Compositio Mathematica最新文献

{"title":"A Hamiltonian ∐n BO(n)-action, stratified Morse theory and the J-homomorphism","authors":"Xin Jin","doi":"10.1112/s0010437x24007279","DOIUrl":"https://doi.org/10.1112/s0010437x24007279","url":null,"abstract":"<p>We use sheaves of spectra to quantize a Hamiltonian <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114013531-0464:S0010437X24007279:S0010437X24007279_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$coprod _n BO(n)$</span></span></img></span></span>-action on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114013531-0464:S0010437X24007279:S0010437X24007279_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$varinjlim _{N}T^*mathbf {R}^N$</span></span></img></span></span> that naturally arises from Bott periodicity. We employ the category of correspondences developed by Gaitsgory and Rozenblyum [<span>A study in derived algebraic geometry, vol. I. Correspondences and duality</span>, Mathematical Surveys and Monographs, vol. 221 (American Mathematical Society, 2017)] to give an enrichment of stratified Morse theory by the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114013531-0464:S0010437X24007279:S0010437X24007279_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$J$</span></span></img></span></span>-homomorphism. This provides a key step in the work of Jin [<span>Microlocal sheaf categories and the</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114013531-0464:S0010437X24007279:S0010437X24007279_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$J$</span></span></img></span></span><span>-homomorphism</span>, Preprint (2020), arXiv:2004.14270v4] on the proof of a claim of Jin and Treumann [<span>Brane structures in microlocal sheaf theory</span>, J. Topol. <span>17</span> (2024), e12325]: the classifying map of the local system of brane structures on an (immersed) exact Lagrangian submanifold <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114013531-0464:S0010437X24007279:S0010437X24007279_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$Lsubset T^*mathbf {R}^N$</span></span></img></span></span> is given by the composition of the stable Gauss map <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114013531-0464:S0010437X24007279:S0010437X24007279_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$Lrightarrow U/O$</span></span></img></span></span> and the delooping of the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912114013531-0464:S0010437X24007279:S0010437X24007279_inline12.png\"><span data-mathjax-type=\"texmath\"><span>$J$</span></span></img></span></span>-homomorphism <sp","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved algebraic fibrings","authors":"Sam P. Fisher","doi":"10.1112/s0010437x24007309","DOIUrl":"https://doi.org/10.1112/s0010437x24007309","url":null,"abstract":"<p>We show that a virtually residually finite rationally solvable (RFRS) group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathtt {FP}_n(mathbb {Q})$</span></span></img></span></span> virtually algebraically fibres with kernel of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathtt {FP}_n(mathbb {Q})$</span></span></img></span></span> if and only if the first <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$ell ^2$</span></span></img></span></span>-Betti numbers of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> vanish, that is, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$b_p^{(2)}(G) = 0$</span></span></img></span></span> for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$0 leqslant p leqslant n$</span></span></img></span></span>. This confirms a conjecture of Kielak. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that amenable virtually RFRS groups of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911163557241-0452:S0010437X24007309:S0010437X24007309_inline9.png\"><span data-","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Gross–Prasad conjecture with its refinement for (SO(5), SO(2)) and the generalized Böcherer conjecture","authors":"Masaaki Furusawa, Kazuki Morimoto","doi":"10.1112/s0010437x24007267","DOIUrl":"https://doi.org/10.1112/s0010437x24007267","url":null,"abstract":"<p>We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912063150983-0859:S0010437X24007267:S0010437X24007267_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$(mathrm {SO}(5), mathrm {SO}(2))$</span></span></img></span></span>. In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240912063150983-0859:S0010437X24007267:S0010437X24007267_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$L$</span></span></img></span></span>-functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cohomological and motivic inclusion–exclusion","authors":"Ronno Das, Sean Howe","doi":"10.1112/s0010437x24007292","DOIUrl":"https://doi.org/10.1112/s0010437x24007292","url":null,"abstract":"<p>We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cyclic base change of cuspidal automorphic representations over function fields","authors":"Gebhard Böckle, Tony Feng, Michael Harris, Chandrashekhar B. Khare, Jack A. Thorne","doi":"10.1112/s0010437x24007243","DOIUrl":"https://doi.org/10.1112/s0010437x24007243","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$G$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; be a split semisimple group over a global function field &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. Given a cuspidal automorphic representation &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$Pi$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$G$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; satisfying a technical hypothesis, we prove that for almost all primes &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$ell$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, there is a cyclic base change lifting of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$Pi$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; along any &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$mathbb {Z}/ell mathbb {Z}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;-extension of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_inline8.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$K$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910160619904-0480:S0010437X24007243:S0010437X24007243_in","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The -invariant over splitting fields of Tits algebras","authors":"Maksim Zhykhovich","doi":"10.1112/s0010437x24007255","DOIUrl":"https://doi.org/10.1112/s0010437x24007255","url":null,"abstract":"<p>We describe the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155753172-0985:S0010437X24007255:S0010437X24007255_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$J$</span></span></img></span></span>-invariant of a semisimple algebraic group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155753172-0985:S0010437X24007255:S0010437X24007255_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> over a generic splitting field of a Tits algebra of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155753172-0985:S0010437X24007255:S0010437X24007255_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> in terms of the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155753172-0985:S0010437X24007255:S0010437X24007255_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$J$</span></span></img></span></span>-invariant over the base field. As a consequence we prove a 10-year-old conjecture of Quéguiner-Mathieu, Semenov, and Zainoulline on the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155753172-0985:S0010437X24007255:S0010437X24007255_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$J$</span></span></img></span></span>-invariant of groups of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155753172-0985:S0010437X24007255:S0010437X24007255_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {D}_n$</span></span></img></span></span>. In the case of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155753172-0985:S0010437X24007255:S0010437X24007255_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {D}_n$</span></span></img></span></span> we also provide explicit formulas for the first component and in some cases for the second component of the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240910155753172-0985:S0010437X24007255:S0010437X24007255_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$J$</span></span></img></span></span>-invariant.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Connes fusion of spinors on loop space","authors":"Peter Kristel, Konrad Waldorf","doi":"10.1112/s0010437x24007188","DOIUrl":"https://doi.org/10.1112/s0010437x24007188","url":null,"abstract":"<p>The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"BPS invariants from p-adic integrals","authors":"Francesca Carocci, Giulio Orecchia, Dimitri Wyss","doi":"10.1112/s0010437x24007176","DOIUrl":"https://doi.org/10.1112/s0010437x24007176","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>$pmathrm {BPS}$</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>$pmathrm {BPS}$</span></span></img></span></span> invariants are integrals of natural <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"2 n2-inequality for cA1 points and applications to birational rigidity","authors":"Igor Krylov, Takuzo Okada, Erik Paemurru, Jihun Park","doi":"10.1112/s0010437x24007164","DOIUrl":"https://doi.org/10.1112/s0010437x24007164","url":null,"abstract":"<p>The <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$4 n^2$</span></span></img></span></span>-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$cA_1$</span></span></img></span></span>, and obtain a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$2 n^2$</span></span></img></span></span>-inequality for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$cA_1$</span></span></img></span></span> points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$1$</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:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {P}^1$</span></span></img></span></span> satisfying the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$K^2$</span></span></img></span></span>-condition, all of which have at most terminal <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$cA_1$</span></span></img></span></span> singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://s","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141169625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"COM volume 160 issue 6 Cover and Front matter","authors":"","doi":"10.1112/s0010437x23007832","DOIUrl":"https://doi.org/10.1112/s0010437x23007832","url":null,"abstract":"","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141108039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}