10.10 Bott–Chern classes and singular loci26958A01D807

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This part of the book returns to closed backgrounds. A closed real $(1,1)$-form has a Bott–Chern class, and positivity can then be asked of the class rather than of a chosen representative. On a compact Kähler manifold, the positivity of a class is described by its non-nef and non-Kähler loci, by intersection numbers and by volumes. The present chapter asks which parts of this description survive on an arbitrary compact complex manifold, where no Kähler metric controls masses, and how closed classes interact with the Hermitian equations of the preceding parts. In section 23.2, closed classes arise from the Lee form of a locally conformally Kähler metric.

Throughout the chapter, $X$ is a connected compact complex manifold of dimension $n\geq 1$ as in the Conventions, and $\omega $ is a fixed Hermitian form on $X$; sections that require a Kähler manifold or a special geometry say so at their beginning. Without a Kähler class, Gauduchon metrics provide the substitute mass control. Pairing a closed current with the $(n-1)$st power of a Gauduchon metric eliminates its potential, so the currents of a fixed class whose negative part is bounded by a multiple of $\omega $ have uniformly bounded mass. With this compactness, regularization defines the non-nef locus through Lelong numbers of minimal currents and the non-Kähler locus through polar sets of Kähler currents. These loci separate the singularities forced by the class from those of a particular current. Restricted volumes, which describe the same loci on Kähler manifolds, behave differently: computed with a Hermitian error, they can be infinite even for the zero class. When $X$ belongs to Fujiki's class $\mathcal C$, pulling a class back to a Kähler resolution of each subvariety gives instead a finite restricted mass; on a smooth subvariety, its positivity detects bigness of the restricted class. For a divisorial trace with bounded remainder, the non-pluripolar mass is the top intersection of a nef residual class; positive mass produces a Kähler current, but possibly with a larger singularity than the trace.

Closed classes also constrain the Hermitian Monge–Ampère equations of earlier chapters. A divisorial component of a positive current in a Hermitian background consumes Gauduchon trace but contributes no density to its non-pluripolar measure, so it can lower the largest coefficient of a subsolution. At the maximal coefficient, weak limits of maximizing Gauduchon tests annihilate the class of the divisor. On the elliptic threefold of section 12.19, closedness forces every positive current to descend to the base surface. No closed class is big there, although small Hermitian perturbations of the boundary classes have positive variational volume, whose degeneration can be computed explicitly.

Smooth solutions raise a metric question. For a big class on a compact Kähler manifold, the solution of the Monge–Ampère equation with a smooth positive density defines a Kähler metric on the ample locus, but its smoothness there does not determine the distance to the complement. Zeros of the density can come from ramification, from collapsed exceptional fibers or from a singular Hessian, and they give different normal lengths even when the determinant vanishes to the same order. For radial equations, boundedness of the potential and finiteness of the distance are governed by two different integrals: a continuous minimal remainder can have a complete end, whereas a Hölder remainder forces finite distance. For exponential equations with a divisorial pole, dividing the determinant by its tangential part gives a lower bound for normal length and a sharp logarithmic threshold for completeness. Explicit surfaces give exponential equations with unique finite-energy solutions and incomplete metrics. One solution has minimal singularities in a big non-nef class; another has a nef and big class but an unbounded cusp.

At the boundary of the nef cone, smooth semipositive representatives have a different metric behavior. On a complex torus, a constant semipositive form whose kernel leaves are dense is the only closed positive current in its class, and adding a small fixed Kähler form produces metrics whose diameters tend to zero. An explicit family of flat metrics collapses in the same way at a class with many semipositive representatives, so the collapse depends on how the class is approached. On the resolution of a Kummer quotient, an irrational nef boundary class has many smooth semipositive representatives. Finally, in a family of blown-up projective planes, degree-zero curves on the nearby fibers force a larger singularity along one component of the central curve, so a prescribed semipositive metric on the special fiber does not extend to the family.