Extension of plurisubharmonic functions with growth control

Suppose that $X$ is an analytic subvariety of a Stein manifold $M$ and that $φ$ is a plurisubharmonic (psh) function on $X$ which is dominated by a continuous psh exhaustion function $u$ of $M$. Given any number $c>1$, we show that $φ$ admits a psh extension to $M$ which is dominated by $cu$ on $M$. We use this result to prove that any $ω$-psh function on a subvariety of the complex projective space is the restriction of a global $ω$-psh function, where $ω$ is the Fubini-Study Kähler form.

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