Pome012278DBB6E8012278DBB6E8
A positive current can be written as a smooth form plus a complex Hessian in more than one way. Changing the decomposition by a smooth $\mathrm{dd}^{\mathrm{c}}$-exact form changes the potential by a bounded function, and positive linear combinations allow us to compare potentials belonging to different backgrounds. In this section, $\eta $ denotes a second smooth real $(1,1)$-form on $X$.
Proposition 5.23.1.237930AECE45 Let $f$ be a smooth real function on $X$, and set $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$. Then
is a bijection. It preserves the positive current and changes each potential by a bounded function. Moreover, $\mathrm{d}\theta '=\mathrm{d}\theta $.
Proof. For every $u\in \mathrm{PSH}(X,\theta )$,
The inverse map is $v\mapsto v+f$, and $f$ is bounded because $X$ is compact. The last assertion follows from $\mathrm{d}\mathrm{dd}^{\mathrm{c}}f=0$. $\square$
Thus the natural object is the class of $\theta $ modulo smooth $\mathrm{dd}^{\mathrm{c}}$-exact forms. This quotient of all smooth real $(1,1)$-forms contains nonclosed classes; the usual Bott–Chern cohomology group is the corresponding quotient of closed forms.
Proposition 5.23.2.B16999654254 If $u\in \mathrm{PSH}(X,\theta )$ and $v\in \mathrm{PSH}(X,\eta )$, then, for $a,b\geq 0$ with $a+b>0$,
In particular, $\mathrm{PSH}(X,\theta )$ is convex. If $\eta \geq \theta $, then $\mathrm{PSH}(X,\theta )\subseteq \mathrm{PSH}(X,\eta )$.
Proof. Omitting a term whose coefficient is zero, we see that $au+bv$ is quasi-plurisubharmonic. Moreover,
For the inclusion, add the semipositive smooth form $\eta -\theta $ to $\theta +\mathrm{dd}^{\mathrm{c}}u$. $\square$
By proposition 5.22.2, any two quasi-plurisubharmonic functions on $X$ belong to $\mathrm{PSH}(X,C\omega )$ for a common sufficiently large constant $C$. This allows us to compare their singularities within one positive background.
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