On the existence of balanced metrics of Hodge-Riemann type

2024/08/26 by Fino, Anna, Mainenti, Asia · 1 citation
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.14377

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 \C.

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• p-Kähler structures on nilmanifolds and holomorphically parallelizable manifolds

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