Structure theorem for compact Vaisman manifolds

A locally conformally Kähler (l.c.K.) manifold is a complex manifold admitting a Kähler covering $\tilde M$, with each deck transformation acting by Kähler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on $\tilde M$. We prove a structure theorem for compact Vaisman manifolds. Every compact Vaisman manifold $M$ is fibered over a circle, the fibers are Sasakian, the fibration is locally trivial, and $M$ is reconstructed from the Sasakian structure on the fibers and the monodromy automorphism induced by this fibration. This construction is canonical and functorial in both directions.

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Published 1 January 2003