{"title":"Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one","authors":"DongSeon Hwang, Shin-young Kim, Kyeong-Dong Park","doi":"10.1007/s10455-023-09915-y","DOIUrl":null,"url":null,"abstract":"<div><p>A horospherical variety is a normal <i>G</i>-variety such that a connected reductive algebraic group <i>G</i> acts with an open orbit isomorphic to a torus bundle over a rational homogeneous manifold. The projective horospherical manifolds of Picard number one are classified by Pasquier, and it turned out that the automorphism groups of all nonhomogeneous ones are non-reductive, which implies that they admit no Kähler–Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be Kähler–Einstein, we compute the greatest Ricci lower bounds of projective horospherical manifolds of Picard number one using the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure based on a recent work of Delcroix and Hultgren. In particular, the greatest Ricci lower bound of the odd symplectic Grassmannian <span>\\(\\text {SGr}(n,2n+1)\\)</span> can be arbitrarily close to zero as <i>n</i> grows.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"64 2","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-023-09915-y.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Global Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10455-023-09915-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A horospherical variety is a normal G-variety such that a connected reductive algebraic group G acts with an open orbit isomorphic to a torus bundle over a rational homogeneous manifold. The projective horospherical manifolds of Picard number one are classified by Pasquier, and it turned out that the automorphism groups of all nonhomogeneous ones are non-reductive, which implies that they admit no Kähler–Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be Kähler–Einstein, we compute the greatest Ricci lower bounds of projective horospherical manifolds of Picard number one using the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure based on a recent work of Delcroix and Hultgren. In particular, the greatest Ricci lower bound of the odd symplectic Grassmannian \(\text {SGr}(n,2n+1)\) can be arbitrarily close to zero as n grows.
期刊介绍:
This journal examines global problems of geometry and analysis as well as the interactions between these fields and their application to problems of theoretical physics. It contributes to an enlargement of the international exchange of research results in the field.
The areas covered in Annals of Global Analysis and Geometry include: global analysis, differential geometry, complex manifolds and related results from complex analysis and algebraic geometry, Lie groups, Lie transformation groups and harmonic analysis, variational calculus, applications of differential geometry and global analysis to problems of theoretical physics.