Pub Date : 2024-05-20DOI: 10.1112/s0010437x24007152
Artem Kotelskiy, Liam Watson, Claudius Zibrowius
When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $delta$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $operatorname {HFT}$ and the Khovanov invariant $widetilde {operatorname {Kh}}$ that were developed by the authors in previous works.
{"title":"Thin links and Conway spheres","authors":"Artem Kotelskiy, Liam Watson, Claudius Zibrowius","doi":"10.1112/s0010437x24007152","DOIUrl":"https://doi.org/10.1112/s0010437x24007152","url":null,"abstract":"<p>When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240517100304785-0216:S0010437X24007152:S0010437X24007152_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$delta$</span></span></img></span></span>-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240517100304785-0216:S0010437X24007152:S0010437X24007152_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {HFT}$</span></span></img></span></span> and the Khovanov invariant <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240517100304785-0216:S0010437X24007152:S0010437X24007152_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$widetilde {operatorname {Kh}}$</span></span></img></span></span> that were developed by the authors in previous works.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"23 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153114","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}
Pub Date : 2024-05-13DOI: 10.1112/s0010437x24007139
Ruotao Yang
<p>Let <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline1.png"><span data-mathjax-type="texmath"><span>$G$</span></span></img></span></span> be a reductive group, and let <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline2.png"><span data-mathjax-type="texmath"><span>$check {G}$</span></span></img></span></span> be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline3.png"><span data-mathjax-type="texmath"><span>${operatorname {Fl}}_G$</span></span></img></span></span> is equivalent to the category of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline4.png"><span data-mathjax-type="texmath"><span>$check {G}$</span></span></img></span></span>-equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. We show that the twisted Whittaker category on <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline5.png"><span data-mathjax-type="texmath"><span>${operatorname {Fl}}_G$</span></span></img></span></span> and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline6.png"><span data-mathjax-type="texmath"><span>$mathsf {O}$</span></span></img></span></span> is equivalent to the twisted Whittaker category on <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline7.png"><span data-mathjax-type="texmath"><span>${operatorname {Fl}}_G$</span></span></img></span></span> in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline8.png"><span data-mathjax-type="texmath"><span>${operatorname {Fl}}_
{"title":"Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group","authors":"Ruotao Yang","doi":"10.1112/s0010437x24007139","DOIUrl":"https://doi.org/10.1112/s0010437x24007139","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> be a reductive group, and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$check {G}$</span></span></img></span></span> be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${operatorname {Fl}}_G$</span></span></img></span></span> is equivalent to the category of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$check {G}$</span></span></img></span></span>-equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. We show that the twisted Whittaker category on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${operatorname {Fl}}_G$</span></span></img></span></span> and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {O}$</span></span></img></span></span> is equivalent to the twisted Whittaker category on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline7.png\"><span data-mathjax-type=\"texmath\"><span>${operatorname {Fl}}_G$</span></span></img></span></span> in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240510094943940-0685:S0010437X24007139:S0010437X24007139_inline8.png\"><span data-mathjax-type=\"texmath\"><span>${operatorname {Fl}}_","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"24 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937655","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}
Pub Date : 2024-05-10DOI: 10.1112/s0010437x24007140
Cédric Pilatte
A set $Ssubset {mathbb {N}}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2in S$, $s_1leqslant s_2$) are distinct. A set $Ssubset {mathbb {N}}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$. In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $mathbb {F}_q[t]$<
{"title":"A solution to the Erdős–Sárközy–Sós problem on asymptotic Sidon bases of order 3","authors":"Cédric Pilatte","doi":"10.1112/s0010437x24007140","DOIUrl":"https://doi.org/10.1112/s0010437x24007140","url":null,"abstract":"<p>A set <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Ssubset {mathbb {N}}$</span></span></img></span></span> is a <span>Sidon set</span> if all pairwise sums <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$s_1+s_2$</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:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$s_1, s_2in S$</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:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$s_1leqslant s_2$</span></span></img></span></span>) are distinct. A set <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$Ssubset {mathbb {N}}$</span></span></img></span></span> is an <span>asymptotic basis of order 3</span> if every sufficiently large integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span> can be written as the sum of three elements of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$S$</span></span></img></span></span>. In 1993, Erdős, Sárközy and Sós asked whether there exists a set <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$S$</span></span></img></span></span> with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240509195826895-0930:S0010437X24007140:S0010437X24007140_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {F}_q[t]$</span><","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"21 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937654","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}
Pub Date : 2024-05-07DOI: 10.1112/s0010437x24007048
Eyal Markman
Let $X$ and $Y$ be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length $n$ subschemes of a $K3$ surface. A class in $H^{p,p}(Xtimes Y,{mathbb {Q}})$ is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let $f:H^2(X,{mathbb {Q}})rightarrow H^2(Y,{mathbb {Q}})$ be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that $f$ is induced by an analytic correspondence. We furthermore lift $f$ to an analytic correspondence
{"title":"Rational Hodge isometries of hyper-Kähler varieties of type are algebraic","authors":"Eyal Markman","doi":"10.1112/s0010437x24007048","DOIUrl":"https://doi.org/10.1112/s0010437x24007048","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$X$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$Y$</span></span></img></span></span> be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span> subschemes of a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$K3$</span></span></img></span></span> surface. A class in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$H^{p,p}(Xtimes Y,{mathbb {Q}})$</span></span></img></span></span> is an <span>analytic correspondence</span>, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$f:H^2(X,{mathbb {Q}})rightarrow H^2(Y,{mathbb {Q}})$</span></span></img></span></span> be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$f$</span></span></img></span></span> is induced by an analytic correspondence. We furthermore lift <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$f$</span></span></img></span></span> to an analytic correspondence <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240506135739299-0180:S0010437X24007048:S0010437X24007048_in","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"17 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140881504","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}
Pub Date : 2024-05-06DOI: 10.1112/s0010437x24007103
P. Koymans, N. Rome
For any abelian group $A$, we prove an asymptotic formula for the number of $A$-extensions $K/mathbb {Q}$ of bounded discriminant such that the associated norm one torus $R_{K/mathbb {Q}}^1 mathbb {G}_m$ satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.
{"title":"Weak approximation on the norm one torus","authors":"P. Koymans, N. Rome","doi":"10.1112/s0010437x24007103","DOIUrl":"https://doi.org/10.1112/s0010437x24007103","url":null,"abstract":"<p>For any abelian group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240503095838219-0161:S0010437X24007103:S0010437X24007103_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$A$</span></span></img></span></span>, we prove an asymptotic formula for the number of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240503095838219-0161:S0010437X24007103:S0010437X24007103_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$A$</span></span></img></span></span>-extensions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240503095838219-0161:S0010437X24007103:S0010437X24007103_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$K/mathbb {Q}$</span></span></img></span></span> of bounded discriminant such that the associated norm one torus <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240503095838219-0161:S0010437X24007103:S0010437X24007103_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$R_{K/mathbb {Q}}^1 mathbb {G}_m$</span></span></img></span></span> satisfies weak approximation. We are also able to produce new results on the Hasse norm principle and to provide new explicit values for the leading constant in some instances of Malle's conjecture.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"26 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140881459","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}
Pub Date : 2024-05-06DOI: 10.1112/s0010437x24007127
Yifei Zhao
We prove the geometric Satake equivalence for étale metaplectic covers of reductive group schemes and extend the Langlands parametrization of V. Lafforgue to genuine cusp forms defined on their associated covering groups.
{"title":"Spectral decomposition of genuine cusp forms over global function fields","authors":"Yifei Zhao","doi":"10.1112/s0010437x24007127","DOIUrl":"https://doi.org/10.1112/s0010437x24007127","url":null,"abstract":"<p>We prove the geometric Satake equivalence for étale metaplectic covers of reductive group schemes and extend the Langlands parametrization of V. Lafforgue to genuine cusp forms defined on their associated covering groups.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"11 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140881767","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}
Pub Date : 2024-05-02DOI: 10.1112/s0010437x24007115
Christopher P. Bendel, Daniel K. Nakano, Cornelius Pillen, Paul Sobaje
In this paper we produce infinite families of counterexamples to Jantzen's question posed in 1980 on the existence of Weyl $p$-filtrations for Weyl modules for an algebraic group and Donkin's tilting module conjecture formulated in 1990. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type ${rm A}_{n}$ or ${rm B}_{2}$.
{"title":"On Donkin's tilting module conjecture II: counterexamples","authors":"Christopher P. Bendel, Daniel K. Nakano, Cornelius Pillen, Paul Sobaje","doi":"10.1112/s0010437x24007115","DOIUrl":"https://doi.org/10.1112/s0010437x24007115","url":null,"abstract":"<p>In this paper we produce infinite families of counterexamples to Jantzen's question posed in 1980 on the existence of Weyl <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430183825590-0689:S0010437X24007115:S0010437X24007115_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-filtrations for Weyl modules for an algebraic group and Donkin's tilting module conjecture formulated in 1990. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240430183825590-0689:S0010437X24007115:S0010437X24007115_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${rm A}_{n}$</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:20240430183825590-0689:S0010437X24007115:S0010437X24007115_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${rm B}_{2}$</span></span></img></span></span>.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"91 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140835390","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}
Pub Date : 2024-04-18DOI: 10.1112/s0010437x24007097
Heng Du, Tong Liu, Yong Suk Moon, Koji Shimizu
We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline étale $mathbf {Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.
{"title":"Completed prismatic F-crystals and crystalline Zp-local systems","authors":"Heng Du, Tong Liu, Yong Suk Moon, Koji Shimizu","doi":"10.1112/s0010437x24007097","DOIUrl":"https://doi.org/10.1112/s0010437x24007097","url":null,"abstract":"<p>We introduce the notion of completed <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417180204316-0193:S0010437X24007097:S0010437X24007097_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$F$</span></span></img></span></span>-crystals on the absolute prismatic site of a smooth <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417180204316-0193:S0010437X24007097:S0010437X24007097_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adic formal scheme. We define a functor from the category of completed prismatic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417180204316-0193:S0010437X24007097:S0010437X24007097_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$F$</span></span></img></span></span>-crystals to that of crystalline étale <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417180204316-0193:S0010437X24007097:S0010437X24007097_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {Z}_p$</span></span></img></span></span>-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"14 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140612424","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}
Pub Date : 2024-04-18DOI: 10.1112/s0010437x24007036
Yeuk Hay Joshua Lam, Arnav Tripathy
The attractor conjecture for Calabi–Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. Using tools from transcendence theory, we provide a family of counterexamples to the attractor conjecture in almost all odd dimensions conditional on a specific case of the Zilber–Pink conjecture in unlikely intersection theory; these Calabi–Yau manifolds were first studied by Dolgachev. We also give constructions of new families of Calabi–Yau varieties, analogous to the mirror quintic family, with all middle Hodge numbers equal to one, which would also give counterexamples to the attractor conjecture.
{"title":"Attractors are not algebraic","authors":"Yeuk Hay Joshua Lam, Arnav Tripathy","doi":"10.1112/s0010437x24007036","DOIUrl":"https://doi.org/10.1112/s0010437x24007036","url":null,"abstract":"<p>The attractor conjecture for Calabi–Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. Using tools from transcendence theory, we provide a family of counterexamples to the attractor conjecture in almost all odd dimensions conditional on a specific case of the Zilber–Pink conjecture in unlikely intersection theory; these Calabi–Yau manifolds were first studied by Dolgachev. We also give constructions of new families of Calabi–Yau varieties, analogous to the mirror quintic family, with all middle Hodge numbers equal to one, which would also give counterexamples to the attractor conjecture.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"14 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140612599","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}
Pub Date : 2024-04-08DOI: 10.1112/s0010437x24007073
Hanci Chi
We prove that there exists at least one positive Einstein metric on $mathbb {HP}^{m+1}sharp overline {mathbb {HP}}^{m+1}$ for $mgeq ~2$. Based on the existence of the first Einstein metric, we give a criterion to check the existence of a second Einstein metric on $mathbb {HP}^{m+1}sharp overline {mathbb {HP}}^{m+1}$. We also investigate the existence of cohomogeneity-one positive Einstein metrics on $mathbb {S}^{4m+4}$ and prove the existence of a non-standard Einstein metric on $mathbb {S}^8$.
{"title":"Positive Einstein metrics with as the principal orbit","authors":"Hanci Chi","doi":"10.1112/s0010437x24007073","DOIUrl":"https://doi.org/10.1112/s0010437x24007073","url":null,"abstract":"<p>We prove that there exists at least one positive Einstein metric on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405172416606-0041:S0010437X24007073:S0010437X24007073_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {HP}^{m+1}sharp overline {mathbb {HP}}^{m+1}$</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:20240405172416606-0041:S0010437X24007073:S0010437X24007073_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mgeq ~2$</span></span></img></span></span>. Based on the existence of the first Einstein metric, we give a criterion to check the existence of a second Einstein metric on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405172416606-0041:S0010437X24007073:S0010437X24007073_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {HP}^{m+1}sharp overline {mathbb {HP}}^{m+1}$</span></span></img></span></span>. We also investigate the existence of cohomogeneity-one positive Einstein metrics on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405172416606-0041:S0010437X24007073:S0010437X24007073_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {S}^{4m+4}$</span></span></img></span></span> and prove the existence of a non-standard Einstein metric on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240405172416606-0041:S0010437X24007073:S0010437X24007073_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {S}^8$</span></span></img></span></span>.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":"21 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562046","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}
$delta$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant 
$operatorname {HFT}$ and the Khovanov invariant 
$widetilde {operatorname {Kh}}$ that were developed by the authors in previous works.
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