PomeF45AA1F2A651For a closed positive current, the construction can be carried out with its own potential in place of the restricted ambient ideals. The trace coefficients then become the generic Lelong numbers of the pulled-back potential along divisors, and divisorial removal gives the positive residual currents directly.
In this section, $T$ is a closed positive $(1,1)$-current on $Y$, written $T=\beta +\mathrm{dd}^{\mathrm{c}}w$ as in lemma 20.4.10. On a smooth compact modification $\pi :Z\to Y$, the pullback $\pi ^*T$ is the positive current defined by composing local plurisubharmonic potentials with $\pi $; since $\pi $ is surjective, these compositions are not identically $-\infty $. With $E$ ranging over the prime divisors on $Z$, set
By divisorial removal (lemma 5.38.1), the second current is positive and closed. The following lemma identifies the coefficients of $N_ Z(T)$ with trace numbers when the ambient manifold is $Y$ itself.
Lemma 20.4.14.691315D9AA1C Let $w$ be a quasi-plurisubharmonic function on $Y$, and take the ambient manifold in (20.3.3) to be $Y$ itself. For every prime divisor $E$ on a smooth compact modification $\pi :Z\to Y$, the resulting self-trace satisfies
In particular, for the potential $w$ of the closed positive current $T$, one has $N_ Z(T)=S_ Z(w;Y)$.
Proof. Since $w$ is not identically $-\infty $, its Lelong number vanishes at almost every point; hence $\nu (w,Y)=0$, and the self-trace is defined. Fix a divisor $E$ on a compact modification and apply lemma 5.38.9 on a coordinate neighborhood of a point of its image in $Y$. With $\nu _ E=\nu (w\circ \pi ,E)$ and $A_ E$ the log discrepancy of $E$, it gives
The limit along the integers defining $\tau _{Y,w}(E)$ is therefore $\nu _ E$. Since all coefficients agree, the two divisorial currents are equal. $\square$
Theorem 20.4.15.25F4A4E3755D The family
is a pseudoeffective Bott–Chern b-divisor class. For closed positive currents $T_1,T_2$ and $a\geq 0$,
On a Kähler manifold, with Kähler models, it agrees with the b-divisor class of the same current in 20.1.1 The notion of b-divisors.
Proof. By the lemma, $N_ Z(T)=S_ Z(w;Y)$, so theorem 20.4.5 gives $g_*N_{Z'}(T)=N_ Z(T)$ for every modification $g:Z'\to Z$ over $Y$. Moreover, $g_*g^*(\pi ^*T)=\pi ^*T$. To see this, write the current as a smooth form plus $\mathrm{dd}^{\mathrm{c}}$ of a potential; the change of variables formula for the degree-one map $g$ applies both to the smooth form and to the locally integrable pulled-back potential, and pushforward commutes with $\mathrm{dd}^{\mathrm{c}}$. Therefore
Taking classes proves compatibility, and positivity of the regular parts proves pseudoeffectivity. Generic Lelong numbers are additive under addition of positive currents and scale under multiplication by nonnegative constants. Hence so are the divisorial parts and the regular parts, which proves the two identities. In the Kähler case, the regular parts and, by the $\partial \bar\partial $-lemma, their cohomology classes are exactly those of the construction cited in the statement. $\square$
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