Manuscripta Mathematica

, Volume 140, Issue 1–2, pp 1–12 | Cite as

An integral invariant from the view point of locally conformally Kähler geometry

  • Akito FutakiEmail author
  • Kota Hattori
  • Liviu Ornea


In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.

Mathematics Subject Classification (2000)

53C55 53C25 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Institute of TechnologyMeguro, TokyoJapan
  2. 2.Faculty of MathematicsUniversity of BucharestBucharestRomania
  3. 3.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania

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