Holomorphic tensors on Vaisman manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909791444533248 |
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| author | Ornea, Liviu Verbitsky, Misha |
| author_facet | Ornea, Liviu Verbitsky, Misha |
| contents | An LCK (locally conformally Kahler) manifold is a complex manifold admitting a Hermitian form $ω$ which satisfies $dω=ω\wedge θ$, where $θ$ is a closed 1-form, called the Lee form. An LCK manifold is called Vaisman if the Lee form is parallel with respect to the Levi-Civita connection. The dual vector field, called the Lee field, is holomorphic and Killing. We prove that any holomorphic tensor on a Vaisman manifold is invariant with respect to the Lee field. This is used to compute the Kodaira dimension of Vaisman manifolds. We prove that the Kodaira dimension of a Vaisman manifold obtained as a $Z$-quotient of an algebraic cone over a projective manifold $X$ is equal to the Kodaira dimension of $X$. This can be applied to prove the deformational stability of the Kodaira dimension of Vaisman manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01077 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Holomorphic tensors on Vaisman manifolds Ornea, Liviu Verbitsky, Misha Algebraic Geometry Complex Variables Differential Geometry 53C55, 32G05 An LCK (locally conformally Kahler) manifold is a complex manifold admitting a Hermitian form $ω$ which satisfies $dω=ω\wedge θ$, where $θ$ is a closed 1-form, called the Lee form. An LCK manifold is called Vaisman if the Lee form is parallel with respect to the Levi-Civita connection. The dual vector field, called the Lee field, is holomorphic and Killing. We prove that any holomorphic tensor on a Vaisman manifold is invariant with respect to the Lee field. This is used to compute the Kodaira dimension of Vaisman manifolds. We prove that the Kodaira dimension of a Vaisman manifold obtained as a $Z$-quotient of an algebraic cone over a projective manifold $X$ is equal to the Kodaira dimension of $X$. This can be applied to prove the deformational stability of the Kodaira dimension of Vaisman manifolds. |
| title | Holomorphic tensors on Vaisman manifolds |
| topic | Algebraic Geometry Complex Variables Differential Geometry 53C55, 32G05 |
| url | https://arxiv.org/abs/2301.01077 |