Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916952432181248 |
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| author | Ornea, Liviu Verbitsky, Misha |
| author_facet | Ornea, Liviu Verbitsky, Misha |
| contents | A complex Hermitian $n$-manifold $(M,I, ω)$ is called locally conformally Kahler (LCK) if $dω=θ\wedgeω$, where $θ$ is a closed 1-form, balanced if $ω^{n-1}$ is closed, and SKT if $dIdω=0$. 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 $-d(Iθ)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds Ornea, Liviu Verbitsky, Misha Differential Geometry 53C55, 32H04 A complex Hermitian $n$-manifold $(M,I, ω)$ is called locally conformally Kahler (LCK) if $dω=θ\wedgeω$, where $θ$ is a closed 1-form, balanced if $ω^{n-1}$ is closed, and SKT if $dIdω=0$. 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 $-d(Iθ)$ 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. |
| title | Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds |
| topic | Differential Geometry 53C55, 32H04 |
| url | https://arxiv.org/abs/2407.04623 |