3.2. Explicit examples [028L]

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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 ${\mathcal{L}}(X)={\mathcal{L}}({\mathbb{C}}^{n})\,|_{{}_{X}}$ .