Hermitian geometrically formal manifolds

Abstract

We study Hermitian geometrically formal metrics on compact complex manifolds, focusing on Dolbeault, Bott-Chern, and Aeppli cohomologies. We establish topological and cohomological obstructions to their existence and we provide a detailed analysis for compact complex surfaces, complex parallelisable solvmanifolds, and Calabi-Eckmann manifolds. We prove that the standard blow-up metric on any blow-up of a Kähler manifold is not geometrically formal, and that Kähler metrics with nonnegative curvature operator are necessarily geometrically formal.