Pome

Corollary 18.24.3F204E0019263

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. $\square$

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