Classification of Conformally Einstein Kähler Manifolds

Definitions and mathematical statement

A Kähler manifold is a complex manifold $(M,J)$ with a Riemannian metric $g_0$ satisfying $g_0(JX,JY)=g_0(X,Y)$ whose associated two-form $\omega_0(X,Y)=g_0(JX,Y)$ is closed. A Riemannian metric $g$ is Einstein if its Ricci tensor satisfies $\operatorname{Ric}_g=\lambda g$ for an Einstein constant $\lambda\in{\mathbb{R}}$. On a disconnected manifold, whether one requires a common constant or allows a separate constant on each connected component is not specified in the supplied source.

The classification problem is to determine all Kähler manifolds $(M,J,g_0)$ for which there exist a globally defined function $\sigma\in C^\infty(M,{\mathbb{R}})$ and an Einstein Riemannian metric $g$ on $M$ such that \[ g=e^{\sigma}g_0. \] Thus the conformal factor must be smooth and strictly positive everywhere on $M$; a factor defined only locally, or vanishing somewhere, does not satisfy this formulation. On a connected $M$, the condition can equivalently be written \[ \exists\,\sigma\in C^\infty(M,{\mathbb{R}})\;\exists\,\lambda\in{\mathbb{R}} \quad \operatorname{Ric}_{e^{\sigma}g_0}=\lambda e^{\sigma}g_0. \] No sign restriction on $\lambda$ or nonconstancy requirement on $\sigma$ is supplied.

The source specifies no complex dimension, compactness, completeness, connectedness, boundary convention, or equivalence relation for the classification. These aspects of the intended classification remain unresolved. The clean registry statement supplies the target above. The original excerpt also contains a subsequent heading about conformally Kähler--Einstein metrics and a slash-separated definition of conformally Kähler/Einstein metrics; that surrounding text does not unambiguously specify an additional hypothesis for this problem. In particular, the clean target requires $g_0$ to be Kähler and $g$ to be Einstein, without requiring either metric to satisfy both conditions.

Short English statement

Classify Kähler metrics that become Einstein after multiplication by a globally defined smooth positive function. The source leaves the dimensional and global setting, boundary convention, and meaning of classification unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID 30005872, source problem code catalogue entry, titled "Classification of Conformally Einstein Kähler Manifolds." The record attributes the problem to the Oberwolfach Reports collection and gives the year 2024; no individual proposer is supplied.
• Source report: Geometrie (2024), Oberwolfach Reports, pp. 1563--1614, DOI 10.4171/owr/2024/28. https://doi.org/10.4171/owr/2024/28. The registry labels its statement corrected_verified and reports a check against the official report. That verification is supplied provenance, not an independent check performed for this draft.
• Supplied literature reference labeled "Derdzinski--Maschler higher-dimensional local classification": https://doi.org/10.1112/S0024611503014175.
• Supplied literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2024/28.
• Supplied literature reference labeled "recent conformal Kähler rigidity in four dimensions": https://arxiv.org/abs/2607.21538.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports strong local classifications in complex dimension greater than two, substantial four-dimensional criteria and classifications, and an incomplete broad classification. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.