Ricci positive metrics on the moment-angle manifolds
type: Item
instance of: scholarly article
title: Ricci positive metrics on the moment-angle manifolds
zbMATH Open document ID: 1445.53025
author: Liman Chen
author: Feifei Fan
published in: Chinese Annals of Mathematics. Series B
publication date: 2019-07-11T00:00:00Z
review text: The study of moment-angle manifolds arises from two sources: complexe quadrics and simple polytopes. In the second case questions of cohomology and geometrical aspects of these manifolds are of interest; one particular metric property is their Ricci curvature. But how does this curvature change if the polytope is altered? By the Euler characteristic each simple 3-dimensional polyhedron can be derived from the tetrahedron by only three processes: cutting off a vertex, cutting off an edge and cutting of two adjacent vertices. The authors consider the first process in the context of (generalized) moment-angle manifolds. Which of them admits positive metrics Ricci? Therefore they start with a simple polytope \(P\) and cut off one vertex \(v\) of \(P\) to get another simple polytope \(P_v\). Then they prove that if the generalized moment-angle manifold corresponding to \(P\) admits a Ricci positive metric, the generalized moment-angle manifold corresponding to \(P_v\) also admits a Ricci positive metric. The proof is based on the result in [\textit{K. Iriye}, Osaka J. Math. 55, No. 4, 587--593 (2018; Zbl 1408.57029)]. The tools of the proof are mainly the connected sum, diffeomorphism on manifolds and some particular embeddings with their formula. In the last chapter Fano polytopes are studied. These polytopes have only vertices that are vertices of primitive lattice vectors. Of course they contain the origin in the interior and further restrictions on the volume of the facets can be given. The authors prove for these that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given.
reviewed by: Lienhard Wimmer
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 52B70
zbMATH DE Number: 7079440
zbMATH Keywords: positive Ricci curvature
zbMATH Keywords: simple polytope
zbMATH Keywords: moment-angle manifolds
zbMATH Keywords: Fano polytope
zbMATH Keywords: cutting off face
MaRDI profile type: Publication
full work available at URL: s11401-019-0145-5
OpenAlex ID: W2943054556
cites work: On the moment-angle manifolds of positive Ricci curvature
cites work: Q4537714
cites work: Toric varieties
cites work: Metric deformations of curvature. I: Local convex deformations
cites work: Invariant metrics of positive Ricci curvature on principal bundles
cites work: Intersections of quadrics, moment-angle manifolds and connected sums
cites work: The classification of simply connected manifolds of positive scalar curvature
cites work: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds
cites work: A new family of complex, compact, non-symplectic manifolds
cites work: A new geometric construction of compact complex manifolds in any dimension
cites work: Holomorphic principal bundles over projective toric varieties
cites work: On the structure of manifolds with positive scalar curvature
cites work: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature
cites work: Positive Ricci curvature on the connected sums of \(S^ n\times S^ m\)
cites work: New connected sums with positive Ricci curvature
cites work: On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I
Wikidata QID: Q127951832
DOI: 10.1007/S11401-019-0145-5
description: scientific article; zbMATH DE number 7079440