24.5 Closed and twisted classesB3F20D9F0DB0

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A volume attached to a closed Bott–Chern class must not depend on the representative, while a volume equation for a locally conformally Kähler (LCK) form must preserve its Lee class and its twisted differential. The examples below show where ordinary Hermitian perturbations fail to respect these requirements. We use the LCK notions of definition 23.2.2: for a closed one-form $\lambda $, $D_\lambda $ is the twisted Hessian of (23.2.19), and $j_ X\colon H^1_{\mathrm{dR}}(X,\mathbb R)\to H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$ is the Lee map (23.2.9); the associated Lee–Gauduchon cone is studied in subsection 23.2.55.

The non-nef and non-Kähler loci of section 10.10 separate closed currents with small negative errors from strictly positive closed currents. Any numerical restricted volume proposed for closed classes on general compact complex manifolds must be compatible with proposition 10.15.1: if a Hermitian form $\omega $ fails the bounded mass property, as on the Hopf threefold, then for every $\varepsilon >0$ the supremum of $\int _ X(T+\varepsilon \omega )^ n$ over smooth closed representatives $T\geq -\varepsilon \omega $ of the zero class is already infinite. Since the failure occurs at each fixed error size, letting the error tend to zero afterwards cannot turn this expression into a class invariant.

For $n\geq 2$, the subsolution threshold also constrains divisorial singularities. Let $\omega $ be a Hermitian form, $\mathrm{d}V$ a smooth positive probability volume form and $0\leq f\in L^1(\mathrm{d}V)$ a nonzero density with $b=b_\omega (f)>0$, and put $p=f^{1/n}$. If a subsolution at the threshold $b$ dominates the current of integration of an effective divisor $E$, then by proposition 10.18.1 the weak limits of every smooth maximizing Gauduchon sequence annihilate the class of every component of $E$. If one such sequence has uniformly integrable determinant-weighted densities, corollary 10.18.4 excludes divisors whose classes have suitable smooth semipositive representatives, for instance the hyperplane divisor on the Hopf threefold, and corollary 10.18.5 shows that a component with nef class has zero variational volume for every smooth density. Both conclusions hold under the root-tail condition $p\in \mathcal D_\omega ^0$. Characterizing the components that can occur would require a realization theorem; note that the weak closure of smooth tests can be smaller than the set of all normalized Gauduchon currents (example 10.18.3).

For a prescribed logarithmic divisor with a positive Hermitian residual, theorem 11.41.6 identifies the threshold on the entire one-sided class with that of the residual background. The ordinary class and relative envelope class have the same attained coefficient for smooth strictly positive data. A nonzero divisor then makes it strictly smaller than the global coefficient, and corollary 11.41.7 excludes divisorial components at the smooth global threshold. The projective product in example 11.41.8 computes this loss explicitly. These positive-residual results do not give a converse from an annihilating face for general integrable data.

For closed classes on manifolds in Fujiki's class $\mathcal C$, section 10.16 defines a finite restricted invariant on each subvariety, independent of the Kähler resolution used to compute it. It is positive exactly when the class is big on the resolved subvariety, and for a nef class on $X$ the union of the subvarieties where it vanishes is the null locus. For a specified divisorial trace with bounded remainder, section 10.17 computes the mass of the trace itself and turns positive residual mass into a strict current, more singular than the trace. The nonclosed ruled family of subsection 20.3.63 goes beyond bounded residuals: theorem 20.3.66 computes both masses over the whole ordinary type, and proposition 20.3.71 identifies its ambient ideal orders, although the unbounded residual is invisible to those ideals. Positive lower volume gives a strict current in that ordinary type downstairs. It is preserved under pure pullback, while proposition 20.3.74 excludes a strict current in the pulled-back ordinary type and still allows a one-sided strict representative. The same obstruction holds after every nonempty finite composition of point blowups, including centers at poles. More generally, theorem 18.27.4 gives an exact ordinary-type strictness criterion for a finite logarithmic divisor plus a residual with arbitrarily small negative curvature. Its ambient ideals and, in the normal crossing case, all intrinsic traces are given by proposition 20.3.8, even when the naive residual restriction is identically $-\infty $. Thus a general restricted invariant must specify which singularity relation its strictness criterion uses. Its construction for arbitrary nonclosed backgrounds, and its relation to intrinsic multiplier-ideal trace data outside these families, remain open.

The LCK setting raises separate questions about equations. By corollary 23.2.10, a compact strictly LCK manifold has no big ordinary Bott–Chern class. Its LCK form belongs instead to a twisted Bott–Chern class, whose representatives are described by the twisted Hessian $D_\lambda $. Prescribing the volume form of a Gauduchon LCK metric in such a class leads to the coupled system (23.2.26), whose second equation is the additional conformal condition imposed by the Gauduchon condition. 23.2.27 asks whether this system has at most one solution when the Lee class and the twisted class are fixed; the nonuniqueness examples for ordinary potentials do not decide it.

On scalar Hopf manifolds the radial case is solved: for every smooth positive radial volume form and every negative Lee period, theorem 23.2.41 gives exactly one Gauduchon metric in the full positive radial family. By proposition 23.2.43, this metric has a positive potential in its own Lee gauge, so its twisted class is zero, and the coupled system determines it among radial candidates. For nonconstant radial densities these metrics are not Vaisman (proposition 23.2.44). Uniqueness among arbitrary LCK candidates is open.

Two geometric conjectures use the Lee form in different ways. 23.2.12 asserts that the class $j_ X([\lambda ])$, for the Lee form $\lambda $ of a strictly LCK metric, is pseudoeffective, that is, pairs nonnegatively with every Gauduchon metric; the LCK metric itself provides one positive pairing. 23.2.54 asks whether a compact complex manifold with vanishing first Bott–Chern class that admits a strictly LCK metric also admits a Vaisman metric. The rigidity theorem theorem 23.2.47 gives this when a single strictly LCK metric is both Chern–Ricci flat and Gauduchon; the remaining existence problem is to arrange these two conditions simultaneously.

There is a pointwise sufficient condition for the first conjecture. If the least Lee-curvature eigenvalue relative to the chosen strictly LCK metric is constant, 23.2.15 makes it zero and gives a smooth semipositive representative of the nonzero ordinary Bott–Chern class. In particular this class is nef, and the balanced obstruction applies; compact invariant LCK quotients satisfy the criterion. The volume polynomial 23.2.14 supplies the argument. The two Hopf families show its scope: 23.2.17 has constant minimum but varying remaining spectrum, whereas 23.2.16 has indefinite Lee curvature and a smooth semipositive correction. In the latter family the first pointwise degeneracy retains positive total volume. A general LCK metric need not have constant minimum, so the unrestricted pseudoeffectivity conjecture stays open.

The Lee–Gauduchon cone of subsection 23.2.55 makes part of this geometry invariant under bimeromorphic maps. Its dual is the cone of classes in $H^1_{\mathrm{dR}}(X,\mathbb R)$ whose image under $j_ X$ is pseudoeffective, and the strict inequalities defining it exclude $\ker j_ X$; the duality and its behavior under modifications are proved in that section. A nonzero pseudoeffective class in the image of $j_ X$ obstructs balanced metrics on every manifold bimeromorphic to $X$. On the Hopf threefold, the classes in $H^1_{\mathrm{dR}}(X,\mathbb R)$ whose image under $j_ X$ is pseudoeffective form a positive half-ray, although the Monge–Ampère masses there are unbounded. This cohomological invariant thus captures a geometric obstruction, while the nonlinear estimates for the scalar equation remain a separate task. The converse of its consequence for balanced metrics fails: the threefold of example 23.2.72 has zero Lee–Gauduchon cone but no strongly Gauduchon metric.