On the existence of balanced metrics of Hodge-Riemann type
Anna Fino, Asia Mainenti
TL;DR
We study the existence of balanced metrics of Hodge-Riemann type on non-Kähler complex manifolds and show that such structures force a manifold of complex dimension $n$ to be $(n-2)$-Kähler, connecting HR-type metrics to $p$-Kähler obstructions. The paper then analyzes invariant structures on compact quotients of Lie groups, proving strong nonexistence results on nilmanifolds and several solvmanifold classes, and showing that complex tori are the only nilmanifolds with invariant Hodge-Riemann balanced metrics. In addition, it provides a non-Kähler obstruction framework for Lie algebras and demonstrates that under many invariant settings HR-balanced metrics imply Kählerity. Finally, the authors construct the first non-Kähler HR-balanced structure in the non-compact case on $M=\mathcal{I}\times \mathbb{C}$, highlighting that HR-balanced metrics can exist beyond the compact, invariant setting. These results narrow the arena where HR-balanced structures can appear and anchor their existence in concrete non-compact constructions.
Abstract
In the paper we study the existence of balanced metrics of Hodge-Riemann type on non-Kähler complex manifolds. We first find some general obstructions, for instance that a Hodge-Riemann balanced manifold of complex dimension $n$ has to be $(n - 2)$-Kähler. 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-Kähler 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-Kähler example of a Hodge-Riemann balanced structure, on a non-compact complex manifold obtained as the product of the Iwasawa manifold by $\mathbb C$.