Pub Date : 2023-12-21DOI: 10.1017/s1474748023000506
Daniel Corey, Jordan Ellenberg, Wanlin Li
The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for nonhyperelliptic curves is in general not easy. We present a ‘combinatorialization’ of this problem, explaining how to define a Ceresa class for a tropical algebraic curve and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over $mathbb {C}(!(t)!)$ and show that the Ceresa class in each of these settings is torsion.
{"title":"THE CERESA CLASS: TROPICAL, TOPOLOGICAL AND ALGEBRAIC","authors":"Daniel Corey, Jordan Ellenberg, Wanlin Li","doi":"10.1017/s1474748023000506","DOIUrl":"https://doi.org/10.1017/s1474748023000506","url":null,"abstract":"<p>The <span>Ceresa cycle</span> is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the <span>Ceresa class</span>. Describing the Ceresa class explicitly for nonhyperelliptic curves is in general not easy. We present a ‘combinatorialization’ of this problem, explaining how to define a Ceresa class for a tropical algebraic curve and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220151957867-0477:S1474748023000506:S1474748023000506_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {C}(!(t)!)$</span></span></img></span></span> and show that the Ceresa class in each of these settings is torsion.</p>","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"32 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138825621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-13DOI: 10.1017/s1474748023000427
Arno Kret, Sug Woo Shin
We prove the existence of $mathrm {GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of ${mathrm {GSO}}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{mathbb {H}}$, arising from forms of ${mathrm {GSO}}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin L-functions and improve on the construction of ${mathrm {SO}}_{2n}$-valued Galois representations by removing the outer automorphism ambiguity.
{"title":"GALOIS REPRESENTATIONS FOR EVEN GENERAL SPECIAL ORTHOGONAL GROUPS","authors":"Arno Kret, Sug Woo Shin","doi":"10.1017/s1474748023000427","DOIUrl":"https://doi.org/10.1017/s1474748023000427","url":null,"abstract":"<p>We prove the existence of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212153014787-0742:S1474748023000427:S1474748023000427_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {GSpin}_{2n}$</span></span></img></span></span>-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212153014787-0742:S1474748023000427:S1474748023000427_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathrm {GSO}}_{2n}$</span></span></img></span></span> under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212153014787-0742:S1474748023000427:S1474748023000427_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$D^{mathbb {H}}$</span></span></img></span></span>, arising from forms of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212153014787-0742:S1474748023000427:S1474748023000427_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathrm {GSO}}_{2n}$</span></span></img></span></span>. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin <span>L</span>-functions and improve on the construction of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212153014787-0742:S1474748023000427:S1474748023000427_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathrm {SO}}_{2n}$</span></span></img></span></span>-valued Galois representations by removing the outer automorphism ambiguity.</p>","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"23 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138578966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-13DOI: 10.1017/s1474748023000476
Rahul Dalal
Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $mathrm {GL}_2$. Here, we use ‘hyperendoscopy’ techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$, eventually getting an analog of the Eichler–Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard and Taïbi to compute dimensions of spaces of level-$1$ quaternionic representations. On the way, we prove a Jacquet–Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$.
The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being ‘regular’. A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series.
We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.
{"title":"COUNTING DISCRETE, LEVEL-","authors":"Rahul Dalal","doi":"10.1017/s1474748023000476","DOIUrl":"https://doi.org/10.1017/s1474748023000476","url":null,"abstract":"<p>Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212120545961-0220:S1474748023000476:S1474748023000476_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {GL}_2$</span></span></img></span></span>. Here, we use ‘hyperendoscopy’ techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212120545961-0220:S1474748023000476:S1474748023000476_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G_2$</span></span></img></span></span>, eventually getting an analog of the Eichler–Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard and Taïbi to compute dimensions of spaces of level-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212120545961-0220:S1474748023000476:S1474748023000476_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$1$</span></span></img></span></span> quaternionic representations. On the way, we prove a Jacquet–Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231212120545961-0220:S1474748023000476:S1474748023000476_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$G_2^c$</span></span></img></span></span>.</p><p>The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being ‘regular’. A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series.</p><p>We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.</p>","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"91 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-04DOI: 10.1017/s1474748023000464
Martin Bays, Omer Ben-Neria, Itay Kaplan, Pierre Simon
We study cofinal systems of finite subsets of $omega _1$ . We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.
{"title":"ON LARGE EXTERNALLY DEFINABLE SETS IN NIP","authors":"Martin Bays, Omer Ben-Neria, Itay Kaplan, Pierre Simon","doi":"10.1017/s1474748023000464","DOIUrl":"https://doi.org/10.1017/s1474748023000464","url":null,"abstract":"We study cofinal systems of finite subsets of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000464_inline1.png\" /> <jats:tex-math> $omega _1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"23 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.1017/s1474748023000403
Hélène Eynard-Bontemps, Andrés Navas
We study conjugacy classes of germs of nonflat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This may be seen as an extension of the fact (also proved in this article) that the value of the Schwarzian derivative at the origin for germs of $C^3$ parabolic diffeomorphisms is invariant under $C^2$ parabolic conjugacy, though it may vary arbitrarily under parabolic $C^1$ conjugacy.
{"title":"ON RESIDUES AND CONJUGACIES FOR GERMS OF 1-D PARABOLIC DIFFEOMORPHISMS IN FINITE REGULARITY","authors":"Hélène Eynard-Bontemps, Andrés Navas","doi":"10.1017/s1474748023000403","DOIUrl":"https://doi.org/10.1017/s1474748023000403","url":null,"abstract":"We study conjugacy classes of germs of nonflat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This may be seen as an extension of the fact (also proved in this article) that the value of the Schwarzian derivative at the origin for germs of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000403_inline1.png\" /> <jats:tex-math> $C^3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> parabolic diffeomorphisms is invariant under <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000403_inline2.png\" /> <jats:tex-math> $C^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> parabolic conjugacy, though it may vary arbitrarily under parabolic <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000403_inline3.png\" /> <jats:tex-math> $C^1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> conjugacy.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"81 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-28DOI: 10.1017/s1474748023000440
Marek Cúth, Martin Doležal, Michal Doucha, Ondřej Kurka
<jats:p>We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces.</jats:p> <jats:p>We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748023000440_inline1.png" /> <jats:tex-math> $F_sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. For <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748023000440_inline2.png" /> <jats:tex-math> $pin left [1,2right )cup left (2,infty right )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that the isometry classes of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748023000440_inline3.png" /> <jats:tex-math> $L_p[0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748023000440_inline4.png" /> <jats:tex-math> $ell _p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748023000440_inline5.png" /> <jats:tex-math> $G_delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete sets and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748023000440_inline6.png" /> <jats:tex-math> $F_{sigma delta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete sets, respectively. Then we show that the isometry class of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748023000440_inline7.png" /> <jats:tex-math> $c_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S1474748023000440_inline8.png" />
{"title":"POLISH SPACES OF BANACH SPACES: COMPLEXITY OF ISOMETRY AND ISOMORPHISM CLASSES","authors":"Marek Cúth, Martin Doležal, Michal Doucha, Ondřej Kurka","doi":"10.1017/s1474748023000440","DOIUrl":"https://doi.org/10.1017/s1474748023000440","url":null,"abstract":"<jats:p>We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces.</jats:p> <jats:p>We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline1.png\" /> <jats:tex-math> $F_sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. For <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline2.png\" /> <jats:tex-math> $pin left [1,2right )cup left (2,infty right )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we show that the isometry classes of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline3.png\" /> <jats:tex-math> $L_p[0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline4.png\" /> <jats:tex-math> $ell _p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline5.png\" /> <jats:tex-math> $G_delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete sets and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline6.png\" /> <jats:tex-math> $F_{sigma delta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-complete sets, respectively. Then we show that the isometry class of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline7.png\" /> <jats:tex-math> $c_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000440_inline8.png\" />","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"54 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-28DOI: 10.1017/s1474748023000312
Jonas Stelzig, Scott O. Wilson
This paper introduces a generalization of the $dd^c$ -condition for complex manifolds. Like the $dd^c$ -condition, it admits a diverse collection of characterizations, and is hereditary under various geometric constructions. Most notably, it is an open property with respect to small deformations. The condition is satisfied by a wide range of complex manifolds, including all compact complex surfaces, and all compact Vaisman manifolds. We show there are computable invariants of a real homotopy type which in many cases prohibit it from containing any complex manifold satisfying such $dd^c$ -type conditions in low degrees. This gives rise to numerous examples of almost complex manifolds which cannot be homotopy equivalent to any of these complex manifolds.
{"title":"A -TYPE CONDITION BEYOND THE KÄHLER REALM","authors":"Jonas Stelzig, Scott O. Wilson","doi":"10.1017/s1474748023000312","DOIUrl":"https://doi.org/10.1017/s1474748023000312","url":null,"abstract":"This paper introduces a generalization of the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000312_inline2.png\" /> <jats:tex-math> $dd^c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-condition for complex manifolds. Like the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000312_inline3.png\" /> <jats:tex-math> $dd^c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-condition, it admits a diverse collection of characterizations, and is hereditary under various geometric constructions. Most notably, it is an open property with respect to small deformations. The condition is satisfied by a wide range of complex manifolds, including all compact complex surfaces, and all compact Vaisman manifolds. We show there are computable invariants of a real homotopy type which in many cases prohibit it from containing any complex manifold satisfying such <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000312_inline4.png\" /> <jats:tex-math> $dd^c$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-type conditions in low degrees. This gives rise to numerous examples of almost complex manifolds which cannot be homotopy equivalent to any of these complex manifolds.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"5 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-28DOI: 10.1017/s147474802300035x
Mahbub Alam, Anish Ghosh, Jiyoung Han
We establish higher moment formulae for Siegel transforms on the space of affine unimodular lattices as well as on certain congruence quotients of $mathrm {SL}_d({mathbb {R}})$ . As applications, we prove functional central limit theorems for lattice point counting for affine and congruence lattices using the method of moments.
{"title":"HIGHER MOMENT FORMULAE AND LIMITING DISTRIBUTIONS OF LATTICE POINTS","authors":"Mahbub Alam, Anish Ghosh, Jiyoung Han","doi":"10.1017/s147474802300035x","DOIUrl":"https://doi.org/10.1017/s147474802300035x","url":null,"abstract":"We establish higher moment formulae for Siegel transforms on the space of affine unimodular lattices as well as on certain congruence quotients of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S147474802300035X_inline1.png\" /> <jats:tex-math> $mathrm {SL}_d({mathbb {R}})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. As applications, we prove functional central limit theorems for lattice point counting for affine and congruence lattices using the method of moments.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"23 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-21DOI: 10.1017/s1474748023000439
Robert D. Gray, Nik Ruškuc
We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form $r=1$ . We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.
{"title":"ON GROUPS OF UNITS OF SPECIAL AND ONE-RELATOR INVERSE MONOIDS","authors":"Robert D. Gray, Nik Ruškuc","doi":"10.1017/s1474748023000439","DOIUrl":"https://doi.org/10.1017/s1474748023000439","url":null,"abstract":"We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S1474748023000439_inline1.png\" /> <jats:tex-math> $r=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"3 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-14DOI: 10.1017/s1474748023000415
Shiho Oi
Abstract We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on $operatorname {Lip}(X,E)$ , where E is a complex Banach space, is a generalized composition operator. Secondly, we give a complete description of unital surjective complex linear isometries on $operatorname {Lip}(X,mathcal {A})$ , where $mathcal {A}$ is a unital factor $C^{*}$ -algebra. These results improve previous results stated by the author.
{"title":"ISOMETRIES AND HERMITIAN OPERATORS ON SPACES OF VECTOR-VALUED LIPSCHITZ MAPS","authors":"Shiho Oi","doi":"10.1017/s1474748023000415","DOIUrl":"https://doi.org/10.1017/s1474748023000415","url":null,"abstract":"Abstract We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on $operatorname {Lip}(X,E)$ , where E is a complex Banach space, is a generalized composition operator. Secondly, we give a complete description of unital surjective complex linear isometries on $operatorname {Lip}(X,mathcal {A})$ , where $mathcal {A}$ is a unital factor $C^{*}$ -algebra. These results improve previous results stated by the author.","PeriodicalId":50002,"journal":{"name":"Journal of the Institute of Mathematics of Jussieu","volume":"82 17","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134901131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$mathbb {C}(!(t)!)$ and show that the Ceresa class in each of these settings is torsion.
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