Holomorphic submersions of locally conformally Kähler manifolds
- 209 Downloads
A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold \(X\) admits a holomorphic submersion with positive-dimensional fibers at least one of which is of Kähler type, then \(X\) is globally conformally Kähler or biholomorphic, up to finite covers, to a small deformation of a Vaisman manifold (i.e., a mapping torus over a circle, with Sasakian fiber). As a consequence, we show that the product of a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.
KeywordsLocally conformally Kähler manifold Holomorphic submersion Vaisman manifold
Mathematics Subject Classification (2010)53C55
Liviu Ornea and Victor Vuletescu are partially supported by CNCS UEFISCDI, Project Number PN-II-ID-PCE-2011-3-0118. All authors thank the anonymous referee for carefully reading their paper and for his very useful comments.
- 1.Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces. 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin (2004)Google Scholar
- 4.Dragomir, S., Ornea, L.: Locally conformal Kähler geometry, Progress in Mathematics 155. Birkhäuser, Boston, Basel (1998)Google Scholar
- 14.Ornea, L., Verbitsky, M.: Topology of locally conformally Kähler manifolds with potential. Int. Math. Res. Not. 4, 717–726 (2010)Google Scholar
- 16.Ornea, L., Verbitsky, M., Vuletescu, V.: Blow-ups of locally conformally Kähler manifolds. Int. Math. Res. Not. (to appear) arxiv:1108.4885. doi: 10.1093/imrn/rns128