3. Algebraic subvarieties of ℂ n [028H]

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3. Algebraic subvarieties of ${\mathbb{C}}^{n}$

If $X$ is an analytic subvariety of ${\mathbb{C}}^{n}$ and $\gamma$ is a positive number, we denote by ${\mathcal{L}}_{\gamma}(X)$ the Lelong class of psh functions $\varphi$ on $X$ which verify $\varphi(z)\leq\gamma\log^{+}\|z\|+C$ for all $z\in X$ , where $C$ is a constant that depends on $\varphi$ . We let ${\mathcal{L}}(X)={\mathcal{L}}_{1}(X)$ . By Theorem A, functions $\varphi\in{\mathcal{L}}(X)$ admit a psh extension in each class ${\mathcal{L}}_{\gamma}({\mathbb{C}}^{n})$ , for every $\gamma>1$ . ¹¹ 1 If $X$ is algebraic this result is claimed in [BL, Proposition 3.3], but there is a gap in their proof.

We assume in the sequel that $X$ is an algebraic subvariety of ${\mathbb{C}}^{n}$ and address the question whether it is necessary to allow the arbitrarily small additional growth. More precisely, is it true that

{\mathcal{L}}(X)\stackrel{{\scriptstyle?}}{{=}}{\mathcal{L}}({\mathbb{C}}^{n})\,|_{{}_{X}},

i.e. is every psh function with logarithmic growth on $X$ the restriction of a globally defined psh function with logarithmic growth? We will give a criterion for this to hold, but show that in general this is not the case.

3.1. Extension preserving the Lelong class

Consider the standard embedding

z\in{\mathbb{C}}^{n}\hookrightarrow[1:z]\in{\mathbb{P}}^{n},

where $[t:z]$ denote the homogeneous coordinates on ${\mathbb{P}}^{n}$ . Let $\omega$ be the Fubini-Study Kähler form and let

\rho(t,z)=\log\sqrt{|t|^{2}+\|z\|^{2}}

be its logarithmically homogeneous potential on ${\mathbb{C}}^{n+1}$ .

We denote by $\overline{X}$ the closure of $X$ in ${\mathbb{P}}^{n}$ , so $\overline{X}$ is an algebraic subvariety of ${\mathbb{P}}^{n}$ . It is well known that the class $PSH({\mathbb{P}}^{n},\omega)$ is in one-to-one correspondence with the Lelong class ${\mathcal{L}}({\mathbb{C}}^{n})$ (see [GZ]). Let us look at the connection between $\omega$ -psh functions on $\overline{X}$ and the class ${\mathcal{L}}(X)$ .

The mapping

F_{X}:PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})\longmapsto{\mathcal{L}}(X),\;(F_{X}\varphi)(z)=\rho(1,z)+\varphi([1:z]),

is well defined and injective. However, it is in general not surjective, as shown by Examples 3.2 and 3.3 that follow.

Conversely, a function $\eta\in{\mathcal{L}}(X)$ induces an upper semicontinuous function $\widetilde{\eta}$ on $\overline{X}$ defined in the obvious way:

\widetilde{\eta}([t:z])=\left\{\begin{array}[]{ll}\eta(z)-\rho(1,z),\;\hskip 68.2866pt{\rm if}\;t=1,\;z\in X,\\ \\ \displaystyle\limsup_{[1:\zeta]\to[0:z],\zeta\in X}(\eta(\zeta)-\rho(1,\zeta)),\;{\rm if}\;t=0,\;[0:z]\in\overline{X}\setminus X.\end{array}\right.

The function $\widetilde{\eta}$ is in general only weakly $\omega$ -psh on $\overline{X}$ , i.e. it is bounded above on $\overline{X}$ and it is $\omega\,|_{{}_{\overline{X}_{r}}}$ -psh on the set $\overline{X}_{r}$ of regular points of $\overline{X}$ . This notion is in direct analogy to that of weakly psh function on an analytic variety (see [D2, section 1]). We do not pursue it any further here.

Note that $\eta\in F_{X}\left(PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})\right)$ if and only if $\widetilde{\eta}\in PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})$ . The following simple characterization is a consequence of Theorem B.

Proposition 3.1.

Let $\eta\in{\mathcal{L}}(X)$ . The following are equivalent:

(i) There exists $\psi\in{\mathcal{L}}({\mathbb{C}}^{n})$ so that $\psi=\eta$ on $X$ .

(ii) $\widetilde{\eta}\in PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})$ .

(iii) For every point $a\in\overline{X}\setminus X$ the following holds: if $(X_{j},a)$ are the irreducible components of the germ $(\overline{X},a)$ then the value

\limsup_{X_{j}\ni[1:\zeta]\to a}(\eta(\zeta)-\rho(1,\zeta))

is independent of $j$ .

In particular, if the germs $(\overline{X},a)$ are irreducible for all points $a\in\overline{X}\setminus X$ then ${\mathcal{L}}(X)={\mathcal{L}}({\mathbb{C}}^{n})\,|_{{}_{X}}$ .

Proof.

Assume that $(i)$ holds. It follows that $\widetilde{\eta}=\varphi\,|_{{}_{\overline{X}}}$ , where

\varphi([t:z]):=\left\{\begin{array}[]{ll}\psi(z)-\rho(1,z),\;\hskip 82.51299pt{\rm if}\;t=1,\\ \limsup_{[1:\zeta]\to[0:z]}(\psi(\zeta)-\rho(1,\zeta)),\;{\rm if}\;t=0,\end{array}\right.

is an $\omega$ -psh function on ${\mathbb{P}}^{n}$ . Hence $\widetilde{\eta}\in PSH(\overline{X},\omega\,|_{{}_{\overline{X}}})$ .

Conversely, if $(ii)$ holds then by Theorem B there exists an $\omega$ -psh function $\varphi$ on ${\mathbb{P}}^{n}$ which extends $\widetilde{\eta}$ . Hence $\psi(z)=\rho(1,z)+\varphi([1:z])$ is an extension of $\eta$ and $\psi\in{\mathcal{L}}({\mathbb{C}}^{n})$ .

The equivalence of $(ii)$ and $(iii)$ follows easily from [D2, Theorem 1.10]. ∎

3.2. Explicit examples

In view of section 3.1, it is easy to construct examples of algebraic curves $X\subset{\mathbb{C}}^{2}$ and functions in ${\mathcal{L}}(X)$ which do not admit an extension in ${\mathcal{L}}({\mathbb{C}}^{2})$ . We write $z=(x,y)\in{\mathbb{C}}^{2}$ .

Example 3.2.

Let $X=\{y=0\}\cup\{y=1\}\subset{\mathbb{C}}^{2}$ and $\eta\in{\mathcal{L}}(X)$ , where

\eta(z)=\left\{\begin{array}[]{ll}\rho(1,z),\;\;\;\;\;\;\;{\rm if}\;z=(x,0),\\ \rho(1,z)+1,\;{\rm if}\;z=(x,1).\end{array}\right.

The function $\widetilde{\eta}$ is not $\omega$ -psh on $\overline{X}=\{y=0\}\cup\{y=t\}$ , hence $\eta$ does not have an extension in ${\mathcal{L}}({\mathbb{C}}^{2})$ . Indeed, the maximum principle is violated along $\{y=0\}$ near the point $a=[0:1:0]$ , since $\widetilde{\eta}([t:1:0])=0$ for $t\neq 0$ , while $\widetilde{\eta}([t:1:t])=1$ .

With a little more effort we can give an example as above where $X$ is an irreducible curve. Let ${\mathbb{C}}^{\star}={\mathbb{C}}\setminus\{0\}$ .

Example 3.3.

Let $X\subset{\mathbb{C}}^{2}$ be the irreducible cubic with equation $xy=x^{3}+1$ . Then

\overline{X}=\{[t:x:y]\in{\mathbb{P}}^{2}:\,xyt=x^{3}+t^{3}\},\;\overline{X}=X\cup\{a\},\;a=[0:0:1].

The germ $(\overline{X},a)$ has two irreducible components $X_{1},\,X_{2}$ , both are smooth at $a$ , $X_{1}$ being tangent to the line $\{x=0\}$ , and $X_{2}$ to the line $\{t=0\}$ .

Note that in fact $X\subset{\mathbb{C}}^{\star}\times{\mathbb{C}}$ is the graph of the rational function $y=x^{2}+x^{-1}$ , $x\in{\mathbb{C}}^{\star}$ . If $(x,y)\in X$ and $x\to 0$ then $(x,y)\to a$ along $X_{1}$ , while as $x\to\infty$ then $(x,y)\to a$ along $X_{2}$ . The function

u(x,y)=\max\{-\log|x|,2\log|x|+1\}

is psh in ${\mathbb{C}}^{\star}\times{\mathbb{C}}$ . It is easy to check that $\eta:=u\,|_{{}_{X}}\in{\mathcal{L}}(X)$ and

\limsup_{X_{1}\ni[1:\zeta]\to a}(\eta(\zeta)-\rho(1,\zeta))=0\;,\;\;\limsup_{X_{2}\ni[1:\zeta]\to a}(\eta(\zeta)-\rho(1,\zeta))=1.

Hence $\eta$ does not admit an extension in ${\mathcal{L}}({\mathbb{C}}^{2})$ .

We conclude this section with an example of a cubic $X$ in ${\mathbb{C}}^{2}$ and a psh function on $X$ of the form $\eta=\log|P|$ , where $P$ is a polynomial, so that $\eta$ admits a “transcendental” extension with exactly the same growth, but small additional growth is necessary if we look for an “algebraic” extension.

Proposition 3.4.

Let $X=\{x=y^{3}\}$ and $\eta(x,y)=\log|1+y|$ , so $\eta\,|_{{}_{X}}\in{\mathcal{L}}_{1/3}(X)$ .

Given $k\geq 1$ , there is a polynomial $Q_{k}(x,y)$ of degree $k+1$ so that $Q_{k}(y^{3},y)=(y+1)^{3k}$ . In particular, $\psi_{k}=\frac{1}{3k}\log|Q_{k}|\in{\mathcal{L}}_{(k+1)/3k}({\mathbb{C}}^{2})$ is an extension of $\eta\,|_{{}_{X}}$ .

There exists no polynomial $Q(x,y)$ of degree $k$ so that $Q(y^{3},y)=(y+1)^{3k}$ . However, $\eta\,|_{{}_{X}}$ has an extension in ${\mathcal{L}}_{1/3}({\mathbb{C}}^{2})$ .

Proof.

We construct $Q_{k}$ by replacing $y^{3}$ by $x$ in the polynomial

(y+1)^{3k}=\sum_{j=0}^{3k}{3k\choose j}y^{j}.

Since $j=3[j/3]+r_{j}$ , $r_{j}\in\{0,1,2\}$ , it follows that

Q_{k}(x,y)=\sum_{j=0}^{3k}{3k\choose j}x^{[j/3]}y^{r_{j}}=3kx^{k-1}y^{2}+l.d.t.\;.

We now check that there is no polynomial $Q(x,y)$ of degree $k$ so that $Q(y^{3},y)=(y+1)^{3k}$ . Indeed, if $Q(x,y)=\sum_{j+l\leq k}c_{jl}x^{j}y^{l}$ then

Q(y^{3},y)=c_{k0}y^{3k}+c_{k-1,1}y^{3k-2}+l.d.t.

does not contain the monomial $y^{3k-1}$ .

Note that $\overline{X}=\{xt^{2}=y^{3}\}=X\cup\{a\}$ , where $a=[0:1:0]$ , so the germ $(\overline{X},a)$ is irreducible. Proposition 3.1 implies that $\eta\,|_{{}_{X}}$ has an extension in ${\mathcal{L}}_{1/3}({\mathbb{C}}^{2})$ . ∎

We conclude with some remarks regarding our last example. If $X$ is an algebraic subvariety of ${\mathbb{C}}^{n}$ and $f$ is a holomorphic function on $X$ , $f$ is said to have polynomial growth if there is an integer $N(f)$ and a constant $A$ so that

|f(z)|\leq A(1+\|z\|)^{N(f)},\;\;\forall\,z\in X.

Then it is well known that there exists a polynomial $P$ of degree at most $N(f)+\varepsilon(X)$ so that $P\,|_{{}_{X}}=f$ , where $\varepsilon(X)>0$ is a constant depending only on $X$ (see e.g. [Bj] and references therein). However, if $\overline{X}\subset{\mathbb{P}}^{N}$ is irreducible at each of its points at infinity then by Proposition 3.1 the psh function $\eta=N(f)^{-1}\log|f|\in{\mathcal{L}}(X)$ has a psh extension in the Lelong class ${\mathcal{L}}({\mathbb{C}}^{n})$ .

On the other hand, Demailly [D1] has shown that in the case of the transcendental curve $X=\{e^{x}+e^{y}=1\}$ any holomorphic function $f$ on $X$ , of polynomial growth, has a polynomial extension of the same degree to ${\mathbb{C}}^{n}$ . Hence it is natural to ask if for this curve one has that ${\mathcal{L}}(X)={\mathcal{L}}({\mathbb{C}}^{n})\,|_{{}_{X}}$ .

3. Algebraic subvarieties of ℂ n [028H]

3. Algebraic subvarieties of ℂn{\mathbb{C}}^{n}

3.1. Extension preserving the Lelong class

Proposition 3.1.

Proof.

3.2. Explicit examples

Example 3.2.

Example 3.3.

Proposition 3.4.

Proof.

3. Algebraic subvarieties of ${\mathbb{C}}^{n}$