Fino-Vezzoni conjecture on Lie algebras with abelian ideals of codimension two (Q6536640)

Complex non-Kähler geometry has received increasing attention recently and one of its open problems is the Fino-Vezzoni conjecture [\textit{A. Fino} and \textit{L. Vezzoni}, J. Geom. Phys. 91, 40--53 (2015; Zbl 1318.53049)], [\textit{A. Fino} and \textit{L. Vezzoni}, Proc. Am. Math. Soc. 144, No. 6, 2455--2459 (2016; Zbl 1345.32017)]. It states that a compact complex manifold admitting two Hermitian metrics -- one balanced and one pluriclosed, must also admit a Kähler metric. The paper under review confirms the conjecture in one new set of examples. They are parallelizable manifolds with the associated unimodular Lie algebras having an abelian ideal of codimension two, which includes some 3-step solvmanifolds. The proof is based on Lie-theoretic methods proving the conjecture for invariant metrics and then a simple averaging argument allows the invariance condition to be dropped.