Integrable LCK manifolds (Q2116421)

A locally conformal Kähler (LCK) manifold is a Hermitian manifold \((M, J , g)\) whose fundamental 2-form \(\Omega\) and Lee 1-form \(\theta\) satisfy the following identities: \[d\Omega = \theta\wedge \Omega\quad\text{and}\quad d\theta = 0.\] In the present paper, the authors study specifically those LCK manifolds (called integrable) admitting a nowhere zero anti-Lee 1-form \(\eta:=-\theta\circ J\), such that \(\ker\eta\) is an integrable distribution. As main results, they provide a necessary and sufficient condition for a LCK manifold to be integrable. They study the possibilities that Lee or anti-Lee vector field (defined as the metric duals of \(\theta\) and \(\eta\) respectively) are Killing. Then, in the last sections, they investigate integrable LCK Lie algebras with a particular focus on the unimodular case. The problem of the classification of unimodular integrable LCK Lie algebras is reduced to the classification of pairs of even dimensional matrices satisfying suitable relations involving the complex structure. Examples of both unimodular and non-unimodular integrable LCK Lie algebras are given.