Fino–Vezzoni conjecture on balanced and pluriclosed Hermitian metrics

Determine whether every compact complex manifold admitting both a balanced Hermitian metric and a pluriclosed Hermitian metric necessarily admits a Kähler metric.

Background

The Fino–Vezzoni conjecture concerns the coexistence of two potentially distinct Hermitian metrics on the same compact complex manifold: one satisfying the balanced condition d(ω^{n−1}) = 0 and another satisfying the pluriclosed condition ∂∂̄ω = 0. The conjecture asserts that this coexistence forces the underlying complex manifold to admit a Kähler metric.

The paper proves the conjecture under the additional assumption that the balanced Hermitian metric is Bismut torsion-parallel, thereby resolving the balanced BTP subclass in every complex dimension. The unrestricted conjecture stated in Conjecture 1 is broader than the theorem proved in the paper and remains the relevant unresolved problem.