Ornea, Liviu; Verbitsky, Misha LCK rank of locally conformally Kähler manifolds with potential. (English) Zbl 1347.53058 J. Geom. Phys. 107, 92-98 (2016). 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. Reviewer: Nicolai K. Smolentsev (Kemerovo) Cited in 5 ReviewsCited in 21 Documents 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 Keywords:locally conformally Kähler; pluricanonical; potential; Vaisman manifold; LCK rank Citations:Zbl 1188.53080; Zbl 1052.53051; Zbl 1238.53054; Zbl 1161.57015 PDFBibTeX XMLCite \textit{L. Ornea} and \textit{M. Verbitsky}, J. Geom. Phys. 107, 92--98 (2016; Zbl 1347.53058) Full Text: DOI arXiv References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.