Examples of projective manifolds not admitting Kähler metric with constant scalar curvature
For \(P^ k=complex\) projective space of (complex) dimension k, \(P^ m\times P^ n\) (\(m\leq n)\) is imbedded in \(P^{mn+m+n}\) by the canonical Segre imbedding. Let \(M_{m,n}\) be a non-singular hyperplane section of \(P^ m\times P^ n\) in \(P^{mn+m+n}\). The author shows: \(M_{m,n}\), \(m<n\), does not admit a Kähler metric with constant scalar curvature; \(M_{m,m}\) admits a homogeneous Einstein-Kähler metric, hence a Kähler metric with constant scalar curvature.
scientific article; zbMATH DE number 3859937