Pome89D86926636ATwo real $(1,1)$-currents on $Z$ with exterior derivative $\pi ^*\mathrm{d}\theta $ differ by a closed current, and modulo $\mathrm{dd}^{\mathrm{c}}$ of global distributions such differences are Bott–Chern classes. Let $\mathcal D'(Z,\mathbb R)$ denote the space of real distributions on $Z$, and set
Write $\{ T\} $ for the class of $T$ in this space. By lemma 20.4.10, every class has a smooth representative, and translation by Bott–Chern classes gives
A smooth real $(1,1)$-form $\eta $ represents an element of $\mathscr A_ Z(\theta )$ exactly when $\mathrm{d}\eta =\pi ^*\mathrm{d}\theta $. As for Bott–Chern classes, pullback of a class is defined on a smooth representative and pushforward on current representatives.
Proposition 10.30.2.5F6B6BF8B562 If $g:Z'\to Z$ is a compact modification over $Y$, pushforward and pullback induce maps
Pushforward preserves classes with positive current representatives.
Proof. Write $\pi '=\pi \circ g$. Translation by the pulled-back background identifies the required maps with the Bott–Chern maps of subsection 20.4.9. Indeed, for a closed current $C'$ on $Z'$ and a smooth closed form $\beta $ on $Z$,
where the first equality uses the change of variables formula for the degree-one map $g$ on the smooth background. Since the Bott–Chern maps are well defined on the closed terms, these identities define the affine maps independently of the representatives. Their composition is the identity because $g_*g^*$ is the identity both on the background and on Bott–Chern classes. Finally, pushforward under a proper map preserves positive currents. $\square$
The prescribed derivative is essential for this pushforward: $g_*\pi '^*\mathrm{d}\theta =\pi ^*\mathrm{d}\theta $ is smooth, whereas the pushforward of an arbitrary smooth exact form can be a singular current.
We call a family $(\mathbb A_ Z)$, with $\mathbb A_ Z\in \mathscr A_ Z(\theta )$ for every model $Z$ and compatible with pushforward, a b-divisor class over the background $\theta $. Pseudoeffectivity and Cartier realizations are defined as in definition 20.4.11, with affine classes in place of Bott–Chern classes. A class in $\mathscr A_ Z(\theta )$ is nef if, for a Hermitian form $\gamma $ on $Z$, it has for every $\epsilon >0$ a smooth representative bounded below by $-\epsilon \gamma $, as in subsection 20.4.16.
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