Pome0CFD6850F9F8Lemma 5.24.2.0CFD6850F9F8 Let $Y$ be a complex manifold, let $\beta $ be a smooth real $(1,1)$-form on $Y$, and let $D\subset Y$ be open. Suppose that $u$ is $\beta $-plurisubharmonic on $Y$, that $v$ is $\beta $-plurisubharmonic on $D$, and that
Then the function equal to $\max (u,v)$ on $D$ and to $u$ on $Y\setminus D$ is $\beta $-plurisubharmonic.
Proof. Denote the glued function by $w$; it is upper semicontinuous by the boundary inequality. Freeze $\beta $ on a small coordinate ball as in the proof of lemma 5.24.1, and choose a quadratic correction $r$ such that $u+r$ and $v+r$ are plurisubharmonic on their respective domains. Off $\partial D$, the function $w+r$ is plurisubharmonic, being locally either $u+r$ or the maximum of two plurisubharmonic functions. At a point of $\partial D$, it equals $u+r$ and dominates $u+r$ on the ball, so it satisfies the submean inequality on every complex line through that point. Thus $w+r$ is plurisubharmonic on the whole ball. Letting the error of the frozen coefficients tend to zero gives $\beta +\mathrm{dd}^{\mathrm{c}}w\geq 0$. $\square$
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