Corollary 21.2.9.0C12F36D1EDD Let $n\geq 1$, let $V\subset \mathbb R^ n$ be a nonempty open convex set, and put
\[ \Omega _ V=\{ z\in (\mathbb C^*)^ n: (\log |z_1|^2,\ldots ,\log |z_ n|^2)\in V\} . \]
Let $\varphi \in \mathrm{PSH}(\Omega _ V)$ be invariant under the action of the compact torus $(S^1)^ n$ by separate rotations of the coordinates, $(z_1,\ldots ,z_ n)\mapsto (\mathrm e^{\mathrm i\theta _1}z_1,\ldots ,\mathrm e^{\mathrm i\theta _ n}z_ n)$. Then there is a finite convex function $f$ on $V$ such that
\[ \varphi (z)=f(\log |z_1|^2,\ldots ,\log |z_ n|^2)\qquad (z\in \Omega _ V). \]
Conversely, for every finite convex function $f$ on $V$ this formula defines an $(S^1)^ n$-invariant plurisubharmonic function on $\Omega _ V$. In particular, an $(S^1)^ n$-invariant plurisubharmonic function on $(\Delta ^*)^ n$ or on $(\mathbb C^*)^ n$ is finite everywhere, where $\Delta ^*=\{ \zeta \in \mathbb C:0<|\zeta |<1\} $.
Proof.
Consider the holomorphic map
\[ E\colon \mathbb C^ n\to (\mathbb C^*)^ n,\qquad E(w)=(\exp (w_1/2),\ldots ,\exp (w_ n/2)). \]
Since $\log |E_ j(w)|^2=\operatorname{Re}w_ j$, it maps the tube $V+\mathrm i\mathbb R^ n$ onto $\Omega _ V$; in particular, $\Omega _ V$ is connected. By proposition 5.9.3, $\varphi \circ E$ is either plurisubharmonic or identically $-\infty $ on the tube, and the second alternative is impossible because $E$ is surjective and $\varphi $ is nontrivial. Since $E(w+\mathrm is')=(\mathrm e^{\mathrm is_1'/2}E_1(w),\ldots ,\mathrm e^{\mathrm is_ n'/2}E_ n(w))$, the rotation invariance of $\varphi $ makes $\varphi \circ E$ invariant under imaginary translations. Proposition 21.2.8 now gives a finite convex function $f$ with $\varphi (E(w))=f(\operatorname{Re}w)$, which is the asserted formula.
Conversely, let $f$ be finite and convex. Then $f(\operatorname{Re}w)$ is plurisubharmonic on the tube by proposition 21.2.8. Near each point of $\Omega _ V$, the map $E$ has the holomorphic local inverse $z\mapsto (2\log z_1,\ldots ,2\log z_ n)$, defined with local branches of the logarithm, and $f(\log |z_1|^2,\ldots ,\log |z_ n|^2)$ is the composition of $f(\operatorname{Re}w)$ with this inverse. By proposition 5.9.3 it is plurisubharmonic near that point, since it is finite. Plurisubharmonicity is a local property, so the function is plurisubharmonic on $\Omega _ V$, and it is $(S^1)^ n$-invariant because it depends only on $|z_1|,\ldots ,|z_ n|$. The domains $(\Delta ^*)^ n$ and $(\mathbb C^*)^ n$ are $\Omega _ V$ for $V=(-\infty ,0)^ n$ and $V=\mathbb R^ n$.
$\square$
Pome
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