{"title":"Weil型的四重与旋量映射","authors":"Bert van Geemen","doi":"10.1016/j.exmath.2023.04.006","DOIUrl":null,"url":null,"abstract":"<div><p>Recent papers by Markman and O’Grady give, besides their main results on the Hodge conjecture and on hyperkähler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.</p><p>In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of <span><math><mrow><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow></math></span>, is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fourfolds of Weil type and the spinor map\",\"authors\":\"Bert van Geemen\",\"doi\":\"10.1016/j.exmath.2023.04.006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Recent papers by Markman and O’Grady give, besides their main results on the Hodge conjecture and on hyperkähler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.</p><p>In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of <span><math><mrow><mi>S</mi><mi>O</mi><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow></math></span>, is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.</p></div>\",\"PeriodicalId\":50458,\"journal\":{\"name\":\"Expositiones Mathematicae\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2023-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Expositiones Mathematicae\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0723086923000427\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Expositiones Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0723086923000427","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Recent papers by Markman and O’Grady give, besides their main results on the Hodge conjecture and on hyperkähler varieties, surprising and explicit descriptions of families of abelian fourfolds of Weil type with trivial discriminant. They also provide a new perspective on the well-known fact that these abelian varieties are Kuga Satake varieties for certain weight two Hodge structures of rank six.
In this paper we give a pedestrian introduction to these results. The spinor map, which is defined using a half-spin representation of , is used intensively. For simplicity, we use basic representation theory and we avoid the use of triality.
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