25.18 Closed classes and singular lociC873148DF75D

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For closed classes on a Hermitian manifold, almost positive representatives keep the cohomology class separate from the auxiliary metric. The convention in section 10.11 follows Boucksom's thesis [Section 2.1.2, BouThesis02]; it does not assign a Bott–Chern class to a nonclosed Hermitian form.

The smooth and analytic regularizations of 5.36.3, used for proposition 10.12.2, have the lower bounds of [Theorem 2.1, Bou04Zariski]; see also [Theorems 1.2.9–1.2.10, BouThesis02]. Smooth regularization loses positivity by a multiple of a continuous approximation of the Lelong numbers, whereas analytic regularization has a curvature error tending uniformly to zero and does not increase Lelong numbers.

The minimal multiplicities of definition 10.12.1 are defined through these almost positive approximating families, as in [Definition 2.1.5, BouThesis02]. They must be distinguished from the Lelong numbers of a single current with minimal singularities: in the example of a nef class on a ruled surface in [Section 1.2.3, BouThesis02], a current with minimal singularities has nonzero Lelong numbers.

The non-Kähler locus of definition 10.13.1 is defined in [Definition 2.1.19, BouThesis02], and the proof of theorem 10.13.2 by finite intersections and regularized maxima is the argument of [Theorem 2.1.20, BouThesis02].

The equivalence between the existence of a Kähler current and the existence of a Kähler modification, used in section 10.10, is the classical criterion recalled in [Theorem 1.2.11, BouThesis02]; it is used as an external geometric input. The equality of the non-Kähler locus and the null locus of a nef and big class is the transcendental Nakamaye theorem of Collins and Tosatti [CT15Loci], which also holds on compact manifolds in Fujiki's class $\mathcal C$; section 10.10 recalls this theorem from the literature.

Restricted volumes give a numerical test for these loci. The numerical restricted volume (10.14.1) follows Collins and Tosatti [Section 2, CT22Restricted]; the requirement that the analytic singularities have a smooth remainder is part of that definition.

The existence of Kähler models for every subvariety uses Fujiki's theorem that class $\mathcal C$ is closed under passing to closed subspaces [Lemma 4.6(2), Fujiki78], followed by Varouchas's theorem on Kähler modifications in the form stated and explained by Barlet [Theorem 10, pp. 31–32, Barlet00]. A common Kähler refinement is obtained by embedded resolution of the graph in a Kähler product, using Bierstone and Milman [Theorem 1.6 and Remark 1.7(2), BM97]. These are classical geometric inputs. The descent of potentials is proved in proposition 5.43.18, so the proof of mass invariance does not repeat it.

In proposition 10.16.5, positive non-pluripolar mass implies bigness by [Proposition 1.22, BEGZ10]. The definition of the positive product, its agreement with the cup product for nef classes, and the fact that it is attained by currents with minimal singularities are in [Definition 1.17, arXiv pp. 17–18, and Proposition 1.20, arXiv p. 20, BEGZ10]. The normalized diagonal current (10.16.7) uses the mass-concentration theorem of Demailly and Păun [Proposition 2.6, DP04] and makes explicit the coefficient in the construction of [Theorem 2.12, DP04]. These results on Kähler manifolds are used as classical inputs. The identity for the mass with a bounded residual in lemma 10.17.3, by contrast, follows directly from the book's formula for bounded masses, by comparison of polynomials. The comparison with the numerical restricted volume for nef classes is [Lemma 2.6, CT22Restricted], and the final equality in (10.16.11) uses the theorem [Theorem 1.1, CT15Loci] recalled above.

The rigidity criterion for nef boundary classes on tori, theorem 10.26.1, develops the flat model of Fu and Ouyang [v1, Section 3, pp. 5–6, FuOuyang26]. Testing against Fourier modes proves the vanishing along the kernel directly, and the metric argument uses a finite covering by dense leaves and a character of the circle. The collapse in this theorem is along a fixed positive perturbation $\theta +t\kappa $, where $\kappa $ is a translation-invariant Kähler form. The calculation with mixed forms in example 10.26.3 shows that the clause in their Lemma 3.1 and Proposition 3.2 allowing arbitrary approaching classes cannot be used in that generality; their Theorem 4.4, which concerns positive perturbations, and the distinction made in their Remark 6.4, p. 18, concern a different setting. Their general theorems on collapse, fibrations and lattice dynamics for K3 surfaces are not used in this argument on tori.

The nonrigid class with an irrational ray in proposition 10.27.2 is the explicit Kummer construction of [v1, Section 7.1, pp. 18–19, proof of Theorem 1.4(a), FuOuyang26]. The resolution of the quotient and its Kähler metric are described in local charts, and the difference of two representatives is given an explicit periodic global $\mathrm{dd}^{\mathrm{c}}$-primitive. Periods over translated real two-dimensional subtori detect the irrational ray without extending cohomology classes across the punctures. Fu and Ouyang's conclusions about diameters of Ricci-flat metrics, spectra and limit spaces require additional inputs and are not used in this construction.

The family obtained from four points and its central metric in section 10.28 are Qin's construction [v1, Theorem A, p. 1, and Section 3, Lemmas 3.1–3.3, pp. 3–4, QinMetricExtension26]. Qin's proof of the nonextension theorem on p. 5 uses global $L^2$ extension for adjoint bundles. Theorem 10.28.4 instead uses the conics of degree zero of lemma 10.28.3: Fubini's theorem, averages over disks and Harnack's inequality force the central weights to be harmonic away from the node, and the remaining curvature has logarithmic coefficient one, whereas the metric constructed here has coefficient one half. This shows that the metric has no semipositive extension whose central restriction is at most as singular as the metric itself, without using global adjoint extension and without asserting that this metric has minimal singularities. Varying the conic gives the exact set of restrictions in theorem 10.28.9: the divisor metric of the negative line multiplied by the pullback of an arbitrary semipositive metric on the base of the pencil. The converse is constructed from the two homogeneous sections of the pencil, and corollary 10.28.10 concerns the least singular metrics among these restrictions.

Inoue surfaces were introduced by Inoue [Inoue74], and their higher-dimensional analogues by Oeljeklaus and Toma [OT05], who proved that the action of $\mathcal O_ K\rtimes U$ used in section 10.29 is free, properly discontinuous and cocompact, and that the resulting manifolds are not Kähler. Ornea and Verbitsky showed that Oeljeklaus–Toma manifolds with $t=1$ contain no positive-dimensional complex subvarieties, using a semipositive closed form with dense null leaves [OV11OT]. The decomposition of Inoue surfaces into three-dimensional tori filled by dense leaves, and a Liouville argument along the leaves for bounded harmonic functions, appear in the work of Fang, Tosatti, Weinkove and Zheng on the Chern–Ricci flow [Section 2, Lemma 2.1, FTWZ16], where the Tricerri metric and the identity $\mathrm{Ric}(\omega _ T)=-\alpha $ are also recorded; their form $\alpha =\mathrm i\, \mathrm dw\wedge \mathrm d\overline w/(4y^2)$ equals $2\pi \theta $ in the normalization of section 10.29, and their $\omega _ T=4\alpha +\beta $ is the Tricerri metric used there. The description of all $\theta $-plurisubharmonic functions in theorem 10.29.4, the mass computations of proposition 10.29.10 and the absence of plurisigned metrics in proposition 10.29.14 are proved directly here. Angella, Guedj and Lu proved that, on a compact threefold, a Hermitian metric with $\mathrm{dd}^{\mathrm{c}}\omega \geq 0$ gives finite upper mass and one with $\mathrm{dd}^{\mathrm{c}}\omega \leq 0$ gives positive lower mass; in particular a pluriclosed metric gives both [Theorem B, AGL23]. The general question of finiteness and positivity of these masses is [Problem 1.8, AGL23]. Angella, Dubickas, Otiman and Stelzig showed that an Oeljeklaus–Toma manifold admits a pluriclosed metric only if $s=t$ [Corollary 3, ADOS24]; proposition 10.29.14 shows that for type $(s,1)$ with $s\geq 2$ there is not even a metric whose $\mathrm{dd}^{\mathrm{c}}$ has a sign.