Special KählerRicci potentials on compact Kähler manifolds
Abstract
A special KählerRicci potential on a Kähler manifold is any nonconstant $C^\infty$ function $\tau$ such that $J(\nabla\tau)$ is a Killing vector field and, at every point with $d\tau\ne 0$, all nonzero tangent vectors orthogonal to $\nabla\tau$ and $J(\nabla\tau)$ are eigenvectors of both $\nabla d\tau$ and the Ricci tensor. For instance, this is always the case if $\tau$ is a nonconstant $C^\infty$ function on a Kähler manifold $(M,g)$ of complex dimension $m>2$ and the metric $\tilde g=g/\tau^2$, defined wherever $\tau\ne 0$, is Einstein. (When such $\tau$ exists, $(M,g)$ may be called {\it almosteverywhere conformally Einstein}.) We provide a complete classification of compact Kähler manifolds with special KählerRicci potentials and use it to prove a structure theorem for compact Kähler manifolds of any complex dimension $m>2$ which are almosteverywhere conformally Einstein.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2002
 DOI:
 10.48550/arXiv.math/0204328
 arXiv:
 arXiv:math/0204328
 Bibcode:
 2002math......4328D
 Keywords:

 Mathematics  Differential Geometry;
 Primary 53C55;
 53C21 (Primary) 53C25 (Secondary)
 EPrint:
 45 pages, AMSTeX, submitted to Journal f\"ur die reine und angewandte Mathematik