Non-Kähler Critical Hermitian Metrics of the Dinew--Popovici Functional

The Dinew–Popovici functional is an energy functional for Hermitian symplectic metrics in a fixed Aeppli cohomology class. Its vanishing characterizes the Kähler metrics in that class, providing a variational approach to Kähler geometry. Dinew and Popovici proved that every critical point is Kähler in complex dimension three. We give a negative answer to Erfan Soheil’s question about higher dimensions by constructing non-Kähler critical metrics on products of two Kähler surfaces $(S_1,{\eta }_1)$ and $(S_2,{\eta }_2)$. These metrics are critical under every variation on the product 4-fold. We characterize the critical product metrics and prove that non-Kähler critical products exist in the Aeppli class $[{\eta }_1+{\eta }_2{]}_A$ precisely when the canonical bundles of the two surface factors are smoothly trivial. As a by-product, we develop a geometric flow driven by the torsion tensor and converging to the fixed Kähler background when this background metric has nonnegative holomorphic bisectional curvature.