Stein manifolds with compact symmetric center
Let M be a complex manifold of dimension n with a strictly plurisubharmonic exhaustion of class \(C^{\infty}\), \(\rho\) : \(M\to [1,\infty]\), such that \(u=\cosh^{-1}\rho\) is a plurisubharmonic function satisfying the homogeneous Monge-Ampère equation \((dd^ cu)^ n=0\) on \(M-\{\rho =1\}.\) Assume that the center \(S=\{\rho =1\}\), with the metric induced by \(dd^ c\rho\), is isomorphic to a compact symmetric space of rank one. The authors prove that M is biholomorphically isometric to \(M_ 0\) with the Kähler metric \(dd^ c\rho_ 0\), is one of the Stein manifolds \(M^ n_ I\), \(M^ n_{II}\), \(M^{2n}_{III}\), \(M^{4n}_{IV}\) and \(M_ V^{16}\), which correspond, respectively, to the n-sphere, the real projective n- space (n\(\geq 2)\), the complex projective n-space, the quaternionic projective n-space (n\(\geq 1)\) and the Cayley projective plane \(P^ 2\). There is a biholomorphic map F: \(M_ 0\to M\) with \(\rho \circ F=\rho_ 0\).
scientific article; zbMATH DE number 4204784
scientific article