Summary: AMBIK¨AHLER GEOMETRY, AMBITORIC SURFACES
AND EINSTEIN 4-ORBIFOLDS
VESTISLAV APOSTOLOV, DAVID M. J. CALDERBANK, AND PAUL GAUDUCHON
Abstract. We give an explicit local classification of conformally equivalent but
oppositely oriented K¨ahler metrics on a 4-manifold which are toric with respect to
a common 2-torus action. In the generic case, these structures have an intriguing
local geometry depending on a quadratic polynomial and two arbitrary functions
of one variable, these two functions being explicit degree 4 polynomials when the
K¨ahler metrics are extremal (in the sense of Calabi).
One motivation for and application of this result is an explicit local description
of Einstein 4-manifolds which are hermitian with respect to either orientation.
This can be considered as a riemannian analogue of a result in General Relativity
due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension
of the classification of self-dual Einstein hermitian 4-manifolds, obtained inde-
pendently by R. Bryant and the first and third authors.
We discuss toric compactifications of these metrics on orbifolds and provide
infinite discrete families of compact toric extremal K¨ahler orbifolds. Our exam-
ples include Bach-flat K¨ahler orbifolds which are conformal to complete smooth
Einstein metrics on an open subset. We illustrate how these examples fit with
recent conjectures relating the existence of extremal toric metrics to various no-