Special Hermitian structures on suspensions

TL;DR

This paper investigates the geometric structures of suspensions of complex manifolds, demonstrating conditions under which certain Hermitian metrics exist or do not exist, and providing explicit examples of non-Kähler manifolds with special structures.

Contribution

It introduces new results on the existence of balanced and pluriclosed metrics on suspensions of Calabi-Yau and hyperk"ahler manifolds, expanding the understanding of complex non-K"ahler geometry.

Findings

Toric suspensions of Calabi-Yau manifolds are balanced.

Suspensions with hyperbolic automorphisms of hyperk"ahler manifolds lack pluriclosed, astheno-K"ahler, or p-pluriclosed metrics.

Modified suspension constructions can produce examples with pluriclosed metrics.

Abstract

Motivated by the construction based on topological suspension of a family of compact non-K\"ahler complex manifolds with trivial canonical bundle given by L. Qin and B. Wang in [QW], 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 hyperk\"ahler manifolds do not admit any pluriclosed, astheno-K\"ahler or p-pluriclosed Hermitian metric. Moreover, we consider natural extensions for hypercomplex manifolds, providing some explicit examples of compact holomorphic symplectic and hypercomplex non-K\"ahler manifolds. We also show that a modified suspension construction provides examples with pluriclosed metrics.