PomeC43F5C12FA6FSince the trace envelope dominates the trace currents on every model, subtracting them from its curvature current gives positive residual representatives, with exterior derivative $\pi ^*\mathrm{d}\theta $. The classes are determined by the prescribed trace currents, even when the envelope has larger Lelong numbers.
In this section, $Y$ is a complex submanifold of a connected compact complex manifold $X$. Let $\Theta $ be a smooth real $(1,1)$-form on $X$ with $\Theta |_ Y=\theta $, and let $u\in \mathrm{PSH}(X,\Theta )$ have generic Lelong number $\nu (u,Y)=0$. We use the trace currents $S_ Z(u;Y)$ of subsection 20.4.1, now with $\Theta $ in the role of the ambient background.
Theorem 10.30.4.28C8F709D1CC The family defined by
is a pseudoeffective b-divisor class over $\theta $. It is unchanged by ambient multiplier-ideal equivalence and, after the resulting affine spaces are identified, by a smooth $\mathrm{dd}^{\mathrm{c}}$ change of the restricted background.
Proof. The currents $S_ Z$ are closed, so the classes belong to the indicated affine spaces, and their compatibility follows from theorem 20.4.5, proposition 10.30.2. Take $v=\mathcal T_\theta (u;Y)$, the trace envelope of (20.3.25). By proposition 20.4.8,
on every model, which proves pseudoeffectivity in the original restricted background. The currents $S_ Z$ depend only on the ambient multiplier ideals. For a smooth real function $f$ on $Y$, the change $\theta \mapsto \theta +\mathrm{dd}^{\mathrm{c}}f$ adds $\mathrm{dd}^{\mathrm{c}}(f\circ \pi )$ to every representative, which is zero in the quotient. This proves both invariances. $\square$
When $\theta $ is closed, this family is the Bott–Chern family of proposition 20.4.12.
On a higher model, the trace current can exceed the pullback of the trace current from a lower model. The effective exceptional difference of theorem 20.4.26 therefore translates the residual class, and the translation preserves the prescribed exterior derivative.
Corollary 10.30.5.9157A5A52988 Let $g:Z'\to Z$ be a compact modification over $Y$, and let $E_ g$ be the effective exceptional divisor of theorem 20.4.26. The trace classes of theorem 10.30.4 satisfy
where the Bott–Chern class on the right acts by translation on the affine space $\mathscr A_{Z'}(\theta )$.
Proof. Let $\pi :Z\to Y$ be the structure map and put $\pi '=\pi \circ g$. Abbreviate $S_ Z=S_ Z(u;Y)$, and use lemma 5.38.4 to write $S_ Z=B_ Z+\mathrm{dd}^{\mathrm{c}}v_ Z$ with $B_ Z$ smooth and closed and $v_ Z$ quasi-plurisubharmonic. The affine class $\mathbb B_ Z$ has the smooth representative $\pi ^*\theta -B_ Z$. Its pullback is therefore the class of $\pi '^*\theta -g^*S_ Z$, since the two representatives differ by $\mathrm{dd}^{\mathrm{c}}(v_ Z\circ g)$. Substituting $S_{Z'}=g^*S_ Z+[E_ g]$ gives the asserted identity. Both representatives have exterior derivative $\pi '^*\mathrm{d}\theta $, because the subtracted currents are closed, so the subtraction takes place within $\mathscr A_{Z'}(\theta )$. $\square$
The cocycle identity (20.4.28) makes these translations compatible with composition of modifications, and by theorem 20.4.26 their analytic-ideal approximants converge in total variation of the trace measures. On the other hand, theorem 20.3.30 identifies the trace envelopes on the models with pullbacks of the original trace envelope. The exceptional translation accounts for the additional divisorial subtraction on the higher model; it is consistent with this exact pullback of the envelope, and it can be nonzero even when the trace data are realized exactly, as in example 20.4.31.
For the rest of the chapter, the ambient manifold is $Y$ itself, and $u\in \mathrm{PSH}(Y,\theta )$. For a smooth compact modification $\pi :Z\to Y$, we write $N_ Z(u)$ for the divisorial current of $u\circ \pi $ and $\mathbb D_\theta (u)$ for the resulting residual family. By the self-trace identity of lemma 20.4.14,
where $E$ ranges over the prime divisors on $Z$. By theorem 10.30.4, $\mathbb D_\theta (u)$ is pseudoeffective. In this case divisorial removal gives the positive representative $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}(u\circ \pi )-N_ Z(u)\geq 0$ directly, by lemma 5.38.1 and passage to the limit of increasing finite sums.
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.