{"title":"Invariant scalar-flat Kähler metrics on line bundles over generalized flag varieties","authors":"Qi Yao","doi":"10.4153/s0008414x24000464","DOIUrl":null,"url":null,"abstract":"<p>Let <span>G</span> be a simply connected semisimple compact Lie group, let <span>X</span> be a simply connected compact Kähler manifold homogeneous under <span>G</span>, and let <span>L</span> be a negative holomorphic line bundle over <span>X</span>. We prove that all <span>G</span>-invariant Kähler metrics on the total space of <span>L</span> arise from the Calabi ansatz. Using this, we show that there exists a unique <span>G</span>-invariant scalar-flat Kähler metric in each <span>G</span>-invariant Kähler class of <span>L</span>. The <span>G</span>-invariant scalar-flat Kähler metrics are automatically asymptotically conical.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4153/s0008414x24000464","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
Let G be a simply connected semisimple compact Lie group, let X be a simply connected compact Kähler manifold homogeneous under G, and let L be a negative holomorphic line bundle over X. We prove that all G-invariant Kähler metrics on the total space of L arise from the Calabi ansatz. Using this, we show that there exists a unique G-invariant scalar-flat Kähler metric in each G-invariant Kähler class of L. The G-invariant scalar-flat Kähler metrics are automatically asymptotically conical.
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