Monge-Ampère exhaustions of almost homogeneous manifolds (Q1654925)

The authors study some classes of complex manifolds \(M\) with a real Lie group \(G\) of biholomorphisms acting with real hypersurfaces as principal orbits (i.e., almost homogeneous complex manifolds). They prove that each such manifold \(M\) has two singular \(G\)-orbits and at least one of them is complex. If \(S\) is such a complex orbit, then \(M_{\mathrm o}:=M\setminus S\) possesses a \(\mathcal C^\infty\)-exhaustion function \(\tau:M_{\mathrm o}\longrightarrow[0,+\infty)\) such that the set \(\{\tau=0\}\) is the other singular \(G\)-orbit \(S_{\mathrm o}\) of \(M\) and whose restriction to \(M_{\mathrm o}\setminus S_o\) is such that \(2i\partial\overline\partial\tau>0\) and there exists a smooth function \(f:(0,+\infty)\longrightarrow\mathbb R\), \(df\neq0\), such that \((\partial\overline\partial(f\circ\tau))^n=0\).