The Ricci curvature of symplectic quotients of Fano manifolds
type: Item
instance of: scholarly article
title: The Ricci curvature of symplectic quotients of Fano manifolds
zbMATH Open document ID: 0629.53057
author: Akito Futaki
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1987-01-01T00:00:00Z
review text: Let M be a Kähler manifold on which a compact connected Lie group K acts as a group of holomorphic isometries. Let k be the Lie algebra of K. If \(\mu\) : \(X\to k^*\) is a momentum mapping and K acts freely on \(\mu^{-1}(0)\) then the symplectic quotient (or Marsden-Weinstein reduction) \(M_ K=\mu^{-1}(0)/K\) has an induced Kähler structure. In this paper a formula is given for the Ricci curvature of \(M_ K\) in terms of the Ricci curvature of M. In particular it is shown that if M is a Fano manifold (i.e. a compact complex manifold of positive first Chern class) then so is the quotient \(M_ K\). When M is a compact Kähler- Einstein manifold of positive Ricci curvature a necessary and sufficient condition is given for \(M_ K\) to be Kähler-Einstein.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32Q99
zbMATH DE Number: 4024241
zbMATH Keywords: Kähler manifold
zbMATH Keywords: symplectic quotient
zbMATH Keywords: Marsden-Weinstein reduction
zbMATH Keywords: Fano manifold
zbMATH Keywords: Kähler-Einstein manifold
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228281
OpenAlex ID: W2042036551
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DOI: 10.2748/TMJ/1178228281
description: scientific article; zbMATH DE number 4024241