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 $a+b>0$,
\[ au+bv\in \mathrm{PSH}(X,a\theta +b\eta ). \]
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,
\[ a\theta +b\eta +\mathrm{dd}^{\mathrm{c}}(au+bv) =a(\theta +\mathrm{dd}^{\mathrm{c}}u)+b(\eta +\mathrm{dd}^{\mathrm{c}}v)\geq 0. \]
For the inclusion, add the semipositive smooth form $\eta -\theta $ to $\theta +\mathrm{dd}^{\mathrm{c}}u$.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.