PomeCorollary 6.3.6.EC970F3402EB Let $m,k\ge 1$, and let $\Omega \subset \mathbb C^ m\times \mathbb C^ k$ be a pseudoconvex domain that is invariant under imaginary translations and whose nonempty real fibers $V_ z$ are convex, in the notation of section 5.12. If $u\in \mathrm{PSH}(\Omega )$ is invariant under imaginary translations, then the function $v(z)=\inf _{t\in V_ z}u(z,t)$ is plurisubharmonic or identically $-\infty $ on $p(\Omega )$.
Corollary 6.3.6EC970F3402EB
Proof. We construct a function $\psi $ with the properties required in theorem 5.12.1; the corollary then follows from that theorem. If $\Omega =\mathbb C^{m+k}$, let $\psi (z,w)=|z|^2+|\operatorname{Re}w|^2$. Otherwise let $d=\operatorname {dist}(\cdot ,\mathbb C^{m+k}\setminus \Omega )$ and put
\[ \psi (z,w)=|z|^2+|\operatorname{Re}w|^2+ \max \{ -\log d(z,w),0\} . \]
Imaginary translations are isometries of $\mathbb C^{m+k}$ that preserve $\Omega $ and its complement, so $d$ and $\psi $ are invariant under them. The function $\psi $ is plurisubharmonic by proposition 5.9.5: $-\log d$ is plurisubharmonic by theorem 6.3.4, and $|z|^2+|\operatorname{Re}w|^2$ is plurisubharmonic by proposition 5.9.2(1), since its complex Hessian matrix is the diagonal matrix with entries $1$ in the $z$-directions and $\frac12$ in the $w$-directions, by the identity $(\operatorname{Re}w_ l)^2=\tfrac 12|w_ l|^2+\tfrac 12\operatorname{Re}(w_ l^2)$ for $1\le l\le k$. Finally, let $c\in \mathbb R$. Since $\psi $ is continuous, the set $\{ (z,t)\in \Omega _{\mathbb R}:\psi (z,t)\le c\} $ is closed in $\Omega _{\mathbb R}$. It is contained in $\{ |z|^2+|t|^2\le c\} $ and, if $\Omega \ne \mathbb C^{m+k}$, in the set $\{ d\ge \mathrm e^{-c}\} $, which is closed in $\mathbb C^{m+k}$ and contained in $\Omega $; it is therefore a closed and bounded subset of $\mathbb C^ m\times \mathbb R^ k$, hence compact. $\square$
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