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ON THE EINSTEIN–KÄHLER METRIC AND THE HOLONOMY OF A LINE BUNDLE

Published online by Cambridge University Press:  05 February 2002

Kenji Tsuboi
Kenji Tsuboi
Affiliation:
Laboratory of Mathematics, Tokyo University of Fisheries, 4-5-7 Kohnan, Minato, Tokyo 108-8477, Japan
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Abstract

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In this paper we give a relation between the Futaki invariant for a compact complex manifold $M$ and the holonomy of a determinant line bundle over a loop in the base space of any principal $G$-bundle, where $G$ is the identity component of the maximal compact subgroup of the complex Lie group consisting of all biholomorphic automorphisms of $M$. Using the property of the Futaki invariant, we show that the holonomy is an obstruction to the existence of the Einstein–Kähler metrics on $M$. Our main result is Theorem 2.1.

AMS 2000 Mathematics subject classification: Primary 32Q20. Secondary 58J28; 58J52

Keywords

Einstein–Kähler metricholonomyFutaki invarianteta invariantindex theorem
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 45 , Issue 1 , February 2002 , pp. 83 - 90
Copyright
Copyright © Edinburgh Mathematical Society 2002