Mathematische Annalen

, Volume 348, Issue 1, pp 25–33

Locally conformal Kähler manifolds with potential


DOI: 10.1007/s00208-009-0463-0

Cite this article as:
Ornea, L. & Verbitsky, M. Math. Ann. (2010) 348: 25. doi:10.1007/s00208-009-0463-0


A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold \({\widetilde M}\) with the deck transformation group acting conformally on \({\widetilde M}\). If M admits a holomorphic flow, acting on \({\widetilde M}\) conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, dim M ≥ 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky (Math Ann 332:121–143, 2005).

Mathematics Subject Classification (2000)

53C55 32G05 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BucharestBucharestRomania
  2. 2.Department of MathematicsUniversity of GlasgowGlasgowScotland
  3. 3.Institute of Theoretical and Experimental PhysicsMoscowRussia