Compatibility between non-K\"ahler structures on complex (nil)manifolds

Liviu Ornea; Alexandra Otiman; Miron Stanciu

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arXiv · 2003.10708

Compatibility between non-K\"ahler structures on complex (nil)manifolds

Liviu Ornea·Alexandra Otiman·Miron Stanciu

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Abstract

We study the interplay between the following types of special non-K\"ahler Hermitian metrics on compact complex manifolds: it locally conformally K\"ahler, $k$-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally K\"ahler compact nilmanifold carrying a balanced or a $k$-Gauduchon metric is necessarily a torus. Combined with a result of Fino and Vezzoni from 2016, this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally K\"ahler is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyperK\"ahler structure.

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BibTeX RIS

Liviu Ornea, Alexandra Otiman, Miron Stanciu. 2020-03-24. Compatibility between non-K\"ahler structures on complex (nil)manifolds. https://arxiv.org/abs/2003.10708

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