{"title":"从 p-二次积分看 BPS 不变量","authors":"Francesca Carocci, Giulio Orecchia, Dimitri Wyss","doi":"10.1112/s0010437x24007176","DOIUrl":null,"url":null,"abstract":"<p>We define <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathrm {BPS}$</span></span></img></span></span> or <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$p\\mathrm {BPS}$</span></span></img></span></span> invariants for moduli spaces <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$\\operatorname {M}_{\\beta,\\chi }$</span></span></img></span></span> of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$F$</span></span></img></span></span>. Our definition relies on a canonical measure <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$\\mu _{\\rm can}$</span></span></img></span></span> on the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$F$</span></span></img></span></span>-analytic manifold associated to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$\\operatorname {M}_{\\beta,\\chi }$</span></span></img></span></span> and the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$p\\mathrm {BPS}$</span></span></img></span></span> invariants are integrals of natural <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline12.png\"><span data-mathjax-type=\"texmath\"><span>${\\mathbb {G}}_m$</span></span></img></span></span> gerbes with respect to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline14.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mu _{\\rm can}$</span></span></span></span>. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline15.png\"/><span data-mathjax-type=\"texmath\"><span>$\\chi$</span></span></span></span>-independence result for these <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline16.png\"/><span data-mathjax-type=\"texmath\"><span>$p\\mathrm {BPS}$</span></span></span></span> invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline17.png\"/><span data-mathjax-type=\"texmath\"><span>$p\\mathrm {BPS}$</span></span></span></span> with usual <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline18.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mathrm {BPS}$</span></span></span></span> invariants through a result of Maulik and Shen [<span>Cohomological</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline19.png\"/><span data-mathjax-type=\"texmath\"><span>$\\chi$</span></span></span></span><span>-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles</span>, Geom. Topol. <span>27</span> (2023), 1539–1586].</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"BPS invariants from p-adic integrals\",\"authors\":\"Francesca Carocci, Giulio Orecchia, Dimitri Wyss\",\"doi\":\"10.1112/s0010437x24007176\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We define <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span></img></span></span>-adic <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm {BPS}$</span></span></img></span></span> or <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\mathrm {BPS}$</span></span></img></span></span> invariants for moduli spaces <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\operatorname {M}_{\\\\beta,\\\\chi }$</span></span></img></span></span> of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline6.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$F$</span></span></img></span></span>. Our definition relies on a canonical measure <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline8.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mu _{\\\\rm can}$</span></span></img></span></span> on the <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline9.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$F$</span></span></img></span></span>-analytic manifold associated to <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline10.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\operatorname {M}_{\\\\beta,\\\\chi }$</span></span></img></span></span> and the <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline11.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\mathrm {BPS}$</span></span></img></span></span> invariants are integrals of natural <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline12.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>${\\\\mathbb {G}}_m$</span></span></img></span></span> gerbes with respect to <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline14.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mu _{\\\\rm can}$</span></span></span></span>. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline15.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\chi$</span></span></span></span>-independence result for these <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline16.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\mathrm {BPS}$</span></span></span></span> invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline17.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\mathrm {BPS}$</span></span></span></span> with usual <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline18.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm {BPS}$</span></span></span></span> invariants through a result of Maulik and Shen [<span>Cohomological</span> <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline19.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\chi$</span></span></span></span><span>-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles</span>, Geom. Topol. <span>27</span> (2023), 1539–1586].</p>\",\"PeriodicalId\":55232,\"journal\":{\"name\":\"Compositio Mathematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-05-30\",\"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/s0010437x24007176\",\"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/s0010437x24007176","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
We define $p$-adic $\mathrm {BPS}$ or $p\mathrm {BPS}$ invariants for moduli spaces $\operatorname {M}_{\beta,\chi }$ of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field $F$. Our definition relies on a canonical measure $\mu _{\rm can}$ on the $F$-analytic manifold associated to $\operatorname {M}_{\beta,\chi }$ and the $p\mathrm {BPS}$ invariants are integrals of natural ${\mathbb {G}}_m$ gerbes with respect to $\mu _{\rm can}$. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a $\chi$-independence result for these $p\mathrm {BPS}$ invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of $p\mathrm {BPS}$ with usual $\mathrm {BPS}$ invariants through a result of Maulik and Shen [Cohomological$\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539–1586].
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.