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18.24 Models pulled back by a finite curve mapEB49D847F199

EB49D847F199

Let $r:C'\to C$ be a finite surjective holomorphic map of degree $d$ between connected compact Riemann surfaces. Fix a smooth real form $\beta $ of area $A\geq 0$ on $C$ and put $\sigma =r^*\beta $. The models pulled back from $C$ occupy a proper subspace of the full model space on $C'$ in general. The fiber maximum of 5.43.15 compares envelope competitors on the two curves. The weighted measure pullback then identifies the metric on this subspace and gives a retraction from the full space.

Proposition 18.24.1.424A6AE6F169 For $v\in \mathrm{PSH}(C,\beta )$ and every smooth real obstacle $F$ on $C$,

\[ \begin{aligned} P_{\sigma ,F\circ r}[v\circ r] & =P_{\beta ,F}[v]\circ r,\\ P_{\sigma ,F\circ r}[v\circ r]_{\mathcal I} & =P_{\beta ,F}[v]_{\mathcal I}\circ r. \end{aligned} \]

In particular, $J(v)=v\circ r$ embeds the ordinary model space $\mathcal M(\beta )$ into $\mathcal M(\sigma )$ and preserves ideal projection.

Proof. Pullback sends every base competitor to an upstairs competitor: it preserves positivity, the obstacle inequality, the ordinary order and, by the Lelong formula and the curve valuative criterion, the all-weight ideal order. Conversely, if $h$ is an upstairs competitor, its fiber maximum $m_ h$ is a base competitor by 5.43.15, since $h\leq F\circ r$ implies $m_ h\leq F$. Also $h\leq m_ h\circ r$. Thus the two unregularized suprema are equal after pullback. A finite nonconstant holomorphic map of curves is open; upper regularization commutes with its pullback, as is seen in the coordinates $t=z^ e$. This proves both identities. Take $F=0$. An ordinary model is fixed by its ordinary envelope, so its pullback is a model; surjectivity of $r$ gives injectivity. Applying the ideal identity to a model proves preservation of ideal projection. $\square$

Let $d_\beta $ and $d_\sigma $ be the polar metrics from proposition 13.29.1. If $q$ is a model on $C'$, denote its polar curvature by $\mu _ q=\mathbf1_{\{ q=-\infty \} }(\sigma +\mathrm{dd}^{\mathrm{c}}q)$. Write $p_\beta (\nu )$ for the base model with polar measure $\nu $, using the same classification. Write $U_\beta (v)=\int _ C\mathrm{MA}_\beta (v)$ and $U_\sigma (q)=\int _{C'}\mathrm{MA}_\sigma (q)$ for their non-pluripolar volumes.

Proposition 18.24.2.C8EE003DDDC9 For $u,v\in \mathcal M(\beta )$,

\[ d_\sigma (J(u),J(v))=d\, d_\beta (u,v),\qquad U_\sigma (J(u))=d\, U_\beta (u). \]

The image $J(\mathcal M(\beta ))$ is closed and contains the linear polar-measure geodesic between each pair of its members. The map

\[ R(q)=J\! \left(p_\beta \! \left(d^{-1}r_*\mu _ q\right)\right) \]

is a $1$-Lipschitz retraction onto this image. Its polar measure is

\[ \mu _{R(q)}=d^{-1}r^*r_*\mu _ q. \]

It preserves the polar mass and the non-pluripolar volume, and commutes with ideal projection on $\mathcal M(\sigma )$. At $A=0$ both model spaces are smooth singletons.

Proof. Change of variables off the finite branch sets gives $\int _{C'}\sigma =dA$. By 5.43.16, the polar measure of $J(v)$ is $r^*\mu _ v$, with total mass $d\mu _ v(C)$. The curve classification gives

\[ \begin{aligned} d_\sigma (J(u),J(v)) & =\| r^*(\mu _ u-\mu _ v)\| _{\mathrm{TV}} =d\| \mu _ u-\mu _ v\| _{\mathrm{TV}},\\ U_\sigma (J(v)) & =dA-d\mu _ v(C)=d\, U_\beta (v). \end{aligned} \]

Completeness of the base model space proves closedness of the image: a convergent sequence in the image comes from a Cauchy sequence downstairs. Linear interpolation of polar measures commutes with $r^*$, giving the stated geodesics.

For any $q\in \mathcal M(\sigma )$, its polar measure has mass at most $dA$. Its pushforward is polar by the image assertion of 5.43.16, so $d^{-1}r_*\mu _ q$ has mass at most $A$ and defines a base model. This proves that $R$ is well defined and gives its displayed polar measure. The identity $r_*r^*=d$ makes $R\circ J=J$. For any finite signed measure $\tau $ on $C'$, pushforward does not increase total variation, hence

\[ \| d^{-1}r^*r_*\tau \| _{\mathrm{TV}} =\| r_*\tau \| _{\mathrm{TV}} \leq \| \tau \| _{\mathrm{TV}}. \]

Apply this to the difference of two polar measures to obtain the Lipschitz estimate. The averaging operator preserves their total mass, so the formula $U_\sigma (q)=dA-\mu _ q(C')$ gives volume preservation. Both measure operations commute with taking the atomic part, which is ideal projection by the curve classification. This proves the commutation assertion. At zero area all polar measures vanish and both spaces are singletons. $\square$

Corollary 18.24.3.F204E0019263 Let $p:X\to C$ be a surjective holomorphic map from a connected compact complex manifold of dimension at least two, with component factorization $p=r\circ q$, $q:X\to C'$. For $\theta =p^*\beta $, the ordinary models that are constant on each $p$-fiber form a closed complete subspace. With the $p$-fiber normalization of 18.23.1, the map $v\mapsto v\circ p$ is an isometry from $(\mathcal M(\beta ),d_\beta )$ onto this subspace.

Every model has the form $U=u\circ q$, and

\[ R_ X(U)=R(u)\circ q \]

is a $1$-Lipschitz retraction onto the subspace. It commutes with ideal projection. The subspace contains the linear polar-measure geodesic between each pair of its members.

Proof. The full model correspondence for $q$ is 18.22.1. A model $u\circ q$ is constant on $p$-fibers exactly when $u$ is constant on $r$-fibers. In that case $u=m_ u\circ r$, and 5.43.15 makes $m_ u$ a base potential. The envelope identity and injectivity of $J$ then show that $m_ u$ is a model. Thus this subspace is exactly the pullback image of $\mathcal M(\beta )$.

By 18.23.1, the $p$-normalized metric is $d^{-1}$ times the polar metric on $C'$. The preceding proposition proves the isometry, closedness, completeness, retraction and geodesic assertions. The ideal-envelope correspondence for $q$ transfers the commutation with ideal projection. $\square$

This retraction averages the singularity data over a fiber. Unlike the ideal retraction, it need not select a nearest model.

Example 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.

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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