On non-K\"ahler compact complex manifolds with balanced and astheno-K\"ahler metrics

TL;DR

This paper constructs examples of non-Kähler compact complex manifolds of any dimension at least 4 that admit both balanced and astheno-K"ahler metrics, with additional Gauduchon properties, expanding understanding of metric coexistence.

Contribution

It provides explicit examples of non-K"ahler manifolds with multiple compatible special metrics, including balanced and astheno-K"ahler, for all dimensions n ≥ 4.

Findings

Existence of non-Kähler manifolds with both balanced and astheno-K"ahler metrics.

Construction of manifolds with k-th Gauduchon metrics for all 1 ≤ k ≤ n-1.

Examples valid for all dimensions n ≥ 4.

Abstract

In this note we present, for every $n \geq 4$, a non-K\"ahler compact complex manifold $X$ of complex dimension $n$ admitting a balanced metric and an astheno-K\"ahler metric which is in addition $k$-th Gauduchon for any $1\leq k\leq n-1$.