A new holomorphic invariant and uniqueness of Kähler-Ricci solitons
type: Item
instance of: scholarly article
title: A new holomorphic invariant and uniqueness of Kähler-Ricci solitons
zbMATH Open document ID: 1036.53053
author: Xiaohua Zhu
author: Gang Tian
published in: Commentarii Mathematici Helvetici
publication date: 2002-09-25T00:00:00Z
review text: A Kähler metric \(g\) on a compact complex manifold \(M\) is called a Kähler-Ricci soliton if there is a holomorphic vector field \(X\) on \(M\) such that the Kähler form \(\omega_g\) of \(g\) satisfies \(\text{Ric} (\omega_g)-\omega_g=L_{X} \omega_g\), where \(L_X\) denotes the Lie derivative along \(X\). In particular, if \(X=0\), \(g\) is a Kähler-Einstein metric. In this paper, a new holomorphic invariant \(F_{X}(.)\) is introduced for any compact Kähler manifold with positive first Chern class and for any holomorphic vector field \(X\). This invariant generalizes the Futaki invariant; if \(X=0\), the invariant is the Futaki invariant. The authors show that \(F_{X}(.)\) is an obstruction to the existence of Kähler-Ricci solitons just as the Futaki invariant is an obstruction to the existence of Kähler-Einstein metrics. Moreover, they consider a class of the compactifications of \(C^*\)-bundles over compact Kähler-Einstein manifolds, and prove that the vanishing of the holomorphic invariant is equivalent to the existence of Kähler-Ricci solitons on these manifolds. Furthermore, the authors solve completely the uniqueness problem of Kähler-Ricci solitons proving that there is at most one Kähler-Ricci soliton on a compact complex manifold \(M\) modulo the identity component \(\text{Aut}^0(M)\) of the holomorphic automorphisms group \(\text{Aut}(M)\) of \(M\). Such a result extends Bando and Mabuchi's theorem on the uniqueness of Kähler-Einstein metrics on complex manifolds with positive first Chern class.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 58E11
zbMATH DE Number: 1807862
zbMATH Keywords: Futaki invariant
zbMATH Keywords: New holomorphic invariant
zbMATH Keywords: Kähler-Ricci solitons
zbMATH Keywords: holomorphic vector field
reviewed by: Marisa Fernández
MaRDI profile type: Publication
full work available at URL: s00014-002-8341-3
OpenAlex ID: W1984156833
DOI: 10.1007/S00014-002-8341-3
description: scientific article; zbMATH DE number 1807862