Do products of compact complex manifolds admit LCK metrics?
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929409607335936 |
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| author | Ornea, Liviu Verbitsky, Misha Vuletescu, Victor |
| author_facet | Ornea, Liviu Verbitsky, Misha Vuletescu, Victor |
| contents | An LCK (locally conformally Kahler) manifold is a Hermitian manifold which admits a Kahler cover with deck group acting by holomorphic homotheties with respect to the Kahler metric. The product of two LCK manifolds does not have a natural product LCK structure. It is conjectured that a product of two compact complex manifolds is never LCK. We classify all known examples of compact LCK manifolds onto three not exclusive classes: LCK with potential, a class of manifolds we call of Inoue type, and those containing a rational curve. In the present paper, we prove that a product of an LCK manifold and an LCK manifold belonging to one of these three classes does not admit an LCK structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_08111 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Do products of compact complex manifolds admit LCK metrics? Ornea, Liviu Verbitsky, Misha Vuletescu, Victor Differential Geometry Algebraic Geometry Complex Variables An LCK (locally conformally Kahler) manifold is a Hermitian manifold which admits a Kahler cover with deck group acting by holomorphic homotheties with respect to the Kahler metric. The product of two LCK manifolds does not have a natural product LCK structure. It is conjectured that a product of two compact complex manifolds is never LCK. We classify all known examples of compact LCK manifolds onto three not exclusive classes: LCK with potential, a class of manifolds we call of Inoue type, and those containing a rational curve. In the present paper, we prove that a product of an LCK manifold and an LCK manifold belonging to one of these three classes does not admit an LCK structure. |
| title | Do products of compact complex manifolds admit LCK metrics? |
| topic | Differential Geometry Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2211.08111 |