A theorem on compact locally conformal Kähler manifolds

Vaisman, Izu

doi:10.1090/S0002-9939-1979-0532151-4

A theorem on compact locally conformal Kähler manifolds
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by Izu Vaisman

Proc. Amer. Math. Soc. 75 (1979), 279-283

DOI: https://doi.org/10.1090/S0002-9939-1979-0532151-4

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Abstract:

We prove that a compact locally conformai Kähler manifold which satisfies either: (1) it has nonpositive conformal invariant $\mu$ [2] and its local conformal Kähler metrics have nonnegative scalar curvature or (2) its local conformal Kähler (l.c.K.) metrics have a positive or negative definite Ricci form is a Kahler manifold. We conjecture that every compact l.c.K. manifold which satisfies all the topological restrictions of a Kähler manifold admits some Kähler metric.

References

Thierry Aubin, Variétés hermitiennes compactes localement conformément kählériennes, C. R. Acad. Sci. Paris 261 (1965), 2427–2430 (French). MR 185555
T. Aubin, The scalar curvature, Differential geometry and relativity, Mathematical Phys. and Appl. Math., Vol. 3, Reidel, Dordrecht, 1976, pp. 5–18. MR 0433500
Samuel I. Goldberg, Curvature and homology, Pure and Applied Mathematics, Vol. XI, Academic Press, New York-London, 1962. MR 0139098
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
Izu Vaisman, On locally conformal almost Kähler manifolds, Israel J. Math. 24 (1976), no. 3-4, 338–351. MR 418003, DOI 10.1007/BF02834764