On the existence of balanced metrics of Hodge-Riemann type

TL;DR

This paper investigates the existence of Hodge-Riemann balanced metrics on non-Kähler complex manifolds, establishing obstructions, non-existence results for certain classes, and providing the first example on a non-compact manifold.

Contribution

It identifies key obstructions and non-existence results for Hodge-Riemann balanced metrics, and constructs the first non-Kähler example on a non-compact manifold.

Findings

Hodge-Riemann balanced manifolds must be (n-2)-Kähler

Non-existence of such structures on certain solvmanifolds and nilmanifolds

First non-Kähler example on a non-compact manifold

Abstract

In the paper we study the existence of balanced metrics of Hodge-Riemann type on non-K\"ahler complex manifolds. We first find some general obstructions, for instance that a Hodge-Riemann balanced manifold of complex dimension $n$ has to be $(n - 2)$-K\"ahler. Then, we focus on the case of compact quotients of Lie groups by lattices, endowed with an invariant complex structure. In particular, we prove non existence results on non-K\"ahler complex parallelizable manifolds and some classes of solvmanifolds, and we show that the only nilmanifolds admitting invariant structures of this type are tori. Finally, we construct the first non-K\"ahler example of a Hodge-Riemann balanced structure, on a non-compact complex manifold obtained as the product of the Iwasawa manifold by $\mathbb C$.