Pub Date : 2022-01-01DOI: 10.4310/acta.2022.v229.n1.a1
Yang Li
{"title":"Strominger–Yau–Zaslow conjecture for Calabi–Yau hypersurfaces in the Fermat family","authors":"Yang Li","doi":"10.4310/acta.2022.v229.n1.a1","DOIUrl":"https://doi.org/10.4310/acta.2022.v229.n1.a1","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71153294","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 : 2021-02-24DOI: 10.4467/20843828am.21.001.14982
Christian Budde
We use a version of the Trotter-Kato approximation theorem for strongly continuous semigroups in order to study ows on growing networks. For that reason we use the abstract notion of direct limits in the sense of category theory
{"title":"Semigroups for flows on limits of graphs","authors":"Christian Budde","doi":"10.4467/20843828am.21.001.14982","DOIUrl":"https://doi.org/10.4467/20843828am.21.001.14982","url":null,"abstract":"We use a version of the Trotter-Kato approximation theorem for strongly continuous semigroups in order to study ows on growing networks. For that reason we use the abstract notion of direct limits in the sense of category theory","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2021-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49157307","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 : 2020-12-08DOI: 10.4310/acta.2022.v229.n2.a3
T. Tao
emph{Sendov's conjecture} asserts that if a complex polynomial $f$ of degree $n geq 2$ has all of its zeroes in closed unit disk ${ z: |z| leq 1 }$, then for each such zero $lambda_0$ there is a zero of the derivative $f'$ in the closed unit disk ${ z: |z-lambda_0| leq 1 }$. This conjecture is known for $n < 9$, but only partial results are available for higher $n$. We show that there exists a constant $n_0$ such that Sendov's conjecture holds for $n geq n_0$. For $lambda_0$ away from the origin and the unit circle we can appeal to the prior work of Degot and Chalebgwa; for $lambda_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when $lambda_0$ is extremely close to the unit circle); and for $lambda_0$ near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.
{"title":"Sendov’s conjecture for sufficiently-high-degree polynomials","authors":"T. Tao","doi":"10.4310/acta.2022.v229.n2.a3","DOIUrl":"https://doi.org/10.4310/acta.2022.v229.n2.a3","url":null,"abstract":"emph{Sendov's conjecture} asserts that if a complex polynomial $f$ of degree $n geq 2$ has all of its zeroes in closed unit disk ${ z: |z| leq 1 }$, then for each such zero $lambda_0$ there is a zero of the derivative $f'$ in the closed unit disk ${ z: |z-lambda_0| leq 1 }$. This conjecture is known for $n < 9$, but only partial results are available for higher $n$. We show that there exists a constant $n_0$ such that Sendov's conjecture holds for $n geq n_0$. For $lambda_0$ away from the origin and the unit circle we can appeal to the prior work of Degot and Chalebgwa; for $lambda_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when $lambda_0$ is extremely close to the unit circle); and for $lambda_0$ near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46209320","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 : 2020-11-09DOI: 10.4467/20843828AM.20.002.13312
Marcin Sroka
We provide the classification of the six-dimensional decomposable Lie algebras, with the dimension of the biggest indecomposable summand less than five, admitting complex structures.
我们给出了六维可分解李代数的分类，其中最大不可分解被加数的维数小于5，允许复杂结构。
{"title":"Existence of complex structures on decomposable Lie algebras","authors":"Marcin Sroka","doi":"10.4467/20843828AM.20.002.13312","DOIUrl":"https://doi.org/10.4467/20843828AM.20.002.13312","url":null,"abstract":"We provide the classification of the six-dimensional decomposable Lie algebras, with the dimension of the biggest indecomposable summand less than five, admitting complex structures.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"2020 1","pages":"25-58"},"PeriodicalIF":3.7,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41533980","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 : 2020-08-17DOI: 10.4310/ACTA.2020.V225.N2.A4
David Gabai, Mehdi Yazdi
In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $mathcal{F'}$ is homotopic to that of $mathcal{F}$. In particular, $mathcal{F}$ and $mathcal{F'}$ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.
Bill Thurston在其1976年的开创性论文中观察到，叶理的闭叶S的欧拉特征与[S]上评估的叶理的欧拉类（由S表示的同源类）一致。本文的主要结果是张叶理的逆：如果在[S]上评估的张叶理$mathcal｛F｝$的欧拉类等于S的欧拉特征，并且下面的流形是双曲的，则存在另一张紧叶理$mathcal｛F’｝$，使得$S$与叶的并集同源。特别是，$mathcal｛F｝$和$mathical｛F’｝$具有相同的Euler类。在同一篇论文中，Thurston证明了闭双曲3-流形上的张叶理具有最多为1的欧拉范数类，并推测反过来，任何范数等于1的积分上同调类都是张叶理的欧拉类。这是两篇论文中的第二篇，这两篇论文共同对瑟斯顿猜想给出了否定的答案。在第一篇论文中，假设本文的主要结果，构造了反例。
{"title":"The fully marked surface theorem","authors":"David Gabai, Mehdi Yazdi","doi":"10.4310/ACTA.2020.V225.N2.A4","DOIUrl":"https://doi.org/10.4310/ACTA.2020.V225.N2.A4","url":null,"abstract":"In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $mathcal{F'}$ is homotopic to that of $mathcal{F}$. In particular, $mathcal{F}$ and $mathcal{F'}$ have the same Euler class. \u0000In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41544577","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 : 2020-08-04DOI: 10.4467/20843828AM.20.001.13311
Punam Gupta, A. Diallo
In this paper, we study the doubly warped product manifolds with semi-symmetric metric connection. We derive the curvature formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for an Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.
{"title":"Einstein doubly warped product manifolds with semi-symmetric metric connection","authors":"Punam Gupta, A. Diallo","doi":"10.4467/20843828AM.20.001.13311","DOIUrl":"https://doi.org/10.4467/20843828AM.20.001.13311","url":null,"abstract":"In this paper, we study the doubly warped product manifolds with semi-symmetric metric connection. We derive the curvature formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for an Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2020-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42124236","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 : 2020-08-01DOI: 10.4310/ACTA.2020.V225.N2.A2
D. Masser, U. Zannier
The main results of this paper involve general algebraic differentials $omega$ on a general pencil of algebraic curves. We show how to determine if $omega$ is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of Andre and Hrushovski and with the Grothendieck-Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin-Mumford type allied to the Zilber-Pink conjectures: we characterize torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations $A^2-DB^2=1$ over a polynomial ring. We determine whether the Pell equation (with squarefree $D$) is solvable for infinitely many members of the pencil.
{"title":"Torsion points, Pell’s equation, and integration in elementary terms","authors":"D. Masser, U. Zannier","doi":"10.4310/ACTA.2020.V225.N2.A2","DOIUrl":"https://doi.org/10.4310/ACTA.2020.V225.N2.A2","url":null,"abstract":"The main results of this paper involve general algebraic differentials $omega$ on a general pencil of algebraic curves. We show how to determine if $omega$ is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of Andre and Hrushovski and with the Grothendieck-Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin-Mumford type allied to the Zilber-Pink conjectures: we characterize torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations $A^2-DB^2=1$ over a polynomial ring. We determine whether the Pell equation (with squarefree $D$) is solvable for infinitely many members of the pencil.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"225 1","pages":"227-312"},"PeriodicalIF":3.7,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43516754","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 : 2020-04-27DOI: 10.4310/acta.2022.v229.n1.a2
A. Naor, Robert Young
We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $nge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $sqrt[4]{log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $nge 2$ is of order $sqrt{log n}$.
{"title":"Foliated corona decompositions","authors":"A. Naor, Robert Young","doi":"10.4310/acta.2022.v229.n1.a2","DOIUrl":"https://doi.org/10.4310/acta.2022.v229.n1.a2","url":null,"abstract":"We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. \u0000The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. \u0000Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $nge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $sqrt[4]{log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $nge 2$ is of order $sqrt{log n}$.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2020-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48326012","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 : 2020-04-18DOI: 10.4310/acta.2023.v230.n2.a1
Younghan Bae, D. Holmes, R. Pandharipande, Johannes Schmitt, Rosa Schwarz
Let $A=(a_1,ldots,a_n)$ be a vector of integers with $d=sum_{i=1}^n a_i$. By partial resolution of the classical Abel-Jacobi map, we construct a universal twisted double ramification cycle $mathsf{DR}^{mathsf{op}}_{g,A}$ as an operational Chow class on the Picard stack $mathfrak{Pic}_{g,n,d}$ of $n$-pointed genus $g$ curves carrying a degree $d$ line bundle. The method of construction follows the log (and b-Chow) approach to the standard double ramification cycle with canonical twists on the moduli space of curves [arXiv:1707.02261, arXiv:1711.10341, arXiv:1708.04471]. Our main result is a calculation of $mathsf{DR}^{mathsf{op}}_{g,A}$ on the Picard stack $mathfrak{Pic}_{g,n,d}$ via an appropriate interpretation of Pixton's formula in the tautological ring. The basic new tool used in the proof is the theory of double ramification cycles for target varieties [arXiv:1812.10136]. The formula on the Picard stack is obtained from [arXiv:1812.10136] for target varieties $mathbb{CP}^n$ in the limit $n rightarrow infty$. The result may be viewed as a universal calculation in Abel-Jacobi theory. As a consequence of the calculation of $mathsf{DR}^{mathsf{op}}_{g,A}$ on the Picard stack $mathfrak{Pic}_{g,n,d}$, we prove that the fundamental classes of the moduli spaces of twisted meromorphic differentials in $overline{mathcal{M}}_{g,n}$ are exactly given by Pixton's formula (as conjectured in the appendix to [arXiv:1508.07940] and in [arXiv:1607.08429]). The comparison result of fundamental classes proven in [arXiv:1909.11981] plays a crucial role in our argument. We also prove the set of relations in the tautological ring of the Picard stack $mathfrak{Pic}_{g,n,d}$ associated to Pixton's formula.
{"title":"Pixton’s formula and Abel–Jacobi theory on the Picard stack","authors":"Younghan Bae, D. Holmes, R. Pandharipande, Johannes Schmitt, Rosa Schwarz","doi":"10.4310/acta.2023.v230.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2023.v230.n2.a1","url":null,"abstract":"Let $A=(a_1,ldots,a_n)$ be a vector of integers with $d=sum_{i=1}^n a_i$. By partial resolution of the classical Abel-Jacobi map, we construct a universal twisted double ramification cycle $mathsf{DR}^{mathsf{op}}_{g,A}$ as an operational Chow class on the Picard stack $mathfrak{Pic}_{g,n,d}$ of $n$-pointed genus $g$ curves carrying a degree $d$ line bundle. The method of construction follows the log (and b-Chow) approach to the standard double ramification cycle with canonical twists on the moduli space of curves [arXiv:1707.02261, arXiv:1711.10341, arXiv:1708.04471]. \u0000Our main result is a calculation of $mathsf{DR}^{mathsf{op}}_{g,A}$ on the Picard stack $mathfrak{Pic}_{g,n,d}$ via an appropriate interpretation of Pixton's formula in the tautological ring. The basic new tool used in the proof is the theory of double ramification cycles for target varieties [arXiv:1812.10136]. The formula on the Picard stack is obtained from [arXiv:1812.10136] for target varieties $mathbb{CP}^n$ in the limit $n rightarrow infty$. The result may be viewed as a universal calculation in Abel-Jacobi theory. \u0000As a consequence of the calculation of $mathsf{DR}^{mathsf{op}}_{g,A}$ on the Picard stack $mathfrak{Pic}_{g,n,d}$, we prove that the fundamental classes of the moduli spaces of twisted meromorphic differentials in $overline{mathcal{M}}_{g,n}$ are exactly given by Pixton's formula (as conjectured in the appendix to [arXiv:1508.07940] and in [arXiv:1607.08429]). The comparison result of fundamental classes proven in [arXiv:1909.11981] plays a crucial role in our argument. We also prove the set of relations in the tautological ring of the Picard stack $mathfrak{Pic}_{g,n,d}$ associated to Pixton's formula.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2020-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44025420","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}
{"title":"The number of closed ideals in $L(L_p)$","authors":"W. Johnson, G. Schechtman","doi":"10.4310/acta.2021.v227.n1.a2","DOIUrl":"https://doi.org/10.4310/acta.2021.v227.n1.a2","url":null,"abstract":"We show that there are $2^{2^{aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<pnot= 2<infty$. This solves a problem in A. Pietsch's 1978 book \"Operator Ideals\". The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{aleph_0}}$ closed ideals in term of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1<q<2$ the space ${frak X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46645010","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}
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