Holomorphic Tensors on Vaisman Manifolds

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.