Actions of special unitary groups on a product of complex projective spaces
type: Item
instance of: scholarly article
title: Actions of special unitary groups on a product of complex projective spaces
zbMATH Open document ID: 0533.57017
author: Fuichi Uchida
published in: Osaka Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let X be a connected closed orientable \(C^{\infty}\) manifold which admits a nontrivial smooth SU(n) action. Suppose \[ H^*(X,{\mathbb{Q}})={\mathbb{Q}}[u,v]/(u^{a+1},v^{b+1}),\quad \deg u=\deg v=2, \] that is, the cohomology ring of X is isomorphic to that of a product \(P_ a(C)\times P_ b(C)\) of complex projective spaces, where \({\mathbb{Q}}\) is the field of rational numbers. For such situation the author shows: Theorem. Suppose \(1\leq b\leq a<n\leq a+b\leq 2n-3\). Then, \(a=n-1\) and X is equivariantly diffeomorphic to \(P_{n-1}(C)\times Y\), where Y is a connected closed orientable manifold whose rational cohomology ring is isomorphic to that of \(P_ b(C)\), and SU(n) acts naturally on \(P_{n- 1}(C)\) and trivially on Y.
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 3846490
zbMATH Keywords: special unitary groups
zbMATH Keywords: complex projective spaces
zbMATH Keywords: actions on cohomology projective spaces
zbMATH Keywords: rational cohomology ring
reviewed by: Karl Heinz Dovermann
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3846490