon variational volume $\mathcal V_\mu $. Both results are stated in section 25.22.

The Hopf threefold fails bounded mass, but it is not a counterexample to the conjecture, since a counterexample would also need a closed nef class with positive top self-intersection. A proof of the conjecture must explain why this closed positivity forces bounded mass.

The elliptic threefold of theorem 12.19.1 is the complementary test case: it has bounded mass but no big closed class. By proposition 10.19.2, its nef cone is the nonnegative span of the classes of two smooth semipositive forms of rank one, and every top self-intersection vanishes. This example of zero lower mass therefore does not bear on the conjecture either. The explicit boundary calculation of proposition 10.19.8 describes its degeneration without identifying a nonclosed metric with a closed class.

The Gauduchon variational volume of definition 12.2.1 splits the conjecture into two possible steps:

Neither implication is known on arbitrary compact complex manifolds. The second concerns closed nef backgrounds; the projective counterexample of theorem 12.14.6 has a nonclosed background and does not decide it. On a compact Kähler manifold both implications follow from the usual volume formula for nef classes and the classical nef-and-big criterion. Without closedness, positivity of $\mathcal V_\mu $ cannot characterize $\mathcal C$, since every Hermitian metric $h$ on every compact complex manifold satisfies $\mathcal V_\mu (h)>0$.

The test metrics can be normalized by $\int _ Xh\wedge \gamma ^{n-1}=1$. By proposition 24.3.2, a positive lower bound for $D_\mu (\gamma )$ along one sequence minimizing the threshold would give the second implication. Solving a Gauduchon equation changes the volume density of a test metric while keeping the Aeppli class of $\gamma ^{n-1}$; since $\alpha $ is closed, it also keeps the pairing with $\alpha $. The missing estimate is control of the trace normalization and of the determinant denominator after this change. Prescribing a volume density alone does not give this estimate.