1. Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds
- Author
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Ornea, Liviu and Verbitsky, Misha
- Subjects
Mathematics - Differential Geometry ,53C55, 32H04 - 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., Comment: 17 pages, version 2.0, added reference Alexandrov-Ivanov, to Dinew-Popovici and Arroyo-Nicolini
- Published
- 2024