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Local Classification of Conformally-Einstein Kähler Metrics in Higher Dimensions

Published online by Cambridge University Press:  05 November 2003

A. Derdzinski  and
G. Maschler
A. Derdzinski
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA. E-mail: andrzej@math.ohio-state.edu
G. Maschler
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3. E-mail: maschler@math.toronto.edu
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Abstract

The requirement that a (non-Einstein) Kähler metric in any given complex dimension $m > 2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m > 2$, on three real parameters along with an arbitrary Kähler–Einstein metric $h$ in complex dimension $m - 1$. We provide an explicit description of all these local-isometry types, for any given $h$. This result is derived from a more general local classification theorem for metrics admitting functions that we call special Kähler–Ricci potentials.

Keywords

conformally Einstein metricKähler metricKähler–Ricci potentialhorizontally homothetic submersion53B3553B2053C2553C55
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 87 , Issue 3 , November 2003 , pp. 779 - 819
Copyright
2003 London Mathematical Society

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Footnotes

The work of the second author was carried out in part at the Max Planck Institute in Bonn, and he is also partially supported by an NSERC Canada individual research grant.