A Derdzinski, G Maschler
Special Kähler-Ricci potentials on compact Kähler manifolds
By a special Kähler-Ricci potential on a Kähler manifold we mean a nonconstant real-valued C∞ function &tgr; such that J(∇&tgr;) is a Killing vector field and, at every point with d&tgr; ≠ 0, all nonzero tangent vectors orthogonal to ∇&tgr; and J(∇&tgr;) are eigenvectors of both ∇ d&tgr; and the Ricci tensor. For instance, this is always the case if &tgr; is a nonconstant C∞ function on a Kähler manifold (M, g) of complex dimension m > 2 and the metric g˜ = g/&tgr;2, defined wherever &tgr; ≠ 0, is Einstein. (When such &tgr; 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 ℂPm. 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.
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2006, 04/2006
Pages: 73 - 116
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