Abstract
By the work of Li, a compact co-Kähler manifold \(M\) is a mapping torus \(K_\varphi \), where \(K\) is a Kähler manifold and \(\varphi \) is a Hermitian isometry. We show here that there is always a finite cyclic cover \(\overline{M}\) of the form \(\overline{M} \cong K \times S^1\), where \(\cong \) is equivariant diffeomorphism with respect to an action of \(S^1\) on \(M\) and the action of \(S^1\) on \(K \times S^1\) by translation on the second factor. Furthermore, the covering transformations act diagonally on \(S^1, K\) and are translations on the \(S^1\) factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle.
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Notes
The authors of [4] use the term cosymplectic for Li’s co-Kähler because they view these manifolds as odd-dimensional versions of symplectic manifolds—even as far as being a convenient setting for time-dependent mechanics [7]. Li’s characterization, however, makes clear the true underlying Kähler structure, so we have chosen to follow his terminology.
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Acknowledgments
We thank Greg Lupton for useful conversations and the referee for several helpful suggestions. The first author thanks the Department of Mathematics at Cleveland State University for its hospitality during his extended visit (funded by CSIC and ICMAT) to Cleveland. In addition, the first author was partially supported by Project MICINN (Spain) MTM2010-17389. The second author was partially supported by a grant from the Simons Foundation: (#244393 to John Oprea).
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Bazzoni, G., Oprea, J. On the structure of co-Kähler manifolds. Geom Dedicata 170, 71–85 (2014). https://doi.org/10.1007/s10711-013-9869-7
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DOI: https://doi.org/10.1007/s10711-013-9869-7