4 research outputs found
Special K\"ahler-Ricci potentials on compact K\"ahler manifolds
A special K\"ahler-Ricci potential on a K\"ahler manifold is any nonconstant
function such that is a Killing vector field
and, at every point with , all nonzero tangent vectors orthogonal
to and are eigenvectors of both and
the Ricci tensor. For instance, this is always the case if is a
nonconstant function on a K\"ahler manifold of complex
dimension and the metric , defined wherever , is Einstein. (When such exists, may be called {\it
almost-everywhere conformally Einstein}.) We provide a complete classification
of compact K\"ahler manifolds with special K\"ahler-Ricci potentials and use it
to prove a structure theorem for compact K\"ahler manifolds of any complex
dimension which are almost-everywhere conformally Einstein.Comment: 45 pages, AMSTeX, submitted to Journal f\"ur die reine und angewandte
Mathemati
Hamiltonian 2-forms in Kahler geometry, III Extremal metrics and stability
This paper concerns the explicit construction of extremal Kaehler metrics on
total spaces of projective bundles, which have been studied in many places. We
present a unified approach, motivated by the theory of hamiltonian 2-forms (as
introduced and studied in previous papers in the series) but this paper is
largely independent of that theory.
We obtain a characterization, on a large family of projective bundles, of
those `admissible' Kaehler classes (i.e., the ones compatible with the bundle
structure in a way we make precise) which contain an extremal Kaehler metric.
In many cases, such as on geometrically ruled surfaces, every Kaehler class is
admissible. In particular, our results complete the classification of extremal
Kaehler metrics on geometrically ruled surfaces, answering several
long-standing questions.
We also find that our characterization agrees with a notion of K-stability
for admissible Kaehler classes. Our examples and nonexistence results therefore
provide a fertile testing ground for the rapidly developing theory of stability
for projective varieties, and we discuss some of the ramifications. In
particular we obtain examples of projective varieties which are destabilized by
a non-algebraic degeneration.Comment: 40 pages, sequel to math.DG/0401320 and math.DG/0202280, but largely
self-contained; partially replaces and extends math.DG/050151