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Special Kähler-Ricci potentials on compact Kähler manifolds

Identifieur interne : 005386 ( Main/Exploration ); précédent : 005385; suivant : 005387

Special Kähler-Ricci potentials on compact Kähler manifolds

Auteurs : A. Derdzinski [États-Unis] ; G. Maschler [États-Unis]

Source :

RBID : ISTEX:17F69D6448F6D697D548C1A3A86D8075C9771D5B

English descriptors

Abstract

By a special Kähler-Ricci potential on a Kähler manifold we mean a nonconstant real-valued C ∞ function τ such that J(∇τ) is a Killing vector field and, at every point with dτ ≠ 0, all nonzero tangent vectors orthogonal to ∇τ and J(∇τ) are eigenvectors of both ∇ dτ and the Ricci tensor. For instance, this is always the case if τ is a nonconstant C∞ function on a Kähler manifold (M, g) of complex dimension m > 2 and the metric g˜ = g/τ2, defined wherever τ ≠ 0, is Einstein. (When such τ exists, (M, g) may be called almost-everywhere conformally Einstein.) We provide a complete classification of compact Kähler manifolds (M, g) with special Kähler-Ricci potentials, showing, in particular, that in any complex dimension m ≧ 2 they form two separate classes: in one, M is the total space of a holomorphic ℂP1 bundle; in the other, M is biholomorphic to ℂP m . We then use this classification to prove a structure theorem for compact Kähler manifolds of any complex dimension m > 2 which are almost-everywhere conformally Einstein.

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DOI: 10.1515/CRELLE.2006.030


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