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Proposition 20.4.12E2830D0CF2C8

Proposition 20.4.12.E2830D0CF2C8 If the ambient background $\theta $ is closed, the classes

\[ \mathbb A_ Z(u;Y,\theta ) =\{ \pi ^*(\theta |_ Y)-S_ Z(u;Y)\} _{\mathrm{BC}} \]

form a pseudoeffective Bott–Chern b-divisor class. They depend only on the ambient multiplier-ideal class of $u$ and the Bott–Chern class of $\theta |_ Y$.

Proof. By proposition 20.4.8, there is one $v$ on $Y$ such that

\[ R_ Z=\pi ^*(\theta |_ Y)+\mathrm{dd}^{\mathrm{c}}(v\circ \pi )-S_ Z(u;Y)\geq 0 \]

on every model. Since $\theta $ and $S_ Z$ are closed, $R_ Z$ is closed, and its class is the class in the statement. This proves pseudoeffectivity. Compatibility with pushforward follows from theorem 20.4.5 together with $g_*g^*(\theta _ Z)=\theta _ Z$. Finally, the trace data do not change under ambient multiplier-ideal equivalence, and replacing $\theta |_ Y$ by $\theta |_ Y+\mathrm{dd}^{\mathrm{c}}f$ with $f$ smooth changes no class; this proves the last assertion. $\square$

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