Special Hermitian structures on suspensions

Motivated by the construction based on topological suspension of a family of compact non-Kähler complex manifolds with trivial canonical bundle given by Qin and Wang (Geom Topol 22:2115–2144, 2018), we study toric suspensions of balanced manifolds by holomorphic automorphisms. In particular, we show that toric suspensions of Calabi–Yau manifolds are balanced. We also prove that suspensions associated with hyperbolic automorphisms of hyperkähler manifolds do not admit any pluriclosed, astheno-Kähler or p-pluriclosed Hermitian metric. Moreover, we consider natural extensions for hypercomplex manifolds, providing some explicit examples of compact holomorphic symplectic and hypercomplex non-Kähler manifolds. We also show that a modified suspension construction provides examples with pluriclosed metrics.