PomeExample 18.24.4.2C40EC6336AD For $r:\mathbb P^1\to \mathbb P^1$, $r(z)=z^3$, and the Fubini–Study background $\beta $ of area one, the retraction $R$ above is not a nearest-point retraction, even on ideal models with analytic singularities.
Example 18.24.42C40EC6336AD
Proof. Put $\sigma =r^*\beta $, of area three, and let $q$ be its model with polar measure $\delta _1$. This is an analytic ideal model: in the affine coordinate, the global function
\[ h(z)=\log |z-1|^2-\tfrac 13\log (1+|z|^6) \]
extends smoothly across infinity and satisfies $\sigma +\mathrm{dd}^{\mathrm{c}}h=\delta _1+\tfrac 23\sigma $. Thus $h$ is $\sigma $-plurisubharmonic with one logarithmic pole, and the curve classification identifies its envelope with $q$; proposition 18.4.8 preserves its analytic type. Writing $\zeta =\exp (2\pi \mathrm i/3)$, one has
\[ \mu _{R(q)}=\tfrac 13(\delta _1+\delta _\zeta +\delta _{\zeta ^2}). \]
The three atoms are distinct, so
\[ d_\sigma (q,R(q))=\tfrac 23+\tfrac 13+\tfrac 13=\tfrac 43. \]
The zero model is $J(0)$ and has zero polar measure, giving $d_\sigma (q,J(0))=1$. Hence the averaged model is strictly farther from $q$ than a member of the same pullback subspace. $\square$
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