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LCK metrics on complex spaces with quotient singularities

  • George-Ionuţ Ioniţă
  • Ovidiu PredaEmail author
Article
  • 6 Downloads

Abstract

In this article we introduce a generalization of locally conformally Kähler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kähler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singularities, then it admits a locally conformally Kähler metric if and only if its universal cover admits a Kähler metric such that the deck automorphisms act by homotheties of the Kähler metric. We also prove that the blow-up at a point of an LCK complex space is also LCK.

Mathematics Subject Classification

32C15 53C55 

Notes

Acknowledgements

Ovidiu Preda is grateful to Professor Liviu Ornea for guiding his learning of geometry which led to the problem studied in this article. Both authors are thankful to Alexandra Otiman for repeatedly taking the time to answer their questions.

References

  1. 1.
    Bierstone, E., Milman, P.D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cartan H.: Quotient d’une variété analytique par un groupe discret d’automor-phismes, Séminaire Henri Cartan, tome 6 (1953–1954), pp. 1–13Google Scholar
  3. 3.
    Grauert, H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331–368 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kollár, J.: Shafarevich maps and plurigenera of algebraic varieties. Invent. Math. 113, 177–215 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Moishezon B.G.: Singular Kählerian spaces. In: Proceedings of the International Conference on Manifolds and Related Topics in Topology (Tokyo 1973), Tokyo, pp. 343–351 (1975)Google Scholar
  6. 6.
    Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publications mathématiques de l’I.H.É.S. tome 9, pp. 5–22 (1961)Google Scholar
  7. 7.
    Ornea L.; Verbitsky M.: A report on locally conformally Kähler manifolds. In: Harmonic Maps and Differential Geometry, pp. 135–149, Contemporary Mathematics, vol. 542. American Mathematical Society, Providence (2011)Google Scholar
  8. 8.
    Tricerri, F.: Some examples of locally conformal Kähler manifolds. Rend. Sem. Mat. Torino 40(1), 81–92 (1982)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Verbitsky, M.: Holomorphic symplectic geometry and orbifold singularities. Asian J. Math. 4(3), 553–564 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Vuletescu, V.: Blowing up points on l.c.K. manifolds. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100)(3), 387–390 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science“Politehnica” University BucharestBucharestRomania
  2. 2.Institute of Mathematics of the Romanian AcademyBucharestRomania

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