Blow-ups of locally conformally Kahler manifolds

TL;DR

This paper investigates the conditions under which blow-ups of locally conformally Kahler (LCK) manifolds retain LCK structures, revealing that only blow-ups along globally conformally Kahler submanifolds do so, and shows twistor spaces generally cannot admit LCK metrics.

Contribution

It establishes a precise criterion for when blow-ups of LCK manifolds preserve the LCK structure and rules out LCK metrics on most twistor spaces.

Findings

Blow-up of a compact LCK manifold along a submanifold admits LCK structure iff the submanifold is globally conformally Kahler.

Twistor spaces of certain manifolds cannot admit LCK metrics unless they are Kahler.

Provides a characterization of LCK structures under complex geometric transformations.

Abstract

A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We also prove that a twistor space (of a compact 4-manifold, a quaternion-Kahler manifold or a Riemannian m anifold) cannot admit an LCK metric, unless it is Kahler.