Bando Futaki invariants and Kähler Einstein metric
type: Item
instance of: scholarly article
title: Bando Futaki invariants and Kähler Einstein metric
zbMATH Open document ID: 0960.53039
author: Naichung Conan Leung
published in: Communications in Analysis and Geometry
publication date: 2001-05-10T00:00:00Z
review text: Narasiham-Seshadri, Donaldson, Uhlenbeck-Yau proved that irreducible holomorphic vector bundles over a Kähler manifold \(X\) are Mumford stable if and only if there exists a Hermitian Einstein metric on \(X\). On a compact Einstein manifold \(M\), the almost Kähler-Einstein metrics are the solutions of: \(\text{Tr}(\omega+ {i\eta\over 2\pi}F)^n= \chi_\eta {\omega^n \over (n-1)!}\), where \(n=\dim M\), \(\eta>0\), \(\omega\) is the Kähler form and \(F\) is the full curvature tensor of \(M\). When \(M\) has positive first Chern class, the existence of almost Kähler-Einstein metrics is shown here to be equivalent to the vanishing of Futaki-Bando invariants [\textit{S. Bando}, ``An obstruction for Chern class forms to be harmonic'', \textit{A. Futaki}, Invent. Math. 73, 437-443 (1983; Zbl 0514.53054)]. At the end, these invariants are related with the supertrace of the heat kernel.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1267477
zbMATH Keywords: positive first Chern class
zbMATH Keywords: Kähler-Einstein metrics
zbMATH Keywords: Futaki-Bando invariants
reviewed by: Cornelia-Livia Bejan
MaRDI profile type: Publication
full work available at URL: cag.1998.v6.n4.a6
OpenAlex ID: W71886967
DOI: 10.4310/CAG.1998.V6.N4.A6
description: scientific article; zbMATH DE number 1267477