Fujiki-Class Limits of Kähler Manifolds

Definitions and mathematical statement

Let $\Delta\subset{\mathbb{C}}$ be an open disc centered at $0$, and put $\Delta^*=\Delta\setminus\{0\}$. Interpret a complex-analytic family of compact complex manifolds in the usual smooth sense: a proper surjective holomorphic submersion $\pi:X\to\Delta$ from a complex manifold, with fibers $X_t=\pi^{-1}(t)$. This interpretation makes the family hypotheses precise; the supplied record does not explicitly specify properness or the submersion condition, so their intended inclusion requires source review.

A compact complex manifold $M$ is Kähler if it admits a smooth real $(1,1)$-form $\omega$ such that ${\,\mathrm{d}}\omega=0$ and $\omega$ is positive definite. It belongs to Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold. Here bimeromorphic means that there is a meromorphic map admitting a meromorphic inverse.

Question. For every family $\pi:X\to\Delta$ as above, does \[ \left(\forall t\in\Delta^*,\;X_t\text{ is K\"ahler}\right) \quad\Longrightarrow\quad X_0\in\mathcal C \] hold?

The supplied dated curation labels the problem partially solved and reports that a projective or integral-class version is proved, while the general question remains open and is related to transcendental Morse inequalities. The precise hypotheses of those special versions are not supplied, so this contextual assessment is not formulated here as a separate theorem or an additional assumption.

Short English statement

If every noncentral fiber of a smooth proper holomorphic family of compact complex manifolds over a disc is Kähler, must the central fiber be bimeromorphic to a compact Kähler manifold? The smooth proper interpretation of "family" requires confirmation from the source. The supplied partially solved label is dated curation, not an independently verified status.

Sources

• Local project problem registry: permanent UnsolvedMath ID $30001229$, source problem code catalogue entry, title Fujiki-Class Limits of Kähler Manifolds. The record assigns the problem to 2009 and supplies no individual proposer.
• Supplied report citation: Multiplier Ideal Sheaves in Algebraic and Complex Geometry (2009), Oberwolfach Reports, pp. 1101--1156, DOI 10.4171/owr/2009/21. Original report: https://doi.org/10.4171/owr/2009/21.
• Supplied literature link labeled "Popovici projective case": https://arxiv.org/abs/0910.2032. The record does not supply its full bibliographic citation or precise theorem hypotheses.
• Supplied literature link labeled "current status": https://arxiv.org/abs/2409.19957. This is the supplied label, not an independently established current assessment.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. It labels the problem partially solved, reports a proved projective or integral-class version, and describes the general Kähler-to-Fujiki-class-$\mathcal C$ limit question as open and related to transcendental Morse inequalities. The record requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that corrupted Kähler and class notation was repaired. These are supplied verification claims; no independent source verification is asserted here.