Algebraic varieties on which the classical Phragmén-Lindelöf estimates hold for plurisubharmonic functions
An algebraic variety \(V\) in \(\mathbb{C}^n\), \(n\geq 2\), is said to satisfy the property (SPL) of Phragmén-Lindelöf type if there exists \(A\geq 1\) such that each plurisubharmonic function \(u\) on \(V\) which satisfies \[ u(z)\leq|z|+ o\bigl(|z|\bigr),\;z\in V\text{ and }u(z)\leq 0,\quad z\in V\cap \mathbb{R}^n \] also satisfies \[ u(z)\leq A|\text{Im} z|,\;z\in V. \] For a given homogeneous polynomial \(P_m\in \mathbb{C}[z_1, \dots,z_n]\) of degree \(m\geq 2\) it is investigated when \[ W:=\bigl\{ (z',z_{n+1}) \in\mathbb{C}^{n+1} :P_m(z')-z_{n+1} =0\bigr\} \] satisfies (SPL). By \textit{R. Meise} and \textit{B. A. Taylor} [Result. Math. 36, No. 1, 121-148 (1989)] it is necessary for \(W\) to satisfy (SPL) that \(P_m\) has real coefficients and is square-free. In the present paper it is shown that even more is necessary: For each \(\Theta\in V(P_m)\cap \mathbb{R}^n\), \(|\Theta |=1\), the localization \((P_m)_\Theta\) of \(P_m\) at \(\Theta\) has to be square-free. \(((P_m)_\Theta\) is the lowest order homogeneous polynomial in the Taylor expansion of \(z\mapsto P_m(z+\Theta))\). As main result it is then proved that for \(n=3\), the variety \(W\) satisfies (SPL) if and only if \(P_m\) has real coefficients, no elliptic factor, is locally hyperbolic in all real characteristics, and the localizations of \(P_m\) in these characteristics are square-free.
scientific article; zbMATH DE number 1351163