Kähler Criteria from Nonintegrable Bundle Spectral Theory

Definitions and mathematical statement

Let $X$ be a compact complex manifold with complex structure $J$ and complex dimension $n\geq 3$. A Kähler form on $X$ is a smooth real differential form $\omega$ of type $(1,1)$ such that \[ \mathrm d\omega=0,\qquad \omega_x(v,Jv)>0\quad\text{for every }x\in X\text{ and }0\ne v\in T_xX. \] The associated Kähler metric is $g(u,v)=\omega(u,Jv)$.

Research problem. For compact complex manifolds $X$ of complex dimension greater than two, develop an analogue of the results of Lamari and Buchdahl concerning the existence of Kähler metrics, in the context of the spectral theory of nonintegrable complex vector bundles.

Unresolved formulation. The supplied record does not state the particular results of Lamari and Buchdahl to be generalized, the hypotheses of the proposed analogue, or whether the requested criteria should be necessary, sufficient, or both. It also does not specify the bundles, the structure whose failure of integrability is intended, the differential operators, or the spectral conditions. Consequently, the record determines a research objective but does not determine a precise conjectured implication or equivalence. These missing choices must be resolved before a theorem-level statement can be formulated.

Short English statement

Seek criteria for the existence of Kähler metrics on compact complex manifolds of complex dimension at least three, extending results of Lamari and Buchdahl through spectral theory for nonintegrable complex vector bundles. The intended bundle structures, spectral operators, hypotheses, and exact criteria remain unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $30004544$, title: Kähler Criteria from Nonintegrable Bundle Spectral Theory; exact source problem code: catalogue entry. The supplied record attributes the problem to the Oberwolfach Reports collection and names no individual proposer.
• Supplied source citation: Mini-Workshop: Almost Complex Geometry (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/33, pp. 1657--1691. https://doi.org/10.4171/owr/2020/33. This is also the supplied literature reference labelled "OWR report." The record gives proposed year 2020; its workshop citation gives 2021. This dating discrepancy is unresolved.
• Supplied literature reference labelled "Siu invariance of plurigenera": https://doi.org/10.1007/s002220050276.
• Supplied literature reference labelled "Transcendental Morse inequalities": https://arxiv.org/abs/2309.04978.
• Literature assessment dated exactly 2026-08-22: this is supplied dated curation, not independently verified. It labels the record partially solved and partly outdated, attributes deformation invariance of plurigenera in the projective general-type setting to Siu, and describes transcendental Morse inequalities and the higher-dimensional Lamari--Buchdahl analogue as open or only partially solved. The record requests independent verification of these time-sensitive claims. The assessment does not identify a precise solved subcase of the research problem stated here.