Pome9F22F2EB9CD1Bott–Chern cohomology records a closed $(1,1)$-current modulo $\mathrm{dd}^{\mathrm{c}}$ of global functions, and so separates the smooth background of the current from its potential. When $\theta $ is closed, subtracting the trace current from the pulled-back background gives a residual class on each model, and compatibility with pushforward organizes these classes into a b-divisor class in the sense of 20.1.1 The notion of b-divisors.
For a complex manifold $Z$, let $\Lambda ^{1,1}_{\mathbb R}$ denote the bundle of real $(1,1)$-forms and define
Lemma 20.4.10.C288B8AF0D4E Replacing smooth forms and smooth functions in this quotient by real currents and real distributions gives the same space. In particular, every closed positive $(1,1)$-current $T$ on $Z$ can be written
with $\beta $ smooth and closed and $w$ a global quasi-plurisubharmonic function.
Proof. The decomposition of lemma 5.38.4 shows that the natural map from the smooth quotient to the current quotient is surjective, and it gives a quasi-plurisubharmonic potential for positive currents. The same lemma shows that two distributional potentials in such decompositions differ by a smooth function; equivalently, a distribution whose $\mathrm{dd}^{\mathrm{c}}$ is smooth is itself smooth. Hence two smooth forms that are identified in the current quotient are already identified in the smooth quotient, and the map is injective. $\square$
From now on, $Z$ is connected and compact. By the lemma, every closed real $(1,1)$-current $T$ has a Bott–Chern class, written $\{ T\} _{\mathrm{BC}}$; the class is pseudoeffective if it contains a closed positive current. Pullback of classes is defined on smooth representatives. Pushforward of classes under a modification is defined on current representatives; it is well defined because pushforward commutes with $\mathrm{dd}^{\mathrm{c}}$, and by the lemma the resulting class is again a smooth Bott–Chern class. For a modification $g$ and every smooth real $(1,1)$-form $\beta $ on its target, $g_*g^*\beta =\beta $ as currents, by the change of variables formula for the degree-one map $g$.
A Gauduchon metric $\gamma $ defines a linear functional on this cohomology:
Integration by parts for currents shows that it does not depend on the representative. On a positive representative it is nonnegative, and it vanishes only if that representative is zero, by the strict positivity of the metric. In particular, the pseudoeffective cone contains no line: if $T_1,T_2\geq 0$ represent opposite classes, their sum has zero Gauduchon pairing, hence $T_1=T_2=0$.
Gauduchon pairings detect closed curvature information that is invisible in de Rham cohomology. Consider the Hopf manifold $X_ n$, $n\geq 2$, with its Gauduchon form $\omega $ and covering coordinates $z\in \mathbb C^ n\setminus \{ 0\} $, as in subsection 19.8.12. The form $\alpha =\mathrm{dd}^{\mathrm{c}}\log |z|^2$ is exact as a de Rham form, because $\mathrm d^{\mathrm c}\log |z|^2$ descends to the quotient: the logarithm changes only by a constant under the contraction. Nevertheless, by the calculation in proposition 19.8.13,
so the Bott–Chern class of $\alpha $ is nonzero.
We return to the smooth compact modifications of the manifold $Y$ fixed at the beginning of the chapter. Two such modifications $\pi :Z\to Y$ and $\pi ':Z'\to Y$ are compared through modifications $g:Z'\to Z$ with $\pi '=\pi \circ g$; we then say that $Z'$ dominates $Z$.
Definition 20.4.11.6EC6FB71DCF4 A Bott–Chern b-divisor class over $Y$ is a family
indexed by the smooth compact modifications of $Y$ and compatible with pushforward: $g_*\mathbb A_{Z'}=\mathbb A_ Z$ whenever $g:Z'\to Z$ is a modification over $Y$. It is pseudoeffective if every $\mathbb A_ Z$ is pseudoeffective. It is Cartier if there are a model $Z$ and a class $a_ Z\in H^{1,1}_{\mathrm{BC}}(Z,\mathbb R)$ such that $\mathbb A_{Z'}=g^*a_ Z$ for every model $g:Z'\to Z$ dominating $Z$; the pair $(Z,a_ Z)$ is then called a Cartier realization of $\mathbb A$.
When the models are Kähler, the $\partial \bar\partial $-lemma identifies this cohomology with the real $(1,1)$-cohomology used in 20.1.1 The notion of b-divisors, and the definition agrees with the one given there.
We now return to the trace data fixed at the beginning of the chapter. When $\theta $ is closed, the common majorant of proposition 20.4.8 provides positive representatives of the residual classes.
Proposition 20.4.12.E2830D0CF2C8 If the ambient background $\theta $ is closed, the classes
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
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. $\square$
If $\theta $ is not closed, the same expression $R_ Z$ is still a positive current, but
It is therefore not closed in general and defines no Bott–Chern class. Classes whose representatives have a prescribed nonzero exterior derivative are treated in section 10.30.
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