The authors solve a long-standing open problem on the extension of plurisubharmonic functions. The general setting of the problem is as follows. Given a manifold \(M\) and it’s analytic subvariety \(X\) when can one extend plurisubharmonic functions from \(X\) to \(M\) and if this is possible then can this be done with growth control (i.e., given some growth constraint on a plurisubharmonic function \(\varphi\) on \(X\) can an extension \(\psi\) be also constrained by a condition that in some sense reflects the constraint on \(\varphi\))?
There were partial results byA. Sadullaev [Dokl. Akad. Nauk UzSSR, No. 5, 3–4 (1982;Zbl 0637.32014)] that every plurisubharmonic function on \(X\) is the restriction of a plurisubharmonic function on \(M\) if \(M\) is a Stein manifold and \(X\) is a complex submanifold (so this covers the nonsingular case and there is no growth control) and byM. Coltoiu [Math. Ann. 290, No. 3, 545–548 (1991;Zbl 0747.32005)] that the same result holds if \(X\) is an analytic subset of \(M=\mathbb C^{n}\).
The authors follow Coltoiu’s approach and show that if \(M\) is a Stein manifold and \(X\) is an analytic subvariety of \(M\) (but otherwise unconstrained) then for every plurisubharmonic \(\varphi\) on \(X\) one can find a plurisubharmonic \(\psi\) on \(M\) such that \(\psi|_{X}=\varphi\). If moreover there is a continuous exhaustion function \(u\) of \(M\) such that \(\varphi(z)<u(z)\) on \(X\) then \(\psi\) can be chosen so that \(\psi(z)< c\max\{u(z),0\}\) on \(M\) for arbitrary \(c>1\).
The proof contains a reduction of the problem to the case \(M=\mathbb C^{m}\) and then embedding the Stein manifold properly to some Euclidean space of high enough dimension. Having done this, the authors explore a similar problem on compact Kähler manifolds. This time quasiplurisubharmonic functions are the object of study (due to the lack of nontrivial plurisubharmonic functions on compact manifolds).
The second theorem says that if \(X\) is a subvariety and \(M\) is projective and equipped with a Hodge form \(\omega\) then any \(\omega\)-plurisubharmonic function on \(X\) is the restriction of an \(\omega\)-plurisubharmonic function on \(M\).
An important corollary to this theorem is that under the same assumptions one can approximate (smoothen) any \(\omega\)-plurisubharmonic \(\varphi\) on \(X\) by a sequence of \(\omega\)-plurisubharmonic functions \(\varphi_{j}\) on \(M\), which is pointwise decreasing on \(M\) and \(\lim \varphi_{j}(z)=\varphi(z)\) for all \(z\in X\).
As an application to the theorems above it is studied when plurisubharmonic functions in the Lelong class on \(X\) allow extensions which belong to the Lelong class in \(\mathbb C^{n}\). The authors provide examples of algebraic curves in \(\mathbb C^{2}\) not enjoying this property.
There is also a discussion why the above theorems cannot (still) be obtained by more straightforward methods of desingularization.

Reviewer: Zywomir Dinew (Kraków)