Ricci positive metrics on the moment-angle manifolds
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.
scientific article; zbMATH DE number 7079440