A Calabi-Yau theorem for Vaisman manifolds
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866910767301787648 |
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| author | Ornea, Liviu Verbitsky, Misha |
| author_facet | Ornea, Liviu Verbitsky, Misha |
| contents | A compact complex Hermitian manifold $(M, I, w)$ is called Vaisman if $dw=w\wedge θ$ and the 1-form $θ$, called the Lee form, is parallel with respect to the Levi-Civita connection. The volume form of $M$ is invariant with respect to the action of the vector field $X$ dual to $θ$ (called the Lee field) and the vector field $I(X)$, called { the anti-Lee field}. The cohomology class of $θ$, called the Lee class, plays the same role as the Kahler class in Kahler geometry. We prove that a Vaisman metric is uniquely determined by its volume form and the Lee class, and, conversely, for each Lee class $[θ]$ and each Lee- and anti-Lee-invariant volume form $V$, there exists a Vaisman structure with the volume form $V$ and the Lee class $c[θ]$. This is an analogue of the Calabi-Yau theorem claiming that the Kahler form is uniquely determined by its volume and the cohomology class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_08808 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Calabi-Yau theorem for Vaisman manifolds Ornea, Liviu Verbitsky, Misha Differential Geometry Algebraic Geometry Complex Variables 53C55, 14J32, 32Q25 A compact complex Hermitian manifold $(M, I, w)$ is called Vaisman if $dw=w\wedge θ$ and the 1-form $θ$, called the Lee form, is parallel with respect to the Levi-Civita connection. The volume form of $M$ is invariant with respect to the action of the vector field $X$ dual to $θ$ (called the Lee field) and the vector field $I(X)$, called { the anti-Lee field}. The cohomology class of $θ$, called the Lee class, plays the same role as the Kahler class in Kahler geometry. We prove that a Vaisman metric is uniquely determined by its volume form and the Lee class, and, conversely, for each Lee class $[θ]$ and each Lee- and anti-Lee-invariant volume form $V$, there exists a Vaisman structure with the volume form $V$ and the Lee class $c[θ]$. This is an analogue of the Calabi-Yau theorem claiming that the Kahler form is uniquely determined by its volume and the cohomology class. |
| title | A Calabi-Yau theorem for Vaisman manifolds |
| topic | Differential Geometry Algebraic Geometry Complex Variables 53C55, 14J32, 32Q25 |
| url | https://arxiv.org/abs/2206.08808 |