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Proceedings of the American Mathematical Society

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A theorem on compact locally conformal Kähler manifolds


Author: Izu Vaisman
Journal: Proc. Amer. Math. Soc. 75 (1979), 279-283
MSC: Primary 53C55
MathSciNet review: 532151
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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.


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  • [1] Thierry Aubin, Variétés hermitiennes compactes localement conformément kählériennes, C. R. Acad. Sci. Paris 261 (1965), 2427–2430 (French). MR 0185555
  • [2] T. Aubin, The scalar curvature, Differential geometry and relativity, Reidel, Dordrecht, 1976, pp. 5–18. Mathematical Phys. and Appl. Math., Vol. 3. MR 0433500
  • [3] Samuel I. Goldberg, Curvature and homology, Pure and Applied Mathematics, Vol. XI, Academic Press, New York-London, 1962. MR 0139098
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  • [5] Izu Vaisman, On locally conformal almost Kähler manifolds, Israel J. Math. 24 (1976), no. 3-4, 338–351. MR 0418003

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DOI: https://doi.org/10.1090/S0002-9939-1979-0532151-4
Article copyright: © Copyright 1979 American Mathematical Society