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.

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 $X$.

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. $\Box$