Hermitian geometrically formal manifolds

TL;DR

This paper investigates the existence and properties of Hermitian geometrically formal metrics on compact complex manifolds, identifying obstructions and analyzing specific classes like surfaces and solvmanifolds.

Contribution

It provides new topological and cohomological obstructions to geometric formality and characterizes when K"ahler metrics are geometrically formal.

Findings

Blow-up metrics on K"ahler manifolds are not geometrically formal

K"ahler metrics with nonnegative curvature operator are geometrically formal

Obstructions are established for the existence of Hermitian geometrically formal metrics

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 K\"ahler manifold is not geometrically formal, and that K\"ahler metrics with nonnegative curvature operator are necessarily geometrically formal.