Inspired by recent work of Farb, Kisin and Wolfson, we develop a method for using actions of finite group schemes over a mixed characteristic dvr R to get lower bounds for the essential dimension of a cover of a variety over K = Frac(R). We then apply this to prove p-incompressibility for congruence covers of a class of unitary Shimura varieties for primes p at which the reduction of the Shimura variety (at any prime of the reflex field over p) does not have any ordinary points. We also make some progress towards a conjecture of Brosnan on the p-incompressibility of the multiplication by p map of an abelian variety.
{"title":"Finite groups scheme actions and incompressibility of Galois covers: beyond the ordinary case","authors":"N. Fakhruddin, Rijul Saini","doi":"10.4171/dm/868","DOIUrl":"https://doi.org/10.4171/dm/868","url":null,"abstract":"Inspired by recent work of Farb, Kisin and Wolfson, we develop a method for using actions of finite group schemes over a mixed characteristic dvr R to get lower bounds for the essential dimension of a cover of a variety over K = Frac(R). We then apply this to prove p-incompressibility for congruence covers of a class of unitary Shimura varieties for primes p at which the reduction of the Shimura variety (at any prime of the reflex field over p) does not have any ordinary points. We also make some progress towards a conjecture of Brosnan on the p-incompressibility of the multiplication by p map of an abelian variety.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85785350","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 the Iwasawa-theoretic version of a Conjecture of Mazur--Rubin and Sano in the case of elliptic units. This allows us to derive the $p$-part of the equivariant Tamagawa number conjecture at $s = 0$ for abelian extensions of imaginary quadratic fields in the semi-simple case and, provided that a standard $mu$-vanishing hypothesis is satisfied, also in the general case.
{"title":"The equivariant Tamagawa number conjecture for abelian extensions of imaginary quadratic fields","authors":"Dominik Bullach, Martin Hofer","doi":"10.4171/dm/907","DOIUrl":"https://doi.org/10.4171/dm/907","url":null,"abstract":"We prove the Iwasawa-theoretic version of a Conjecture of Mazur--Rubin and Sano in the case of elliptic units. This allows us to derive the $p$-part of the equivariant Tamagawa number conjecture at $s = 0$ for abelian extensions of imaginary quadratic fields in the semi-simple case and, provided that a standard $mu$-vanishing hypothesis is satisfied, also in the general case.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82586835","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 this paper we construct a Ricci de Turck flow on any incomplete Riemannian manifold with bounded curvature. The central property of the flow is that it stays uniformly equivalent to the initial incomplete Riemannian metric, and in that sense preserves any given initial singularity structure. Together with the corresponding result by Shi for complete manifolds [Shi89], this gives that any (complete or incomplete) manifold of bounded curvature can be evolved by the Ricci de Turck flow for a short time.
本文构造了曲率有界的不完全黎曼流形上的Ricci de Turck流。流的中心性质是它与初始的不完全黎曼度规保持一致的等价，从这个意义上说，它保留了任何给定的初始奇点结构。结合Shi对完全流形的相应结果[Shi89]，给出了任何有界曲率的(完全或不完全)流形都可以被Ricci de Turck流在短时间内演化。
{"title":"Ricci DeTurck flow on incomplete manifolds","authors":"Tobias Marxen, Boris Vertman","doi":"10.4171/dm/894","DOIUrl":"https://doi.org/10.4171/dm/894","url":null,"abstract":"In this paper we construct a Ricci de Turck flow on any incomplete Riemannian manifold with bounded curvature. The central property of the flow is that it stays uniformly equivalent to the initial incomplete Riemannian metric, and in that sense preserves any given initial singularity structure. Together with the corresponding result by Shi for complete manifolds [Shi89], this gives that any (complete or incomplete) manifold of bounded curvature can be evolved by the Ricci de Turck flow for a short time.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86253573","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}
A k-differential on a Riemann surface is a section of the k-th power of the canonical bundle. Loci of k-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of k-differentials. The classification of connected components of the strata of k-differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich–Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the strata of k-differentials for general k. As an application, we give a complete classification of connected components of the strata of quadratic differentials with arbitrary poles. Moreover, we distinguish certain components of the strata of kdifferentials by generalizing the hyperelliptic structure and spin parity for higher k. We also describe an approach to determine explicitly parities of k-differentials in genus zero and one, which inspires an amusing conjecture in number theory. A key viewpoint we use is the notion of multi-scale k-differentials introduced by Bainbridge– Chen–Gendron–Grushevsky–Möller for k = 1 and extended by Costantini–Möller– Zachhuber for all k.
{"title":"Towards a classification of connected components of the strata of $k$-differentials","authors":"Dawei Chen, Q. Gendron","doi":"10.4171/dm/892","DOIUrl":"https://doi.org/10.4171/dm/892","url":null,"abstract":"A k-differential on a Riemann surface is a section of the k-th power of the canonical bundle. Loci of k-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of k-differentials. The classification of connected components of the strata of k-differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich–Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the strata of k-differentials for general k. As an application, we give a complete classification of connected components of the strata of quadratic differentials with arbitrary poles. Moreover, we distinguish certain components of the strata of kdifferentials by generalizing the hyperelliptic structure and spin parity for higher k. We also describe an approach to determine explicitly parities of k-differentials in genus zero and one, which inspires an amusing conjecture in number theory. A key viewpoint we use is the notion of multi-scale k-differentials introduced by Bainbridge– Chen–Gendron–Grushevsky–Möller for k = 1 and extended by Costantini–Möller– Zachhuber for all k.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88378470","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 show that Chow groups of low dimension cycles are torsion free for a class of sufficiently generic Severi-Brauer varieties. Using a recent result of Karpenko, this allows us to compute the algebraic connective K-theory in low degrees for the same class of varieties. Independently of these results, we show that the associated graded ring for the topological filtration on the Grothendieck ring is torsion free in the same degrees for arbitrary SeveriBrauer varieties.
{"title":"Algebraic connective $K$-theory of a Severi-Brauer variety with prescribed reduced behavior","authors":"Eoin Mackall","doi":"10.4171/dm/820","DOIUrl":"https://doi.org/10.4171/dm/820","url":null,"abstract":"We show that Chow groups of low dimension cycles are torsion free for a class of sufficiently generic Severi-Brauer varieties. Using a recent result of Karpenko, this allows us to compute the algebraic connective K-theory in low degrees for the same class of varieties. Independently of these results, we show that the associated graded ring for the topological filtration on the Grothendieck ring is torsion free in the same degrees for arbitrary SeveriBrauer varieties.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80794312","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}
{"title":"Erratum to: \"The minimal exact crossed product\"","authors":"Alcides Buss, S. Echterhoff, R. Willett","doi":"10.4171/dm/851","DOIUrl":"https://doi.org/10.4171/dm/851","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77243615","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 : 2021-01-01DOI: 10.25537/DM.2021V26.661-673
A. Stavrova
Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{pm 1},...,x_n^{pm 1}]. We prove that G has isotropic rank >=1 over R iff it has isotropic rank >=1 over the field of fractions k(x_1,...,x_n) of R, and if this is the case, then the natural map H^1_{et}(R,G)to H^1_{et}(k(x_1,...,x_n),G) has trivial kernel, and G is loop reductive, i.e. contains a maximal R-torus. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that H^1_{Zar}(R,G)=* for such groups G. We also deduce that if G is a reductive group over R of isotropic rank >=2, then the natural map of non-stable K_1-functors K_1^G(R)to K_1^G( k((x_1))...((x_n)) ) is injective, and an isomorphism if G is moreover semisimple.
{"title":"Torsors of isotropic reductive groups over Laurent polynomials","authors":"A. Stavrova","doi":"10.25537/DM.2021V26.661-673","DOIUrl":"https://doi.org/10.25537/DM.2021V26.661-673","url":null,"abstract":"Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{pm 1},...,x_n^{pm 1}]. We prove that G has isotropic rank >=1 over R iff it has isotropic rank >=1 over the field of fractions k(x_1,...,x_n) of R, and if this is the case, then the natural map H^1_{et}(R,G)to H^1_{et}(k(x_1,...,x_n),G) has trivial kernel, and G is loop reductive, i.e. contains a maximal R-torus. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that H^1_{Zar}(R,G)=* for such groups G. We also deduce that if G is a reductive group over R of isotropic rank >=2, then the natural map of non-stable K_1-functors K_1^G(R)to K_1^G( k((x_1))...((x_n)) ) is injective, and an isomorphism if G is moreover semisimple.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85759938","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}
{"title":"On the local regularity theory for the magnetohydrodynamic equations","authors":"D. Chamorro, F. Cortez, Jiao He, Oscar Jarŕın","doi":"10.4171/dm/811","DOIUrl":"https://doi.org/10.4171/dm/811","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83179191","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}
This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton–Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results ΣΩΣX ≃ ΣX ∨ (X ∧ ΣΩΣX) and Ω(X ∨ Y ) ≃ ΩX × ΩY × ΩΣ(ΩX ∧ ΩY ) in the maximal generality of an ∞-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather’s Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton–Milnor Splittings. Moreover, working in this generality shows that the James and Hilton–Milnor splittings hold in many new contexts, for example in: elementary ∞-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting results in this last context extend Wickelgren and Williams’ splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of ∞-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers–Massey Theorem, while the second reduces to the classical computational proof. 2020 Mathematics Subject Classification: 55P35, 55P40, 55P99, 55Q20, 18N60, 14F42
{"title":"On the James and Hilton-Milnor splittings, and the metastable EHP sequence","authors":"Sanath K. Devalapurkar, Peter J. Haine","doi":"10.4171/dm/845","DOIUrl":"https://doi.org/10.4171/dm/845","url":null,"abstract":"This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton–Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results ΣΩΣX ≃ ΣX ∨ (X ∧ ΣΩΣX) and Ω(X ∨ Y ) ≃ ΩX × ΩY × ΩΣ(ΩX ∧ ΩY ) in the maximal generality of an ∞-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather’s Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton–Milnor Splittings. Moreover, working in this generality shows that the James and Hilton–Milnor splittings hold in many new contexts, for example in: elementary ∞-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting results in this last context extend Wickelgren and Williams’ splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of ∞-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers–Massey Theorem, while the second reduces to the classical computational proof. 2020 Mathematics Subject Classification: 55P35, 55P40, 55P99, 55Q20, 18N60, 14F42","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88014671","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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