Non-K\"ahler Critical Hermitian Metrics of the Dinew--Popovici Functional

Abstract

The Dinew--Popovici functional is an energy functional for Hermitian symplectic metrics in a fixed Aeppli cohomology class. Its vanishing characterizes the K\"ahler metrics in that class, providing a variational approach to K\"ahler geometry. Dinew and Popovici proved that every critical point is K\"ahler in complex dimension three. We give a negative answer to Erfan Soheil's question about higher dimensions by constructing non-K\"ahler critical metrics on products of two K\"ahler 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-K\"ahler 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 K\"ahler background when this background metric has nonnegative holomorphic bisectional curvature.