  Building on the five-dimensional constructions in hep-th/0601177, we provide
a unified description of four-dimensional N = 2 superconformal off-shell
multiplets in projective superspace, including a realization in terms of N = 1
superfields. In particular, superconformal polar multiplets are consistently
defined for the first time. We present new 4D N = 2 superconformal sigma-models
described by polar multiplets. Such sigma-models realize general superconformal
couplings in projective superspace, but involve an infinite tale of auxiliary N
= 1 superfields. The auxiliaries should be eliminated by solving infinitely
many algebraic nonlinear equations, and this is a nontrivial technical problem.
We argue that the latter can be avoided by making use of supersymmetry
considerations. All information about the resulting superconformal model (and
hence the associated superconformal cone) is encoded in the so-called canonical
coordinate system for a Kaehler metric, which was introduced by Bochner and
Calabi in the late 1940s.
