Holomorphic Tensors on Vaisman Manifolds

Liviu Ornea () and Misha Verbitsky ()
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Liviu Ornea: University of Bucharest, Faculty of Mathematics and Informatics
Misha Verbitsky: Instituto Nacional de Matemática Pura e Aplicada (IMPA)

Chapter Chapter 10 in Essays on Topology, 2025, pp 161-173 from Springer

Abstract: Abstract An LCK (locally conformally Kähler)ManifoldLCK manifold is a complex manifold admitting a Hermitian form ω $$\omega $$ which satisfies dω = ω ∧ θ $$d\omega =\omega \wedge \theta $$ , where θ $$\theta $$ is a closed 1-form, called the Lee formLee 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,Lee field is holomorphic and Killing. We prove that any holomorphic tensorTensorholomorphic 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 $${\mathbb Z}$$ -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 DimensionKodairaof Vaisman manifoldsManifoldVaisman.

Keywords: Locally conformally Kähler; LCK potential; Algebraic group; Zariski closure; 53C55; 32G05 (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-81414-3_10

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DOI: 10.1007/978-3-031-81414-3_10

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