Introduction [027Z]

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Introduction

Let $X\subset{\mathbb{C}}^{n}$ be a (closed) analytic subvariety. In the case when $X$ is smooth it is well known that a plurisubharmonic (psh) function on $X$ extends to a psh function on ${\mathbb{C}}^{n}$ [Sa] (see also [BL, Theorem 3.2]). Using different methods, Coltoiu generalized this result to the case when $X$ is singular [Co, Proposition 2].

In this article we follow Coltoiu’s approach and show that it is possible to obtain extensions with global growth control:

Theorem A. Let $X$ be an analytic subvariety of a Stein manifold $M$ and let $\varphi$ be a psh function on $X$ . Assume that $u$ is a continuous psh exhaustion function on $M$ so that $\varphi(z)<u(z)$ for all $z\in X$ . Then for every $c>1$ there exists a psh function $\psi=\psi_{c}$ on $M$ so that $\psi\,|_{{}_{X}}=\varphi$ and $\psi(z)<c\max\{u(z),0\}$ for all $z\in M$ .

We recall that a function $\varphi:X\to[-\infty,+\infty)$ is called psh if $\varphi\not\equiv-\infty$ on $X$ and if every point $z\in X$ has a neighborhood $U$ in ${\mathbb{C}}^{n}$ so that $\varphi=u\,|_{{}_{U}}$ for some psh function $u$ on $U$ . We refer to [FN] and [D2, section 1] for a detailed discussion of this notion. We note here that if $\varphi$ is not identically $-\infty$ on an irreducible component $Y$ of $X$ then $\varphi$ is locally integrable on $Y$ with respect to the area measure of $Y$ . Let us stress that the more general notion of weakly psh function is not appropriate for the extension problem (see section 3).

We then look at a similar problem on a compact Kähler manifold $V$ . Here psh functions have to be replaced by quasiplurisubharmonic (qpsh) ones. Given a Kähler form $\omega$ , we let

PSH(V,\omega)=\left\{\varphi\in L^{1}(V,[-\infty,+\infty)):\,\varphi\text{ upper semicontinuous, }dd^{c}\varphi\geq-\omega\right\}

denote the set of $\omega$ -plurisubharmonic ( $\omega$ -psh) functions. If $X\subset V$ is an analytic subvariety, we define similarly the class $PSH(X,\omega\,|_{{}_{X}})$ of $\omega$ -psh functions on $X$ (see section 2 for precise definitions).

By restriction, $\omega$ -psh functions on $V$ yield $\omega\,|_{{}_{X}}$ -psh functions on $X$ . Assuming that $\omega$ is a Hodge form, i.e. a Kähler form with integer cohomology class, our second result is that every $\omega\,|_{{}_{X}}$ -psh function on $X$ arises in this way.

Theorem B. Let $X$ be a subvariety of a projective manifold $V$ equipped with a Hodge form $\omega$ . Then any $\omega\,|_{{}_{X}}$ -psh function on $X$ is the restriction of an $\omega$ -psh function on $V$ .

Note that in the assumptions of Theorem B there exists a positive holomorphic line bundle $L$ on $V$ whose first Chern class $c_{1}(L)$ is represented by $\omega$ . In this case the $\omega$ -psh functions are in one-to-one correspondence with the set of (singular) positive metrics of $L$ (see [GZ]). Thus an alternate formulation of Theorem B is the following:

Theorem B’. Let $X$ be a subvariety of a projective manifold $V$ and $L$ be an ample line bundle on $V$ . Then any (singular) positive metric of $L\,|_{{}_{X}}$ is the restriction of a (singular) positive metric of $L$ on $V$ .

Recall that it is possible to regularize qpsh functions on ${\mathbb{P}}^{n}$ , since it is a homogeneous manifold. Hence Theorem B has the following immediate corollary:

Corollary C. Let $X$ be a subvariety of a projective manifold $V$ equipped with a Hodge form $\omega$ . If $\varphi\in PSH(X,\omega\,|_{{}_{X}})$ then there exists a sequence of smooth functions $\varphi_{j}\in PSH(V,\omega)$ which decrease pointwise on $V$ so that $\lim\,\varphi_{j}=\varphi$ on $X$ .

When $X$ is smooth this regularization result is well known to hold even when the cohomology class of $\omega$ is not integral (see [D3], [BK]).

Corollary C allows to show that the singular Kähler-Einstein currents constructed in [EGZ1] have continuous potentials, a result that has been obtained recently in [EGZ2] by completely different methods (see also [DZ] for partial results in this direction).

We prove Theorem A in section 1. The compact setting is considered in section 2, where Theorem B is derived from Theorem A. In section 3 we discuss the special situation when $X$ is an algebraic subvariety of ${\mathbb{C}}^{n}$ . As an application of Theorem B, we give a characterization of those psh functions in the Lelong class ${\mathcal{L}}(X)$ which admit an extension in the Lelong class ${\mathcal{L}}({\mathbb{C}}^{n})$ (see section 3 for the necessary definitions). In particular, we give simple examples of algebraic curves $X\subset{\mathbb{C}}^{2}$ and of functions $\eta\in{\mathcal{L}}(X)$ which do not have extensions in ${\mathcal{L}}({\mathbb{C}}^{2})$ .