2. Extension of qpsh functions [0288]

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2. Extension of qpsh functions

Let $V$ be a compact Kähler manifold equipped with a Kähler form $\omega$ . We let $PSH(V,\omega)$ denote the set of $\omega$ -psh functions on $V$ . These are upper semicontinuous functions $\varphi\in L^{1}(V,[-\infty,+\infty))$ such that $\omega+dd^{c}\varphi\geq 0$ , where $d=\partial+\overline{\partial}$ and $d^{c}=\frac{1}{2\pi i}(\partial-\overline{\partial})$ . We refer the reader to [GZ] for basic properties of $\omega$ -psh functions.

Let $X$ be an analytic subvariety of $V$ . Recall that an upper semicontinuous function $\varphi:X\to[-\infty,+\infty)$ is called $\omega\,|_{{}_{X}}$ -psh if $\varphi\not\equiv-\infty$ on $X$ and if there exist an open cover $\{U_{i}\}_{i\in I}$ of $X$ and psh functions $\varphi_{i},\rho_{i}$ defined on $U_{i}$ , where $\rho_{i}$ is smooth and $dd^{c}\rho_{i}=\omega$ , so that $\rho_{i}+\varphi=\varphi_{i}$ holds on $X\cap U_{i}$ , for every $i\in I$ . Moreover, $\varphi$ is called strictly $\omega\,|_{{}_{X}}$ -psh if it is $(1-\varepsilon)\omega\,|_{{}_{X}}$ -psh for some small $\varepsilon>0$ . The current $\omega\,|_{{}_{X}}+dd^{c}\varphi$ is then called a Kähler current on $X$ (see [EGZ1, section 5.2]). We denote by $PSH(X,\omega\,|_{{}_{X}})$ , resp. $PSH^{+}(X,\omega\,|_{{}_{X}})$ , the class of $\omega\,|_{{}_{X}}$ -psh, resp. strictly $\omega\,|_{{}_{X}}$ -psh functions on $X$ .

Every $\omega$ -psh function $\varphi$ on $V$ yields, by restriction, an $\omega\,|_{{}_{X}}$ -psh function $\varphi\,|_{{}_{X}}$ on $X$ , as soon as $\varphi\,|_{{}_{X}}\not\equiv-\infty$ . The question we address here is whether this restriction operator is surjective. In other words, is there equality

PSH(X,\omega\,|_{{}_{X}})\stackrel{{\scriptstyle?}}{{=}}PSH(V,\omega)\,|_{{}_{X}}.

2.1. The smooth case

We start with the elementary observation that smooth strictly $\omega$ -psh functions can easily be extended.

Proposition 2.1.

Let $V$ be a compact Kähler manifold equipped with a Kähler form $\omega$ , and let $X$ be a complex submanifold of $V$ . Then

PSH^{+}(X,\omega\,|_{{}_{X}})\cap{\mathcal{C}}^{\infty}(X,{\mathbb{R}})=\left(PSH^{+}(V,\omega)\cap{\mathcal{C}}^{\infty}(V,\mathbb{R})\right)\,|_{{}_{X}}.

We include a proof for the convenience of the reader, although this is probably part of the “folklore” (see e.g. [Sch] for the case where $\omega$ is a Hodge form).

Proof.

Let $\varphi\in{\mathcal{C}}^{\infty}(X,\mathbb{R})$ be such that $(1-\varepsilon)\omega\,|_{{}_{X}}+dd^{c}\varphi\geq 0$ on $X$ , for some $\varepsilon>0$ . We first choose $\tilde{\varphi}$ to be any smooth extension of $\varphi$ to $V$ . Consider

\psi:=\tilde{\varphi}+A\chi\,\text{dist}(\cdot,X)^{2},

where $\chi$ is a test function supported in a small neighborhood of $X$ and such that $\chi\equiv 1$ near $X$ . Here ${dist}$ is any Riemannian distance on $V$ , for instance the distance associated to the Kähler metric $\omega$ . Then $\psi$ is yet another smooth extension of $\varphi$ to $V$ , which now satisfies $(1-\varepsilon/2)\omega+dd^{c}\psi\geq 0$ near $X$ , if $A$ is chosen large enough.

The function $\log(\text{dist}(\cdot,X)^{2})$ is well defined and qpsh in a neighborhood of $X$ . Let $\chi$ be a test function supported in this neighborhood so that $\chi\equiv 1$ near $X$ . The function $u=\chi\log(\text{dist}(\cdot,X)^{2})$ is $N\omega$ -psh on $V$ for a large integer $N$ . Moreover, $\exp(u)$ is smooth and $X=\{u=-\infty\}$ . Replacing $\omega$ by $N\omega$ , $\varphi$ by $N\varphi$ , and $\psi$ by $N\psi$ , we may assume that $N=1$ . Set now

\psi_{C}:=\frac{1}{2}\,\log\left[e^{2\psi}+e^{u+C}\right].

This again is a smooth extension of $\varphi$ , and a straightforward computation yields

dd^{c}\psi_{C}\geq\frac{2e^{2\psi}dd^{c}\psi+e^{u+C}dd^{c}u}{2(e^{2\psi}+e^{u+C})}\;.

Hence

\left(1-\frac{\varepsilon}{2}\right)\omega+dd^{c}\psi_{C}\geq\frac{2e^{2\psi}\left[\left(1-\frac{\varepsilon}{2}\right)\omega+dd^{c}\psi\right]+(1-\varepsilon)e^{u+C}\omega}{2(e^{2\psi}+e^{u+C})}\geq 0,

if $C$ is chosen large enough. ∎

This proof breaks down when $\varphi$ is singular and hence a different approach is needed. We consider in the next section the particular case when $\omega$ is a Hodge form.

2.2. Proof of Theorem B

We assume here that $\omega$ is a Hodge form, i.e. that the cohomology class $\{\omega\}$ belongs to $H^{2}(V,\mathbb{Z})$ (more precisely to the image of $H^{2}(V,\mathbb{Z})$ in $H^{2}(V,\mathbb{R})$ under the mapping induced by the inclusion $\mathbb{Z}\hookrightarrow\mathbb{R}$ ). We prove the following more precise version of Theorem B.

Theorem 2.2.

Let $X$ be a subvariety of a projective manifold $V$ equipped with a Hodge form $\omega$ . If $\varphi\in PSH(X,\omega\,|_{{}_{X}})$ then given any constant $a>0$ there exists $\psi\in PSH(V,\omega)$ so that $\psi\,|_{{}_{X}}=\varphi$ and $\max_{V}\psi<\max_{X}\varphi+a$ .

In the assumptions of Theorem 2.2 there exists a positive holomorphic line bundle $L$ on $V$ whose first Chern class $c_{1}(L)$ is represented by $\omega$ . By Kodaira’s embedding theorem $L$ is ample, hence for large $k$ there exists an embedding $\pi:V\hookrightarrow{\mathbb{P}}^{n}$ such that $L^{k}=\pi^{*}{\mathcal{O}}(1)$ .

Replacing $\omega$ by $k\omega$ , $\varphi$ by $k\varphi$ , we can assume that $L={\mathcal{O}}(1)$ , $V$ is an algebraic submanifold of the complex projective space ${\mathbb{P}}^{n}$ , and $\omega=\omega_{FS}\,|_{{}_{V}}$ is the Fubini-Study Kähler form. Hence $X$ is an algebraic subvariety of ${\mathbb{P}}^{n}$ , and Theorem 2.2 follows if we show that $\omega_{FS}$ -psh functions on $X$ extend to $\omega_{FS}$ -psh functions on ${\mathbb{P}}^{n}$ .

Therefore we assume in the sequel that $X\subset V={\mathbb{P}}^{n}$ and $\omega$ is the Fubini-Study Kähler form on ${\mathbb{P}}^{n}$ . Let $[z_{0}:\ldots:z_{n}]$ denote the homogeneous coordinates. Without loss of generality, we may assume that they are chosen so that no coordinate hyperplane $\{z_{j}=0\}$ contains any irreducible component of $X$ .

Let

\theta(z)=\log\frac{\max\{|z_{0}|,\dots,|z_{n}|\}}{\sqrt{|z_{0}|^{2}+\ldots+|z_{n}|^{2}}}\,,\;z=[z_{0}:\ldots:z_{n}]\in{\mathbb{P}}^{n}.

This is an $\omega$ -psh function and for all $z\in{\mathbb{P}}^{n}$ ,

-m\leq\theta(z)\leq 0\,,\;\;{\rm where}\;m=\log\sqrt{n+1}.

We start by noting that Theorem A yields special subextensions of $\omega$ -psh functions on $X$ .

Lemma 2.3.

Let $\varepsilon\geq 0$ and $u$ be a continuous $(1+\varepsilon)\omega$ -psh function on ${\mathbb{P}}^{n}$ so that $u(z)\leq 0$ for all $z\in{\mathbb{P}}^{n}$ . If $c>1$ and $\varphi$ is an $\omega$ -psh function on $X$ so that $\varphi<u$ , then there exists a $c\omega$ -psh function $\psi$ on ${\mathbb{P}}^{n}$ so that

\frac{1}{c}\,\psi(z)\leq\frac{1}{1+\varepsilon}\,u(z),\;\forall z\in{\mathbb{P}}^{n},

and

\psi(z)=\varphi(z)+(c-1)\theta(z)+(c-1)\min_{\zeta\in{\mathbb{P}}^{n}}u(\zeta),\;\forall z\in X.

Proof.

Let

M=-\min_{\zeta\in{\mathbb{P}}^{n}}u(\zeta)\geq 0.

We work first in an affine chart $\{z_{j}=1\}\equiv{\mathbb{C}}^{n}$ . Let $X_{j}=X\cap\{z_{j}=1\}$ and let $\rho_{j}\geq 0$ be the potential of $\omega$ in this chart with $\rho_{j}(0)=0$ . Then $\varphi+\rho_{j}$ is psh on $X_{j}$ and since $u\leq 0$ ,

\varphi+\rho_{j}+M<u+\rho_{j}+M\leq\frac{1}{1+\varepsilon}\,u+\rho_{j}+M\;{\rm on}\;X_{j}.

Note that $(1+\varepsilon)^{-1}u+\rho_{j}+M\geq 0$ is a continuous psh exhaustion function on ${\mathbb{C}}^{n}$ . Theorem A yields a psh function $\widetilde{\psi}$ on ${\mathbb{C}}^{n}$ so that

\widetilde{\psi}<\frac{c}{1+\varepsilon}\,u+c\rho_{j}+cM\;{\rm on}\;{\mathbb{C}}^{n}\;,\;\;\widetilde{\psi}=\varphi+\rho_{j}+M\;{\rm on}\;X_{j}.

The function $\psi_{j}=\widetilde{\psi}-c\rho_{j}-cM$ extends uniquely to a $c\omega$ -psh function on ${\mathbb{P}}^{n}$ which verifies

\psi_{j}\leq\frac{c}{1+\varepsilon}\,u\;\;{\rm on}\;{\mathbb{P}}^{n}.

Moreover on $X\cap\{z_{j}=1\}$ we have

\psi_{j}=\varphi-(c-1)\rho_{j}-(c-1)M=\varphi+(c-1)\theta_{j}-(c-1)M,

where

\theta_{j}(z)=\log\frac{|z_{j}|}{\sqrt{|z_{0}|^{2}+\ldots+|z_{n}|^{2}}}\;.

Hence $\psi_{j}=-\infty$ on $X\cap\{z_{j}=0\}$ .

We finally let $\psi=\max\{\psi_{0},\ldots,\psi_{n}\}$ . This is a $c\omega$ -psh function on ${\mathbb{P}}^{n}$ which verifies the desired conclusions, since $\theta=\max\{\theta_{0},\ldots,\theta_{n}\}$ . ∎

Proof of Theorem 2.2. Fix $a>0$ . Replacing $\varphi$ by $\varphi-\max_{X}\varphi-a$ we may assume that $\max_{X}\varphi=-a$ . We will show that there exists a sequence of smooth $\omega$ -psh functions $\varphi_{j}$ on ${\mathbb{P}}^{n}$ which decrease pointwise on ${\mathbb{P}}^{n}$ to a negative $\omega$ -psh function $\psi$ so that $\psi=\varphi$ on $X$ .

Let $X^{\prime}$ be the union of the irreducible components $W$ of $X$ so that $\varphi\,|_{{}_{W}}\not\equiv-\infty$ . We first construct by induction on $j\geq 1$ a sequence of numbers $\varepsilon_{j}\searrow 0$ and a sequence of negative smooth $(1+\varepsilon_{j})\omega$ -psh functions $\psi_{j}$ on ${\mathbb{P}}^{n}$ so that for all $j\geq 2$

\frac{\psi_{j}}{1+\varepsilon_{j}}<\frac{\psi_{j-1}}{1+\varepsilon_{j-1}}\;\;{\rm on}\;{\mathbb{P}}^{n}\;,\;\;\psi_{j-1}>\varphi\;{\rm on}\;X\;,\;\;\int_{X^{\prime}}(\psi_{j}-\varphi)<\frac{1}{j}\;,\;\;\int_{W}\psi_{j}<-j\,,

for every irreducible component $W$ of $X$ where $\varphi\,|_{{}_{W}}\equiv-\infty$ . Here the integrals are with respect to the area measure on each irreducible component $X_{j}$ of $X$ , i.e.

\int_{X}f:=\sum_{X_{j}}\int_{X_{j}}f\,\omega^{\dim X_{j}}.

Let $\varepsilon_{1}=1$ , $\psi_{1}=0$ , and assume that $\varepsilon_{j-1},\,\psi_{j-1}$ , where $j\geq 2$ , are constructed with the above properties. Since $\varphi<\psi_{j-1}\,|_{{}_{X}}$ and the latter is continuous on the compact set $X$ , we can find $\delta>0$ so that $\varphi<\psi_{j-1}-\delta$ on $X$ .

Let $c>1$ . By Lemma 2.3, there exists a $c\omega$ -psh function $\psi_{c}$ so that

\frac{\psi_{c}}{c}\leq\frac{\psi_{j-1}-\delta}{1+\varepsilon_{j-1}}\;\;{\rm on}\;{\mathbb{P}}^{n}\;,\;\;\psi_{c}=\varphi+(c-1)\theta-(c-1)M_{j-1}\;\;{\rm on}\;X,

where

M_{j-1}=\delta-\min_{\zeta\in{\mathbb{P}}^{n}}\psi_{j-1}(\zeta)\geq 0.

We can regularize $\psi_{c}$ on ${\mathbb{P}}^{n}$ : there exists a sequence of smooth $c\omega$ -psh functions decreasing to $\psi_{c}$ on ${\mathbb{P}}^{n}$ . Therefore we can find a smooth $c\omega$ -psh function $\psi^{\prime}_{c}$ on ${\mathbb{P}}^{n}$ so that

\frac{\psi^{\prime}_{c}}{c}<\frac{\psi_{j-1}-\frac{\delta}{2}}{1+\varepsilon_{j-1}}\;{\rm on}\;{\mathbb{P}}^{n},\;\;\psi^{\prime}_{c}>\varphi+(c-1)\theta-(c-1)M_{j-1}\geq\varphi-(c-1)(m+M_{j-1})\;\;{\rm on}\;X.

By dominated, resp. monotone convergence, we can in addition ensure that

\int_{X^{\prime}}(\psi^{\prime}_{c}-\varphi)\leq\int_{X^{\prime}}(\psi^{\prime}_{c}-\varphi-(c-1)\theta+(c-1)M_{j-1})<c-1,

\int_{W}\psi^{\prime}_{c}<-j-(c-1)(m+M_{j-1})|W|,

for every irreducible component $W$ of $X$ where $\varphi\,|_{{}_{W}}\equiv-\infty$ . Here $|W|$ denotes the (projective) area of $W$ .

Now let $\psi^{\prime\prime}_{c}=\psi^{\prime}_{c}+(c-1)(m+M_{j-1})$ . Then on ${\mathbb{P}}^{n}$ we have

\frac{\psi^{\prime\prime}_{c}}{c}<\frac{\psi_{j-1}-\frac{\delta}{2}}{1+\varepsilon_{j-1}}+\frac{(c-1)(m+M_{j-1})}{c}<\frac{\psi_{j-1}}{1+\varepsilon_{j-1}}-\frac{\delta}{4}+(c-1)(m+M_{j-1}).

Moreover, $\psi^{\prime\prime}_{c}>\varphi$ on $X$ and

$\displaystyle\int_{X^{\prime}}(\psi^{\prime\prime}_{c}-\varphi)$	$\displaystyle=$	$\displaystyle\int_{X^{\prime}}(\psi^{\prime}_{c}-\varphi)+(c-1)(m+M_{j-1})\|X^{\prime}\|$
	$\displaystyle<$	$\displaystyle(c-1)(1+m\|X^{\prime}\|+M_{j-1}\|X^{\prime}\|)\;,$
$\displaystyle\int_{W}\psi^{\prime\prime}_{c}$	$\displaystyle=$	$\displaystyle\int_{W}\psi^{\prime}_{c}+(c-1)(m+M_{j-1})\|W\|<-j\;,$

for every irreducible component $W$ of $X$ where $\varphi\,|_{{}_{W}}\equiv-\infty$ .

We take $c=1+\varepsilon_{j}$ and $\psi_{j}=\psi^{\prime\prime}_{c}$ , where $\varepsilon_{j}>0$ is so that

\varepsilon_{j}<\varepsilon_{j-1}/2\;,\;\;\varepsilon_{j}(m+M_{j-1})<\frac{\delta}{4}\;,\;\;\varepsilon_{j}(1+m|X^{\prime}|+M_{j-1}|X^{\prime}|)<\frac{1}{j}\;.

Then $\varepsilon_{j},\,\psi_{j}$ have the desired properties.

We conclude that $\varphi_{j}=(1+\varepsilon_{j})^{-1}\psi_{j}$ is a decreasing sequence of smooth negative $\omega$ -psh function on ${\mathbb{P}}^{n}$ , so that $\varphi_{j}>(1+\varepsilon_{j})^{-1}\varphi>\varphi$ on $X$ . Hence $\psi=\lim_{j\to\infty}\varphi_{j}$ is a negative $\omega$ -psh function on ${\mathbb{P}}^{n}$ and $\psi\geq\varphi$ on $X$ . Note that

\int_{X^{\prime}}(\varphi_{j}-\varphi)=\frac{1}{1+\varepsilon_{j}}\int_{X^{\prime}}(\psi_{j}-\varphi)-\frac{\varepsilon_{j}}{1+\varepsilon_{j}}\int_{X^{\prime}}\varphi<\frac{1}{j}-\frac{\varepsilon_{j}}{1+\varepsilon_{j}}\int_{X^{\prime}}\varphi\;,

\int_{W}\varphi_{j}=\frac{1}{1+\varepsilon_{j}}\int_{W}\psi_{j}<-\frac{j}{2}\;,

for every irreducible component $W$ of $X$ where $\varphi\,|_{{}_{W}}\equiv-\infty$ . It follows that $\psi=\varphi$ on $X$ and the proof of Theorem 2.2 is finished. $\Box$