Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds

Abstract

A complex Hermitian $n$-manifold $(M,I,\omega )$ is called locally conformally Kahler (LCK) if $d\omega =\theta \wedge \omega$, where $\theta$ is a closed 1-form, balanced if ${\omega }^{n-1}$ is closed, and SKT if $dId\omega =0$. We conjecture that any compact complex manifold admitting two of these three types of Hermitian forms (balanced, SKT, LCK) also admits a Kahler metric, and prove partial results towards this conjecture. We conjecture that the (1,1)-form $-d(I\theta )$ is Bott--Chern homologous to a positive (1,1)-current. This conjecture implies that $(M,I)$ does not admit a balanced Hermitian metric. We verify this conjecture for all known classes of LCK manifolds.

17 pages, version 2.0, added reference Alexandrov-Ivanov, to Dinew-Popovici and Arroyo-Nicolini

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• Blowing-up points on l.c. K. manifolds
• Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds
• Non-linear Hopf manifolds are locally conformally Kahler
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Cited by in corpus (1)

• The Lee--Gauduchon cone on complex manifolds