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arXiv · 2301.01077

Holomorphic tensors on Vaisman manifolds

Liviu Ornea·Misha Verbitsky
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Abstract

An LCK (locally conformally Kahler) manifold is a complex manifold admitting a Hermitian form $\omega$ which satisfies $d\omega =\omega\wedge \theta$, where $\theta$ 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.

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Liviu Ornea, Misha Verbitsky. 2023-01-03. Holomorphic tensors on Vaisman manifolds. https://doi.org/10.1007/978-3-031-81414-3_10

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