Given a double cover between 3-manifolds branched along a nullhomologous link, we establish an inequality between the dimensions of their Heegaard Floer homologies. We discuss the relationship with the L-space conjecture and give some other topological applications, as well as an analogous result for sutured Floer homology.
{"title":"Rank inequalities for the Heegaard Floer homology of branched covers","authors":"Kristen Hendricks, Tye Lidman, Robert Lipshitz","doi":"10.4171/dm/878","DOIUrl":"https://doi.org/10.4171/dm/878","url":null,"abstract":"Given a double cover between 3-manifolds branched along a nullhomologous link, we establish an inequality between the dimensions of their Heegaard Floer homologies. We discuss the relationship with the L-space conjecture and give some other topological applications, as well as an analogous result for sutured Floer homology.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80793913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub-Laplacian, we prove that it is possible to split any QL into several pieces which can be studied separately, and which come from well-characterized parts of the associated sequence of eigenfunctions. Secondly, building upon this result, we classify all QLs of a particular family of sub-Laplacians defined on products of compact quotients of Heisenberg groups. We express the QLs through a disintegration of measure result which follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear.Both results are based on the construction of an adequate elliptic operator commuting with the sub-Laplacian, and on the associated joint spectral calculus. They illustrate the fact that, because of the possibly high degeneracy of the spectrum, the spectral theory of sub-Laplacians can be very rich.
{"title":"Quantum limits of sub-Laplacians via joint spectral calculus","authors":"Cyril Letrouit Dma, Ljll, Cage","doi":"10.4171/dm/908","DOIUrl":"https://doi.org/10.4171/dm/908","url":null,"abstract":"We establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub-Laplacian, we prove that it is possible to split any QL into several pieces which can be studied separately, and which come from well-characterized parts of the associated sequence of eigenfunctions. Secondly, building upon this result, we classify all QLs of a particular family of sub-Laplacians defined on products of compact quotients of Heisenberg groups. We express the QLs through a disintegration of measure result which follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear.Both results are based on the construction of an adequate elliptic operator commuting with the sub-Laplacian, and on the associated joint spectral calculus. They illustrate the fact that, because of the possibly high degeneracy of the spectrum, the spectral theory of sub-Laplacians can be very rich.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74313404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that there are infinitely many pairwise non-commensurable hyperbolic $n$-manifolds that have the same ambient group and trace ring, for any $n geq 3$. The manifolds can be chosen compact if $n geq 4$.
{"title":"Non-commensurable hyperbolic manifolds with the same trace ring","authors":"Olivier Mila","doi":"10.4171/dm/828","DOIUrl":"https://doi.org/10.4171/dm/828","url":null,"abstract":"We prove that there are infinitely many pairwise non-commensurable hyperbolic $n$-manifolds that have the same ambient group and trace ring, for any $n geq 3$. The manifolds can be chosen compact if $n geq 4$.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73427572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-30DOI: 10.25537/DM.2020V25.2445-2471
M. Lim
Let $A$ be an abelian variety defined over a number field $F$. We prove a control theorem for the fine Selmer group of the abelian variety $A$ which essentially says that the kernel and cokernel of the natural restriction maps in a given $mathbb{Z}_p$-extension $F_infty/F$ are finite and bounded. We emphasise that our result does not have any constraints on the reduction of $A$ and the ramification of $F_infty/F$. As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary $mathbb{Z}_p$-extension has trivial $Lambda$-corank. We then derive an asymptotic growth formula for the $p$-torsion subgroup of the dual fine Selmer group in a $mathbb{Z}_p$-extension. However, as the fine Mordell-Weil group needs not be $p$-divisible in general, the fine Tate-Shafarevich group needs not agree with the $p$-torsion of the dual fine Selmer group, and so the asymptotic growth formula for the dual fine Selmer groups do not carry over to the fine Tate-Shafarevich groups. Nevertheless, we do provide certain sufficient conditions, where one can obtain a precise asymptotic formula.
{"title":"On the Control Theorem for Fine Selmer Groups and the Growth of Fine Tate-Shafarevich Groups in $mathbb{Z}_p$-Extensions","authors":"M. Lim","doi":"10.25537/DM.2020V25.2445-2471","DOIUrl":"https://doi.org/10.25537/DM.2020V25.2445-2471","url":null,"abstract":"Let $A$ be an abelian variety defined over a number field $F$. We prove a control theorem for the fine Selmer group of the abelian variety $A$ which essentially says that the kernel and cokernel of the natural restriction maps in a given $mathbb{Z}_p$-extension $F_infty/F$ are finite and bounded. We emphasise that our result does not have any constraints on the reduction of $A$ and the ramification of $F_infty/F$. As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary $mathbb{Z}_p$-extension has trivial $Lambda$-corank. We then derive an asymptotic growth formula for the $p$-torsion subgroup of the dual fine Selmer group in a $mathbb{Z}_p$-extension. However, as the fine Mordell-Weil group needs not be $p$-divisible in general, the fine Tate-Shafarevich group needs not agree with the $p$-torsion of the dual fine Selmer group, and so the asymptotic growth formula for the dual fine Selmer groups do not carry over to the fine Tate-Shafarevich groups. Nevertheless, we do provide certain sufficient conditions, where one can obtain a precise asymptotic formula.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74525689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-26DOI: 10.25537/dm.2021v26.1121-1144
Tom Bachmann
We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In particular we establish the analog of Voevodsky's cancellation theorem.
{"title":"Cancellation theorem for motivic spaces with finite flat transfers","authors":"Tom Bachmann","doi":"10.25537/dm.2021v26.1121-1144","DOIUrl":"https://doi.org/10.25537/dm.2021v26.1121-1144","url":null,"abstract":"We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In particular we establish the analog of Voevodsky's cancellation theorem.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89155493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings), which says that $widetilde{CH}^*(mathbb{P}(E))$ is determined by $widetilde{CH}^*(X)$ and $widetilde{CH}^*(Xtimesmathbb{P}^2)$ for smooth quasi-projective schemes $X$ and vector bundles $E$ over $X$ with odd rank. If the rank of $E$ is even, the theorem is still true under a new kind of orientability, which we call it by projective orientability. As an application, we compute the MW-motives of blow-up over smooth centers.
{"title":"Projective bundle theorem in MW-motivic cohomology","authors":"N. Yang","doi":"10.4171/dm/835","DOIUrl":"https://doi.org/10.4171/dm/835","url":null,"abstract":"We present a version of projective bundle theorem in MW-motives (resp. Chow-Witt rings), which says that $widetilde{CH}^*(mathbb{P}(E))$ is determined by $widetilde{CH}^*(X)$ and $widetilde{CH}^*(Xtimesmathbb{P}^2)$ for smooth quasi-projective schemes $X$ and vector bundles $E$ over $X$ with odd rank. If the rank of $E$ is even, the theorem is still true under a new kind of orientability, which we call it by projective orientability. \u0000As an application, we compute the MW-motives of blow-up over smooth centers.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78523634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $pi:mathbb{C}^ntimesmathbb{C}rightarrowmathbb{C}$ be the projection map onto the second factor and let $D$ be a domain in $mathbb{C}^{n+1}$ such that for $yinpi(D)$, every fiber $D_y:=Dcappi^{-1}(y)$ is a smoothly bounded strongly pseudoconvex domain in $mathbb{C}^n$ and is diffeomorphic to each other. By Chau's theorem, the Kahler-Ricci flow has a long time solution $omega_y(t)$ on each fiber $D_y$. This family of flows induces a smooth real (1,1)-form $omega(t)$ on the total space $D$ whose restriction to the fiber $D_y$ satisfies $omega(t)vert_{D_y}=omega_y(t)$. In this paper, we prove that $omega(t)$ is positive for all $t>0$ in $D$ if $omega(0)$ is positive. As a corollary, we also prove that the fiberwise Kahler-Einstein metric is positive semi-definite on $D$ if $D$ is pseudoconvex in $mathbb{C}^{n+1}$.
{"title":"Fiberwise Kähler-Ricci flows on families of bounded strongly pseudoconvex domains","authors":"Youngook Choi, Sungmin Yoo","doi":"10.4171/dm/886","DOIUrl":"https://doi.org/10.4171/dm/886","url":null,"abstract":"Let $pi:mathbb{C}^ntimesmathbb{C}rightarrowmathbb{C}$ be the projection map onto the second factor and let $D$ be a domain in $mathbb{C}^{n+1}$ such that for $yinpi(D)$, every fiber $D_y:=Dcappi^{-1}(y)$ is a smoothly bounded strongly pseudoconvex domain in $mathbb{C}^n$ and is diffeomorphic to each other. By Chau's theorem, the Kahler-Ricci flow has a long time solution $omega_y(t)$ on each fiber $D_y$. This family of flows induces a smooth real (1,1)-form $omega(t)$ on the total space $D$ whose restriction to the fiber $D_y$ satisfies $omega(t)vert_{D_y}=omega_y(t)$. In this paper, we prove that $omega(t)$ is positive for all $t>0$ in $D$ if $omega(0)$ is positive. As a corollary, we also prove that the fiberwise Kahler-Einstein metric is positive semi-definite on $D$ if $D$ is pseudoconvex in $mathbb{C}^{n+1}$.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88004548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the special value conjecture for the Zeta function of a proper, regular arithmetic scheme X that we formulated in our previous article [8] is compatible with the functional equation of the Zeta function provided that the factor C(X,n) we were not able to compute in loc. cit. has the simple explicit form suggested in [9].
{"title":"Compatibility of special value conjectures with the functional equation of zeta functions","authors":"M. Flach, B. Morin","doi":"10.4171/dm/852","DOIUrl":"https://doi.org/10.4171/dm/852","url":null,"abstract":"We prove that the special value conjecture for the Zeta function of a proper, regular arithmetic scheme X that we formulated in our previous article [8] is compatible with the functional equation of the Zeta function provided that the factor C(X,n) we were not able to compute in loc. cit. has the simple explicit form suggested in [9].","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89143384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible modules over the crystalline distribution algebra of G in terms of overconvergent isocrystals on locally closed subspaces in the (formal) flag variety of G. We treat the case of SL(2) as an example.
{"title":"Intermediate extensions and crystalline distribution algebras","authors":"Christine Huyghe, Tobias Schmidt","doi":"10.4171/dm/863","DOIUrl":"https://doi.org/10.4171/dm/863","url":null,"abstract":"Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible modules over the crystalline distribution algebra of G in terms of overconvergent isocrystals on locally closed subspaces in the (formal) flag variety of G. We treat the case of SL(2) as an example.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84117771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present article, we study the overconvergent cohomology groups related to $text{GSp}_{2g}$. We construct a pairing on the cohomology groups. On the other hand, by considering the parabolic cohomology groups and applying the strategy in [JN19], we constructed the cuspidal eigenvariety for $text{GSp}_{2g}$. The pairing on the cohomology groups then induces a pairing on some coherent sheaves of the cuspidal eigenvariety. As an application, we follow the strategy in [Bel10, Chapter VI] to study the ramification locus of the cuspidal eigenvariety for $text{GSp}_{4}$ over the corresponding weight space.
{"title":"A pairing on the cuspidal eigenvariety for $text{GSp}_{2g}$ and the ramification locus","authors":"Ju-Feng Wu","doi":"10.4171/dm/826","DOIUrl":"https://doi.org/10.4171/dm/826","url":null,"abstract":"In the present article, we study the overconvergent cohomology groups related to $text{GSp}_{2g}$. We construct a pairing on the cohomology groups. On the other hand, by considering the parabolic cohomology groups and applying the strategy in [JN19], we constructed the cuspidal eigenvariety for $text{GSp}_{2g}$. The pairing on the cohomology groups then induces a pairing on some coherent sheaves of the cuspidal eigenvariety. As an application, we follow the strategy in [Bel10, Chapter VI] to study the ramification locus of the cuspidal eigenvariety for $text{GSp}_{4}$ over the corresponding weight space.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78637851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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