On locally and globally conformal Kähler manifolds

Author:
Izu Vaisman

Journal:
Trans. Amer. Math. Soc. **262** (1980), 533-542

MSC:
Primary 53C55; Secondary 53B35

DOI:
https://doi.org/10.1090/S0002-9947-1980-0586733-7

MathSciNet review:
586733

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Abstract: Some relations between the locally conformal Kähler (l.c.K.) and the globally conformal Kähler (g.c.K.) properties are established. Compact l.c.K. manifolds which are not g.c.K. do not have Kähler metrics. l.c.K. manifolds which are not g.c.K. are analytically irreducible. Various curvature restrictions on l.c.K. manifolds imply the g.c.K. property. Total spaces of induced Hopf fibrations are l.c.K. and not g.c.K. manifolds.

Conjecture. A compact l.c.K. manifold which is not g.c.K. has at least one odd odd-dimensional Betti number.

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0586733-7

Article copyright:
© Copyright 1980
American Mathematical Society