{"title":"Higher Chow cycles on a family of Kummer surfaces","authors":"Ken Sato","doi":"10.4153/s0008414x24000415","DOIUrl":null,"url":null,"abstract":"<p>We construct a collection of families of higher Chow cycles of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918000838399-0173:S0008414X24000415:S0008414X24000415_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(2,1)$</span></span></img></span></span> on a two-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918000838399-0173:S0008414X24000415:S0008414X24000415_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\ge 18$</span></span></img></span></span> in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard–Fuchs differential operators.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-02","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/s0008414x24000415","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We construct a collection of families of higher Chow cycles of type $(2,1)$ on a two-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank $\ge 18$ in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard–Fuchs differential operators.
$(2,1)$ on a two-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank 
$\ge 18$ in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard–Fuchs differential operators.
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