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Locally conformally Kähler manifolds with holomorphic Lee field Moroianu, Andrei


A locally conformally Kähler (LCK) manifold is a compact Hermitian manifold $(M,g,J)$ whose fundamental 2-form $\omega:=g(J\cdot,\cdot)$ verifies $d\omega=\theta\wedge\omega$ for a certain closed 1-form $\theta$ called the Lee form. We study here LCK manifolds whose Lee vector field (the metric dual of $\theta$) is holomorphic. We will show that if its norm is constant or if its divergence vanishes, then the metric is Vaisman, i.e. the Lee form is parallel with respect to the Levi-Civita connection of $g$. We will then give examples of non-Vaisman LCK manifolds with holomorphic Lee field, and we classify all such structures on manifolds of Vaisman type. These results have been obtained in collaboration with F. Madani, S. Moroianu, L. Ornea and M. Pilca.

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