Proposition 16.29.5.7DD8E9433762 Let $\beta \geq \theta $ be a smooth real $(1,1)$-form on $X$. Slicewise projection, $\Gamma \mapsto (P_\beta [\Gamma _\tau ]_{\mathcal I})_{\tau <c}$, defines an injective and order-preserving map
\[ P_\beta [\cdot ]_{\mathcal I}:\mathscr T_{a,c}(\theta )\longrightarrow \mathscr T_{a,c}(\beta ). \]
For every smooth real $(1,1)$-form $\beta '$ with $\beta \geq \beta '\geq \theta $, these maps satisfy
\[ P_\beta \bigl [P_{\beta '}[\Gamma ]_{\mathcal I}\bigr ]_{\mathcal I}=P_\beta [\Gamma ]_{\mathcal I}. \]
Proof.
Since $\beta \geq \theta $, every slice is $\beta $-plurisubharmonic, and by theorem 18.4.2 its projection is a $\beta $-plurisubharmonic function fixed by $P_\beta [\cdot ]_{\mathcal I}$. Decreasingness, constancy for $\tau \leq a$, and order preservation follow from proposition 18.4.6.
For concavity, let $r,t<c$ and $0<s<1$, and put $\tau =(1-s)r+s t$. By the same proposition and the concavity of the original curve,
\[ \begin{split} (1-s)P_\beta [\Gamma _ r]_{\mathcal I}+sP_\beta [\Gamma _ t]_{\mathcal I} & \leq P_\beta [(1-s)\Gamma _ r+s\Gamma _ t]_{\mathcal I}\\ & \leq P_\beta [\Gamma _\tau ]_{\mathcal I}. \end{split} \]
Here the first inequality follows by combining the scaling and addition formulas of that proposition, with backgrounds $(1-s)\beta $ and $s\beta $.
Each projection preserves the multiplier ideals at every weight. Hence the two sides of the transition identity are $P_\beta [\cdot ]_{\mathcal I}$-fixed representatives of the same multiplier-ideal class, and they are equal. If two projected curves agree, their original slices have the same multiplier ideals; since those slices are $P_\theta [\cdot ]_{\mathcal I}$-fixed, they agree as well. This proves injectivity.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.