一阶几何 p-adic Simpson 对应关系

IF 1.3 1区 数学 Q1 MATHEMATICS Compositio Mathematica Pub Date : 2024-05-21 DOI:10.1112/s0010437x24007024
Ben Heuer
{"title":"一阶几何 p-adic Simpson 对应关系","authors":"Ben Heuer","doi":"10.1112/s0010437x24007024","DOIUrl":null,"url":null,"abstract":"<p>For any smooth proper rigid space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$X$</span></span></img></span></span> over a complete algebraically closed extension <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$K$</span></span></img></span></span> of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb {Q}_p$</span></span></img></span></span> we give a geometrisation of the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adic Simpson correspondence of rank one in terms of analytic moduli spaces: the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$v$</span></span></img></span></span>-line bundles. As an application, we study a major open question in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$\\pi ^{{\\mathrm {\\acute {e}t}}}_1(X)\\to K^\\times$</span></span></img></span></span> in terms of moduli spaces: for projective <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$X$</span></span></img></span></span> over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline12.png\"/><span data-mathjax-type=\"texmath\"><span>$K=\\mathbb {C}_p$</span></span></span></span>, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline13.png\"/><span data-mathjax-type=\"texmath\"><span>$\\operatorname {Pic}(X)$</span></span></span></span>.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A geometric p-adic Simpson correspondence in rank one\",\"authors\":\"Ben Heuer\",\"doi\":\"10.1112/s0010437x24007024\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For any smooth proper rigid space <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$X$</span></span></img></span></span> over a complete algebraically closed extension <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$K$</span></span></img></span></span> of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbb {Q}_p$</span></span></img></span></span> we give a geometrisation of the <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span></img></span></span>-adic Simpson correspondence of rank one in terms of analytic moduli spaces: the <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline6.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span></img></span></span>-adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline7.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$v$</span></span></img></span></span>-line bundles. As an application, we study a major open question in <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline8.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span></img></span></span>-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline9.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span></img></span></span>-adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline10.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\pi ^{{\\\\mathrm {\\\\acute {e}t}}}_1(X)\\\\to K^\\\\times$</span></span></img></span></span> in terms of moduli spaces: for projective <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline11.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$X$</span></span></img></span></span> over <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline12.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$K=\\\\mathbb {C}_p$</span></span></span></span>, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240520155950870-0374:S0010437X24007024:S0010437X24007024_inline13.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\operatorname {Pic}(X)$</span></span></span></span>.</p>\",\"PeriodicalId\":55232,\"journal\":{\"name\":\"Compositio Mathematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-05-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Compositio Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1112/s0010437x24007024\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Compositio Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/s0010437x24007024","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

对于在$\mathbb {Q}_p$ 的完整代数封闭扩展$K$上的任何光滑适当刚性空间$X$,我们给出了秩为一的$p$-adic Simpson对应的解析模空间的几何解析:$p$-adic character variety典型地是希钦基上拓扑扭转希格斯线束的模空间的褶曲。这也消除了指数的选择。关键的思路是把两边都与 $v$ 线束的模空间联系起来。作为应用,我们研究了法尔廷斯提出的p$-adic非阿贝尔霍奇理论中的一个主要未决问题,即哪些希格斯束对应于p$-adic辛普森对应下的连续表示。我们通过描述连续字符 $\pi ^{\mathrm {\acute {e}t}}_1(X)\to K^\times$ 在模空间方面的本质映像来回答这个问题:对于在 $K=\mathbb {C}_p$ 上的投影 $X$,它是由希格斯线束给出的,就像复几何学中的奇恩类消失一样。然而,一般来说,正确的条件是更严格的假设,即底层线束是拓扑群 $\operatorname {Pic}(X)$ 中的拓扑扭转元素。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
A geometric p-adic Simpson correspondence in rank one

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb {Q}_p$ we give a geometrisation of the $p$-adic Simpson correspondence of rank one in terms of analytic moduli spaces: the $p$-adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$-line bundles. As an application, we study a major open question in $p$-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the $p$-adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $\pi ^{{\mathrm {\acute {e}t}}}_1(X)\to K^\times$ in terms of moduli spaces: for projective $X$ over $K=\mathbb {C}_p$, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group $\operatorname {Pic}(X)$.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Compositio Mathematica
Compositio Mathematica 数学-数学
CiteScore
2.10
自引率
0.00%
发文量
62
审稿时长
6-12 weeks
期刊介绍: Compositio Mathematica is a prestigious, well-established journal publishing first-class research papers that traditionally focus on the mainstream of pure mathematics. Compositio Mathematica has a broad scope which includes the fields of algebra, number theory, topology, algebraic and differential geometry and global analysis. Papers on other topics are welcome if they are of broad interest. All contributions are required to meet high standards of quality and originality. The Journal has an international editorial board reflected in the journal content.
期刊最新文献
Cohomological and motivic inclusion–exclusion Improved algebraic fibrings On the Gross–Prasad conjecture with its refinement for (SO(5), SO(2)) and the generalized Böcherer conjecture A Hamiltonian ∐n BO(n)-action, stratified Morse theory and the J-homomorphism The -invariant over splitting fields of Tits algebras
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1