TY - JOUR

T1 - An integral invariant from the view point of locally conformally Kähler geometry

AU - Futaki, Akito

AU - Hattori, Kota

AU - Ornea, Liviu

PY - 2013/1

Y1 - 2013/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00229-011-0527-9

DO - 10.1007/s00229-011-0527-9

M3 - Article

AN - SCOPUS:84871956706

VL - 140

SP - 1

EP - 12

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 1-2

ER -