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LCK rank of locally conformally Kähler manifolds with potential. (English) Zbl 1347.53058

Authors’ abstract: An LCK manifold with potential is a quotient of a Kähler manifold \(X\) equipped with a positive Kähler potential \(f\), such that the monodromy group acts on \(X\) by holomorphic homotheties and multiplies \(f\) by a character. The LCK rank is the rank of the image of this character, considered as a function from the monodromy group to real numbers. We prove that an LCK manifold with potential can have any rank between 1 and \(b_1(M)\). Moreover, LCK manifolds with proper potential (ones with rank 1) are dense. Two errata to our previous work [Zbl 1188.53080; Zbl 1052.53051; Zbl 1238.53054; Zbl 1161.57015] are given in the last section.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D15 Almost contact and almost symplectic manifolds
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[2] Ornea, L.; Verbitsky, M., Locally conformally Kahler metrics obtained from pseudoconvex shells, Proc. Amer. Math. Soc., 144, 325-335 (2016) · Zbl 1327.53098
[3] Verbitsky, M., Theorems on the vanishing of cohomology for locally conformally hyper-Kähler manifolds, Proc. Steklov Inst. Math., 246, 3, 54-78 (2004) · Zbl 1101.53027
[4] Belgun, F. A., On the metric structure of non-Kähler complex surfaces, Math. Ann., 317, 1, 1-40 (2000) · Zbl 0988.32017
[5] Ornea, L.; Verbitsky, M., Locally conformal Kähler manifolds with potential, Math. Ann., 348, 25-33 (2010) · Zbl 1213.53090
[6] Ornea, L.; Verbitsky, M., Topology of locally conformally Kähler manifolds with potential, Int. Math. Res. Not. IMRN, 717-726 (2010) · Zbl 1188.53080
[7] Kokarev, G., On pseudo-harmonic maps in conformal geometry, Proc. Lond. Math. Soc., 99, 168-194 (2009) · Zbl 1175.53073
[9] Ornea, L.; Verbitsky, M., Structure theorem for compact Vaisman manifolds, Math. Res. Lett., 10, 799-805 (2003) · Zbl 1052.53051
[10] Gini, R.; Ornea, L.; Parton, M.; Piccinni, P., Reduction of Vaisman structures in complex and quaternionic geometry, J. Geom. Phys., 56, 2501-2522 (2006) · Zbl 1109.53070
[11] Parton, M.; Vuletescu, V., Examples of non-trivial rank in locally conformal Kähler geometry, Math. Z., 270, 1-2, 179-187 (2012) · Zbl 1242.32011
[12] Ornea, L.; Verbitsky, M., Automorphisms of locally conformally Kähler manifolds with potential, Int. Math. Res. Not. IMRN, 4, 894-903 (2012) · Zbl 1238.53054
[13] Ornea, L.; Verbitsky, M., Morse-Novikov cohomology of locally conformally Kähler manifolds, J. Geom. Phys., 59, 3, 295-305 (2009) · Zbl 1161.57015
[14] Boyer, C.; Galicki, K., Sasakian Geometry (2008), Oxford Univ. Press · Zbl 1155.53002
[15] Vaisman, I., Generalized Hopf manifolds, Geom. Dedicata, 13, 3, 231-255 (1982) · Zbl 0506.53032
[16] Tsukada, K., The canonical foliation of a compact generalized Hopf manifold, Differential Geom. Appl., 11, 1, 13-28 (1999) · Zbl 0941.53043
[17] Tsukada, K., Holomorphic forms and holomorphic vector fields on compact generalized Hopf manifolds, Compos. Math., 93, 1, 1-22 (1994) · Zbl 0811.53032
[18] El Kacimi Alaoui, A.; Gmira, B., Stabilité du caractère kählérien transverse, Israel J. Math., 101, 323-347 (1997) · Zbl 0948.53041
[19] Kamishima, Y.; Ornea, L., Geometric flow on compact locally conformally Kähler manifolds, Tohoku Math. J., 57, 2, 201-221 (2005) · Zbl 1083.53068
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