Holomorphic tensors on Vaisman manifolds

TL;DR

This paper studies the properties of holomorphic tensors on Vaisman manifolds, showing their invariance under the Lee field and relating their Kodaira dimension to that of associated algebraic cones, with implications for stability under deformation.

Contribution

It proves the invariance of holomorphic tensors under the Lee field on Vaisman manifolds and relates their Kodaira dimension to that of algebraic cones over projective manifolds.

Findings

Holomorphic tensors on Vaisman manifolds are Lee field invariant.

Kodaira dimension of Vaisman manifolds matches that of the base projective manifold.

Deformational stability of the Kodaira dimension is established.

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.