{"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}
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 $\operatorname {Pic}(X)$.
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.