Example 5.47.12174FD6E0DD10

Example 5.47.12.174FD6E0DD10 The torus $\mathbb T^2=\{ z\in \mathbb C^2:|z_1|=|z_2|=1\} $ is real polar, in the sense of section 5.10, but not pluripolar.

To see that $\mathbb T^2$ is real polar, let $\mu $ be the probability measure on $\mathbb T^2$ obtained as the image of $(2\pi )^{-2}\, \mathrm d\theta $ on $[-\pi ,\pi )^2$ under $\theta \mapsto (\mathrm e^{\mathrm i\theta _1},\mathrm e^{\mathrm i\theta _2})$, and put

If we identify $\mathbb C^2$ with $\mathbb R^4$, then $u$ is a positive multiple of the fundamental potential $E_4*\mu $, so it is subharmonic on $\mathbb R^4$ by proposition 3.4.11. It is finite off $\mathbb T^2$, since for $z\notin \mathbb T^2$ the distances $|z-\zeta |$, $\zeta \in \mathbb T^2$, are bounded below by the distance from $z$ to the compact set $\mathbb T^2$. Now let $z=(\mathrm e^{\mathrm i\alpha _1},\mathrm e^{\mathrm i\alpha _2})\in \mathbb T^2$. By rotation invariance, $\mu $ is also the image of $(2\pi )^{-2}\, \mathrm d\theta $ under $\theta \mapsto \zeta (\theta )=(\mathrm e^{\mathrm i(\alpha _1+\theta _1)},\mathrm e^{\mathrm i(\alpha _2+\theta _2)})$, and

Hence

so $\mathbb T^2\subset \{ u=-\infty \} $, and $\mathbb T^2$ is real polar.

Suppose that $\mathbb T^2$ were pluripolar. By theorem 5.47.10 there would be $V\in \mathrm{PSH}(\mathbb C^2)$ with $\mathbb T^2\subset \{ V=-\infty \} $. Fix $c\in \mathbb C$ with $|c|=1$. By proposition 5.9.3, the function $f(\zeta )=V(\zeta ,c)$ is either identically $-\infty $ or subharmonic on $\mathbb C$, and $f=-\infty $ on the unit circle. In the second case, upper semicontinuity gives $\varlimsup _{\zeta \to \zeta _0}f(\zeta )\le f(\zeta _0)=-\infty $ at every point $\zeta _0$ of the unit circle, and the maximum principle (proposition 3.4.7), applied to $f+M$ on the unit disc for every $M>0$, shows that $f=-\infty $ on the unit disc. Hence $V=-\infty $ on $\{ |z_1|\le 1,\ |z_2|=1\} $. Now fix $z_1$ with $|z_1|<1$; the function $w\mapsto V(z_1,w)$ equals $-\infty $ on the unit circle, and the same argument shows that it equals $-\infty $ on the unit disc. Thus $V=-\infty $ on the open bidisc $\{ |z_1|<1,\ |z_2|<1\} $, contradicting the local integrability of $V$ (theorem 5.9.1).

For $n\ge 3$, the same arguments show that $\mathbb T^2\times \mathbb C^{n-2}\subset \mathbb C^ n$ is real polar but not pluripolar. The function $U(z)=u(z_1,z_2)$ is subharmonic on $\mathbb R^{2n}$, by lemma 4.2.7 with $k=4$ and $d=2n$. The function $U$ is finite off $\mathbb T^2\times \mathbb C^{n-2}$ and equal to $-\infty $ on it. If the set were pluripolar, theorem 5.47.10 would give $V\in \mathrm{PSH}(\mathbb C^ n)$ with poles on it, and the argument above, with the last $n-2$ coordinates fixed, would show that $V=-\infty $ on the product of the open bidisc with $\mathbb C^{n-2}$, which is again impossible.