On existence of Kähler metrics with constant scalar curvature
type: Item
instance of: scholarly article
title: On existence of Kähler metrics with constant scalar curvature
zbMATH Open document ID: 0828.53057
author: Andrew D. Hwang
published in: Osaka Journal of Mathematics
publication date: 1995-05-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 749316
zbMATH Keywords: generalized Einstein-Kähler metric
zbMATH Keywords: Chern class
zbMATH Keywords: constant scalar curvature
zbMATH Keywords: extremal Kähler metric
reviewed by: Neda Bokan
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 749316