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arXiv · 2408.14377

On the existence of balanced metrics of Hodge-Riemann type

Anna Fino·Asia Mainenti
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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 $\mathbb C$.

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Anna Fino, Asia Mainenti. 2024-08-26. On the existence of balanced metrics of Hodge-Riemann type. https://doi.org/10.1007/s13398-026-01833-5

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