On existence of Kähler metrics with constant scalar curvature
The main purpose of this paper is to study classes of manifolds and the corresponding line bundles over these manifolds that can be endowed with a Kähler metric with constant scalar curvature. The author proves that a compact almost-homogeneous Kählerian manifold \(N\), whose maximal compact subgroup \(K \subset \text{Aut}^0 (N)\) acts holomorphically on \(N\) with real cohomogeneity one, admits a Kähler metric with constant positive scalar curvature if any (finite-dimensional) representation of \(\text{Aut}^0 (N)\) is completely reducible. He studies a Kählerian compactification \(\widehat {L}\) of \(L^0\) \((L \setminus\) zero section), where \((L,h)\) is a line bundle over a product of Ricci-positive Einstein-Kähler manifolds with \(b_2 = 1\). He proves that if \(\widehat {L} = {\mathbf P} (L \oplus 1)\) and \(c_1 (L.h)\) is neither definite nor semidefinite, then \(\widehat {L}\) admits a Kähler metric with constant scalar curvature. The author obtains some classes of manifolds as well as the corresponding line bundles over these manifolds which admit an extremal metric in each Kähler class. An extremal Kähler metric is a critical point of the functional \(\Phi_\Omega\) that assigns to a Kähler form the square of the \(L^2\)-norm of the scalar curvature. Any Kähler metric with constant scalar curvature is extremal.
scientific article; zbMATH DE number 749316