{"title":"Almost complex structures on connected sums of complex projective spaces","authors":"Oliver Goertsches, Panagiotis Konstantis","doi":"10.2140/akt.2019.4.139","DOIUrl":"https://doi.org/10.2140/akt.2019.4.139","url":null,"abstract":"We show that the m-fold connected sum $m#mathbb{C}mathbb{P}^{2n}$ admits an almost complex structure if and only if m is odd.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.139","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67938485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Gamma$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for $mathrm{B} Gamma$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $Gamma$ with coefficients in the $mathbb{C}Gamma$-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey's work on explicitly inverting this Chern character in low homological degrees.
{"title":"Positive scalar curvature and low-degree group homology","authors":"No'e B'arcenas, Rudolf Zeidler","doi":"10.2140/akt.2018.3.565","DOIUrl":"https://doi.org/10.2140/akt.2018.3.565","url":null,"abstract":"Let $Gamma$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for $mathrm{B} Gamma$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $Gamma$ with coefficients in the $mathbb{C}Gamma$-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey's work on explicitly inverting this Chern character in low homological degrees.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2018.3.565","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43588572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.
{"title":"On derived categories of arithmetic toric varieties","authors":"Matthew R. Ballard, A. Duncan, P. McFaddin","doi":"10.2140/akt.2019.4.211","DOIUrl":"https://doi.org/10.2140/akt.2019.4.211","url":null,"abstract":"We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.211","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42117171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The congruence subgroup problem for a finitely generated group $Gamma$ and $Gleq Aut(Gamma)$ asks whether the map $hat{G}to Aut(hat{Gamma})$ is injective, or more generally, what is its kernel $Cleft(G,Gammaright)$? Here $hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $Cleft(IA(Phi_{n}),Phi_{n}right)$, where $Phi_{n}$ is a free metabelian group on $ngeq4$ generators, and $IA(Phi_{n})=ker(Aut(Phi_{n})to GL_{n}(mathbb{Z}))$. We introduce surjective representations of $IA(Phi_{n})$ onto the group $ker(GL_{n-1}(mathbb{Z}[x^{pm1}])overset{xmapsto1}{longrightarrow}GL_{n-1}(mathbb{Z}))$ which come via the classical Magnus representation of $IA(Phi_{n})$. Using this representations combined with some methods and results from Algebraic K-theory, we prove that for every $ngeq4$, $Cleft(IA(Phi_{n}),Phi_{n}right)$ contains a product of $n$ copies of the congruence kernel $ker(widehat{SL_{n-1}(mathbb{Z}[x^{pm1}]})to SL_{n-1}(widehat{mathbb{Z}[x^{pm1}]}))$ which is central in $widehat{IA(Phi_{n})}$. It enables us to show that contrary to free nilpotent cases, $Cleft(IA(Phi_{n}),Phi_{n}right)$ is not trivial and not even finitely generated. We note that using some results of this paper we show in an upcoming paper that actually, all the elements of $Cleft(IA(Phi_{n}),Phi_{n}right)$ lie in the center of $widehat{IA(Phi_{n})}$.
{"title":"The IA-congruence kernel of high rank free metabelian groups","authors":"David el-Chai Ben-Ezra","doi":"10.2140/akt.2019.4.383","DOIUrl":"https://doi.org/10.2140/akt.2019.4.383","url":null,"abstract":"The congruence subgroup problem for a finitely generated group $Gamma$ and $Gleq Aut(Gamma)$ asks whether the map $hat{G}to Aut(hat{Gamma})$ is injective, or more generally, what is its kernel $Cleft(G,Gammaright)$? Here $hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $Cleft(IA(Phi_{n}),Phi_{n}right)$, where $Phi_{n}$ is a free metabelian group on $ngeq4$ generators, and $IA(Phi_{n})=ker(Aut(Phi_{n})to GL_{n}(mathbb{Z}))$. \u0000We introduce surjective representations of $IA(Phi_{n})$ onto the group $ker(GL_{n-1}(mathbb{Z}[x^{pm1}])overset{xmapsto1}{longrightarrow}GL_{n-1}(mathbb{Z}))$ which come via the classical Magnus representation of $IA(Phi_{n})$. Using this representations combined with some methods and results from Algebraic K-theory, we prove that for every $ngeq4$, $Cleft(IA(Phi_{n}),Phi_{n}right)$ contains a product of $n$ copies of the congruence kernel $ker(widehat{SL_{n-1}(mathbb{Z}[x^{pm1}]})to SL_{n-1}(widehat{mathbb{Z}[x^{pm1}]}))$ which is central in $widehat{IA(Phi_{n})}$. It enables us to show that contrary to free nilpotent cases, $Cleft(IA(Phi_{n}),Phi_{n}right)$ is not trivial and not even finitely generated. \u0000We note that using some results of this paper we show in an upcoming paper that actually, all the elements of $Cleft(IA(Phi_{n}),Phi_{n}right)$ lie in the center of $widehat{IA(Phi_{n})}$.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45632187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend earlier work of Waldhausen which defines operations on the algebraic $K$-theory of the one-point space. For a connected simplicial abelian group $X$ and symmetric groups $Sigma_n$, we define operations $theta^n colon A(X) rightarrow A(X{times}BSigma_n)$ in the algebraic $K$-theory of spaces. We show that our operations can be given the structure of $E_{infty}$-maps. Let $phi_n colon A(X{times}BSigma_n) rightarrow A(X{times}ESigma_n) simeq A(X)$ be the $Sigma_n$-transfer. We also develop an inductive procedure to compute the compositions $phi_n circ theta^n$, and outline some applications.
{"title":"Segal operations in the algebraic K-theory of\u0000topological spaces","authors":"T. Gunnarsson, R. Staffeldt","doi":"10.2140/akt.2019.4.1","DOIUrl":"https://doi.org/10.2140/akt.2019.4.1","url":null,"abstract":"We extend earlier work of Waldhausen which defines operations on the algebraic $K$-theory of the one-point space. For a connected simplicial abelian group $X$ and symmetric groups $Sigma_n$, we define operations $theta^n colon A(X) rightarrow A(X{times}BSigma_n)$ in the algebraic $K$-theory of spaces. We show that our operations can be given the structure of $E_{infty}$-maps. Let $phi_n colon A(X{times}BSigma_n) rightarrow A(X{times}ESigma_n) simeq A(X)$ be the $Sigma_n$-transfer. We also develop an inductive procedure to compute the compositions $phi_n circ theta^n$, and outline some applications.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44031777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we construct a Hecke-equivariant Chow motive whose realizations equal intersection cohomology of Siegel threefolds with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Siegel modular forms.
{"title":"On the intersection motive of certain Shimura varieties: the case of Siegel threefolds","authors":"J. Wildeshaus","doi":"10.2140/akt.2019.4.525","DOIUrl":"https://doi.org/10.2140/akt.2019.4.525","url":null,"abstract":"In this article, we construct a Hecke-equivariant Chow motive whose realizations equal intersection cohomology of Siegel threefolds with regular algebraic coefficients. As a consequence, we are able to define Grothendieck motives for Siegel modular forms.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.525","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48050008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We use descent theoretic methods to solve the homotopy limit problem for Hermitian $K$-theory over very general Noetherian base schemes, assuming that the natural map from Hermitian $K$-theory to algebraic $K$-theory is a map of commutative motivic ring spectra. As another application of these descent theoretic methods, we compute the cellular Picard group of 2-complete Hermitian $K$-theory over $mathop{Spec}(mathbb{C})$, showing that the only invertible cellular spectra are suspensions of the tensor unit.
{"title":"The homotopy limit problem and the cellular\u0000Picard group of Hermitian K-theory","authors":"Drew Heard","doi":"10.2140/akt.2021.6.137","DOIUrl":"https://doi.org/10.2140/akt.2021.6.137","url":null,"abstract":"We use descent theoretic methods to solve the homotopy limit problem for Hermitian $K$-theory over very general Noetherian base schemes, assuming that the natural map from Hermitian $K$-theory to algebraic $K$-theory is a map of commutative motivic ring spectra. As another application of these descent theoretic methods, we compute the cellular Picard group of 2-complete Hermitian $K$-theory over $mathop{Spec}(mathbb{C})$, showing that the only invertible cellular spectra are suspensions of the tensor unit.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44396240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0) provided that n is nilpotent on X (resp. is invertible on X). Our descent and vanishing results apply more generally to certain Artin stacks whose stabilizers are extensions of finite group schemes by group schemes of multiplicative type.
{"title":"Vanishing theorems for the negative K-theory of\u0000stacks","authors":"Marc Hoyois, A. Krishna","doi":"10.2140/akt.2019.4.439","DOIUrl":"https://doi.org/10.2140/akt.2019.4.439","url":null,"abstract":"We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0) provided that n is nilpotent on X (resp. is invertible on X). Our descent and vanishing results apply more generally to certain Artin stacks whose stabilizers are extensions of finite group schemes by group schemes of multiplicative type.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2019.4.439","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67938680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of "stability relative to a class of functors", which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.
{"title":"Generalized stability for abstract homotopy theories","authors":"Moritz Groth, Michael Shulman","doi":"10.2140/akt.2021.6.1","DOIUrl":"https://doi.org/10.2140/akt.2021.6.1","url":null,"abstract":"We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of \"stability relative to a class of functors\", which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"78 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41263306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case has already been given several other proofs. -- On montre que sur toute hypersurface cubique complexe de dimension au moins 2, le groupe de Chow des cycles de dimension 1 est engendre par les droites. Le cas lisse est un theoreme connu. La demonstration ici donnee repose sur un resultat sur les surfaces geometriquement rationnelles sur un corps quelconque (1983), obtenu via la K-theorie algebrique.
{"title":"Droites sur les hypersurfaces cubiques","authors":"Jean-Louis Colliot-Th'elene","doi":"10.2140/akt.2018.3.723","DOIUrl":"https://doi.org/10.2140/akt.2018.3.723","url":null,"abstract":"Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case has already been given several other proofs. \u0000-- \u0000On montre que sur toute hypersurface cubique complexe de dimension au moins 2, le groupe de Chow des cycles de dimension 1 est engendre par les droites. Le cas lisse est un theoreme connu. La demonstration ici donnee repose sur un resultat sur les surfaces geometriquement rationnelles sur un corps quelconque (1983), obtenu via la K-theorie algebrique.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2018.3.723","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48742395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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