{"title":"Projectively induced Kähler cones over regular Sasakian manifolds","authors":"Stefano Marini, Nicoletta Tardini, Michela Zedda","doi":"10.1007/s10711-024-00935-x","DOIUrl":null,"url":null,"abstract":"<p>Motivated by a conjecture in Loi et al. (Math Zeit 290:599–613, 2018) we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to <span>\\(\\mathcal D_a\\)</span>—homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10711-024-00935-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Motivated by a conjecture in Loi et al. (Math Zeit 290:599–613, 2018) we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to \(\mathcal D_a\)—homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric.
受Loi等人（Math Zeit 290:599-613，2018）中一个猜想的启发，我们证明了规则完整萨萨基流形上的凯勒锥是里奇平坦的，并且只有当它是平坦的时候，它才是投影诱导的。我们还得到，根据同调变换，同质紧凑萨萨基流形上的凯勒锥是投影诱导的。作为主要工具，我们提供了横向凯勒度量的凯勒势与锥形度量的凯勒势之间的关系。
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