Ricci Positive Metrics on the Moment-Angle Manifolds

Abstract

In this paper, the authors consider the problem of which (generalized) moment-angle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polytope Pv, and prove that if the generalized moment-angle manifold corresponding to P admits a Ricci positive metric, the generalized moment-angle manifold corresponding to Pv also admits a Ricci positive metric. For a special class of polytope called Fano polytopes, the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given.