Let X be a compact complex manifold of complex dimension n. If X admits a balanced Hermitian metric with fundamental form omega_B satisfying d(omega_B^n-1)=0 and a pluriclosed Hermitian metric with fundamental form omega_P satisfying dd^comega_P=0, then X admits a Kähler metric.

Fino–Vezzoni conjecture on balanced Bismut torsion-parallel manifolds

The Fino–Vezzoni conjecture predicts that a compact complex manifold carrying both balanced and pluriclosed Hermitian metrics must be Kähler. The new work gives this conclusion for balanced Bismut-torsion-parallel manifolds when a pluriclosed metric exists, but does not settle the unrestricted conjecture.

Known results

Compact Bismut-torsion-parallel threefolds satisfy the conjecture, including the balanced case (Chen–Zheng, 2024–2025).
In the balanced threefold case, a manifold is Kähler or admits no pluriclosed metric.
The balanced case in complex dimension $n \ge 4$ was reported as open before the latest development.

August 2026 conditional theorem

Shuwen Chen and Fangyang Zheng establish the Kähler conclusion for balanced Bismut-torsion-parallel manifolds admitting a pluriclosed Hermitian metric. This substantially extends the known special cases, but the general conjecture without that extra assumption remains unresolved.

Current status (as of August 2026): The conjecture is proved for compact Bismut-torsion-parallel threefolds and, more generally, in the new pluriclosed-metric case; the unrestricted balanced problem remains open, especially in complex dimension $n \ge 4$ .

Fino–Vezzoni conjecture on balanced Bismut torsion-parallel manifolds

Progress summary

Primary source

Additional references

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