Goldberg conjecture for compact Einstein almost Kähler manifolds

Prove that the almost complex structure of every compact Einstein almost Kähler manifold is integrable; equivalently, establish that any compact Einstein almost Kähler manifold is Kähler.

Background

The paper discusses the classical Goldberg conjecture, asserting that compact Einstein almost Kähler manifolds should be Kähler. The authors review partial results confirming the conjecture under specific curvature conditions—such as nonnegative scalar curvature, certain four-dimensional cases, and constant sectional curvature—while noting that the general case has not been resolved.

Their work develops statistical characterizations of Kähler and co-Kähler structures and links Kähler metrics to Fisher information metrics, but they explicitly acknowledge that the general Goldberg conjecture remains open outside the known special cases.

References

Goldberg conjecture: "The almost complex structure of a compact Einstein almost Kähler manifold is integrable (and therefore the manifold is Kähler)". It is still open otherwise.

— Any Kähler metric is a Fisher information metric  (2405.19020 - Gnandi, 2024) in Subsection “A new characterisation of Kähler metrics,” within Section “Kähler and co-Kähler metrics are always Fisher information metrics”

An obstructed ICD, however, could conceivably integrate through Einstein metrics that cease to be Kähler, and holonomy rigidity for nonzero Kähler-Einstein metrics is not known in full generality \S6.3.

— A negative Kähler-Einstein threefold with non-integrable infinitesimal Einstein deformations  (2608.13481 - Krishna, 13 Aug 2026) in Section 1, Introduction

Goldberg conjectured that every compact almost-Kähler Einstein manifold is in fact Kähler.

— Kählerity of complete almost-Kähler gradient shrinking Ricci solitons  (2609.00840 - Xie, 1 Sep 2026) in Section 1, Introduction