Local classification of conformally-Einstein Kähler metrics in higher dimensions

Details Local classification of conformally-Einstein Kähler metrics in higher dimensions Local classification of conformally-Einstein Kähler metrics in higher dimensions

2002-03-31

arxiv.org/abs/math/0204013v1

Proc. London Math. Soc. (3) 87 (2003), no. 3, 779-819

Mathematics

Differential Geometry

38 pages, AMSTeX, submitted to the Proceedings of the London Mathematical Society

Scientific paper

10.1112/S0024611503014175

The requirement that a (non-Einstein) K\"ahler 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 K\"ahler-Einstein metric $h$ in complex dimension $m-1$. We provide an explicit description of all these local-isometry types, for any given $h$. That result is derived from a more general local classification theorem for metrics admitting functions we call {\it special K\"ahler-Ricci potentials}.

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