Structure theorem for compact Vaisman manifolds

Details Structure theorem for compact Vaisman manifolds Structure theorem for compact Vaisman manifolds

2003-05-18

arxiv.org/abs/math/0305259v1

Math. Res. Lett, 10(2003), no. 5-6, 799-805

Mathematics

Differential Geometry

8 pages

Scientific paper

A locally conformally Kaehler (l.c.K.) manifold is a complex manifold admitting a Kaehler covering $\tilde M$, with each deck transformation acting by Kaehler 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 as a Riemannian suspension 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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