Extension of plurisubharmonic functions with growth control

TL;DR

This paper proves that plurisubharmonic functions on subvarieties of Stein manifolds can be extended to the whole manifold with controlled growth, and applies this to functions on projective space, enhancing understanding of psh extension properties.

Contribution

It establishes a growth-controlled extension method for psh functions from subvarieties to Stein manifolds and applies this to projective space, broadening extension theory.

Findings

Extension of psh functions with growth control on Stein manifolds.

Any ω-psh function on a subvariety of projective space is a restriction of a global ω-psh function.

Abstract

Suppose that $X$ is an analytic subvariety of a Stein manifold $M$ and that $\varphi$ 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 $\varphi$ admits a psh extension to $M$ which is dominated by $cu$ on $M$. We use this result to prove that any $\omega$-psh function on a subvariety of the complex projective space is the restriction of a global $\omega$-psh function, where $\omega$ is the Fubini-Study K\"ahler form.