On locally and globally conformal Kähler manifolds

Vaisman, Izu

doi:10.1090/S0002-9947-1980-0586733-7

On locally and globally conformal Kähler manifolds
HTML articles powered by AMS MathViewer

by Izu Vaisman PDF

Trans. Amer. Math. Soc. 262 (1980), 533-542 Request permission

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.

References

Toyoko Kashiwada, Some properties of locally conformal Kähler manifolds, Hokkaido Math. J. 8 (1979), no. 2, 191–198. MR 551550, DOI 10.14492/hokmj/1381758270
Toyoko Kashiwada and Shizuko Sato, On harmonic forms in compact locally conformal Kähler manifolds with the parallel Lee form, Ann. Fac. Sci. Univ. Nat. Zaï re (Kinshasa) Sect. Math.-Phys. 6 (1980), no. 1-2, 17–29. MR 678639
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751–798. MR 187255, DOI 10.2307/2373157
Yozo Matsushima, Holomorphic vector fields on compact Kähler manifolds, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 7, American Mathematical Society, Providence, R.I., 1971. MR 0412485, DOI 10.1090/cbms/007
Bruce L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119–132. MR 107279, DOI 10.2307/1970097
Shigeo Sasaki, On almost contact manifolds, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. 128–136. MR 0215232
J. A. Schouten, Der Ricci-Kalkül, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 10, Springer-Verlag, Berlin-New York, 1978 (German). Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie; Reprint of the 1924 original. MR 516659
Izu Vaisman, On the analytic distributions and foliations of a Kähler manifold, Proc. Amer. Math. Soc. 58 (1976), 221–228. MR 428233, DOI 10.1090/S0002-9939-1976-0428233-5

On locally conformal almost Kähler manifolds

24

Izu Vaisman, A theorem on compact locally conformal Kähler manifolds, Proc. Amer. Math. Soc. 75 (1979), no. 2, 279–283. MR 532151, DOI 10.1090/S0002-9939-1979-0532151-4
Izu Vaisman, Some curvature properties of locally conformal Kähler manifolds, Trans. Amer. Math. Soc. 259 (1980), no. 2, 439–447. MR 567089, DOI 10.1090/S0002-9947-1980-0567089-2
Izu Vaisman, A geometric condition for an l.c.K. manifold to be Kähler, Geom. Dedicata 10 (1981), no. 1-4, 129–134. MR 608134, DOI 10.1007/BF01447416

Kähler manifolds with parallel Lee form

12