Corollary 18.14.10.20390B807B79 Let $p:M\to Y$ be a surjective holomorphic submersion between connected compact complex manifolds, with $1\leq \dim Y<\dim M$; its fibers may be disconnected. Use the weighted divisor data $A,H,\Psi ,\kappa _ A$ above, let $\beta $ be a smooth real $(1,1)$-form on $Y$, and let $w\in \mathrm{PSH}(Y,\beta )$. Put
\[ \Theta =p^*\beta +\kappa _ A,\qquad u=\Psi +w\circ p. \]
For the factorization $p=r\circ \pi $ of lemma 5.43.4, set $\beta '=r^*\beta $ and $w'=w\circ r$. For $F\in C(M,\mathbb R)$ define
\[ \widehat F_{\mathrm{comp}}(y') =\inf _{x\in \pi ^{-1}(y')\setminus H}\{ F(x)-\Psi (x)\} , \qquad \inf \varnothing =+\infty . \]
This obstacle is continuous in the extended real sense on $Y'$, and
\[ \begin{split} P_{\Theta ,F}[u] & =\Psi +P_{\beta ',\widehat F_{\mathrm{comp}}}[w']\circ \pi ,\\ P_{\Theta ,F}[u]_{\mathcal I} & =\Psi +P_{\beta ',\widehat F_{\mathrm{comp}}}[w']_{\mathcal I}\circ \pi . \end{split} \]
Every representative of the ordinary type of $u$ or of $P_\Theta [u]_{\mathcal I}$ has zero extended mass, and both singularity volumes of these types are zero. The ordinary and ideal envelopes of $u$ agree below every smooth obstacle on $M$ if and only if those of $w'$ agree below every smooth obstacle on $Y'$.
For $\varphi \in \mathrm{PSH}(Y,\beta )$, put $v=\Psi +\varphi \circ p$. Then
\[ u\in \mathcal E_{\mathrm{env}}(M,\Theta ,v) \quad \Longleftrightarrow \quad w\in \mathcal E_{\mathrm{env}}(Y,\beta ,\varphi ). \]
For $q_ M=P_\Theta [u]_{\mathcal I}$ and $q_ Y=P_\beta [w]_{\mathcal I}$, one also has
\[ u\in \mathcal E_{\mathrm{env}}(M,\Theta ,q_ M) \quad \Longleftrightarrow \quad w\in \mathcal E_{\mathrm{env}}(Y,\beta ,q_ Y). \]
If $Y$ is a curve, let $\mu _ w,D_ w$ be the polar and atomic measures of corollary 18.14.5. Then equality of the envelopes of $u$ below zero, the equality $\mu _ w=D_ w$, and equality of the envelopes of $u$ below every smooth obstacle are equivalent. They are also equivalent to $u\in \mathcal E_{\mathrm{env}}(M,\Theta ,q_ M)$.
Proof.
The covering map $r$ preserves the curvature inequality, so $w'\in \mathrm{PSH}(Y',\beta ')$. Moreover $\dim Y'=\dim Y<\dim M$, and
\[ \Theta =\pi ^*\beta '+\kappa _ A,\qquad u=\Psi +w'\circ \pi . \]
Applying proposition 18.14.1 to the submersion $\pi $, which has connected fibers, gives the obstacle identities and the mass assertions, and corollary 18.14.5 for the same submersion gives the equivalence for all obstacles. In example 18.14.9, this factorization has $Y'=E$, $r=q$ and $\pi =a$, so it recovers the computed envelope.
For the assertions on relative classes, apply proposition 18.14.4 to $\pi $, with reference $\varphi \circ r$ on $Y'$, and then transfer membership through $r$ by proposition 18.12.4. That proposition, applied with the zero obstacle, also gives
\[ P_{\beta '}[w']_{\mathcal I}=q_ Y\circ r. \]
Applying the same two equivalences with these ideal references proves the assertion for $q_ M$ and $q_ Y$.
In the curve case, $Y'$ is a curve. The covering is locally biholomorphic, so curvature measures and point Lelong numbers pull back without multiplicities. The pole set of $w'$ is $r^{-1}\{ w=-\infty \} $, and therefore
\[ \mu _{w'}=r^*\mu _ w,\qquad D_{w'}=r^*D_ w, \]
where the pullback of a measure is defined on each sheet through the local inverse of $r$. Such pullbacks reflect equality: over every sufficiently small ball in $Y$, the restriction of a measure is recovered from any sheet over that ball. Hence $\mu _{w'}=D_{w'}$ if and only if $\mu _ w=D_ w$. The curve case of corollary 18.14.5, applied to $\pi $, identifies the former equality with the equality of the envelopes upstairs below zero, with their equality below every smooth obstacle and with membership in the relative envelope class upstairs. This proves the last assertion.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.