Pub Date : 2023-11-03DOI: 10.2140/pjm.2023.325.281
Wade Hindes, Joseph H. Silverman
Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $Pin V(K)$. For each prime $mathfrak{p}$ of $K$, let $m_{mathfrak{p}}(S,P)$ denote the number of points in the orbit of $Pbmodmathfrak{p}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{mathfrak{p}}(S,P)$ and use it to show that the set of primes for which $m_{mathfrak{p}}(S,P)$ grows subexponentially as a function of $operatorname{mathsf{N}}_{K/mathbb{Q}}mathfrak{p}$ is a set of density zero. For $V=mathbb{P}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.
{"title":"The size of semigroup orbits modulo primes","authors":"Wade Hindes, Joseph H. Silverman","doi":"10.2140/pjm.2023.325.281","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.281","url":null,"abstract":"Let $V$ be a projective variety defined over a number field $K$, let $S$ be a polarized set of endomorphisms of $V$ all defined over $K$, and let $Pin V(K)$. For each prime $mathfrak{p}$ of $K$, let $m_{mathfrak{p}}(S,P)$ denote the number of points in the orbit of $Pbmodmathfrak{p}$ for the semigroup of maps generated by $S$. Under suitable hypotheses on $S$ and $P$, we prove an analytic estimate for $m_{mathfrak{p}}(S,P)$ and use it to show that the set of primes for which $m_{mathfrak{p}}(S,P)$ grows subexponentially as a function of $operatorname{mathsf{N}}_{K/mathbb{Q}}mathfrak{p}$ is a set of density zero. For $V=mathbb{P}^1$ we show that this holds for a generic set of maps $S$ provided that at least two of the maps in $S$ have degree at least four.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"27 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819346","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 : 2023-11-03DOI: 10.2140/pjm.2023.325.191
Michelle Chu, Alan W. Reid
{"title":"Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in S4","authors":"Michelle Chu, Alan W. Reid","doi":"10.2140/pjm.2023.325.191","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.191","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"25 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135820399","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 : 2023-11-03DOI: 10.2140/pjm.2023.325.353
Luka Zwaan
In arXiv1312.7267, the first non-trivial example of a Poisson manifold of strong compact type is given. The construction uses the theory of K3 surfaces and results in a Poisson manifold with leaf space $S^1$. We modify the construction to obtain a new class of examples. Specifically, we obtain for each strongly integral affine 2-torus a Poisson manifold of strong compact type with said torus as leaf space.
{"title":"Poisson manifolds of strong compact type over 2-tori","authors":"Luka Zwaan","doi":"10.2140/pjm.2023.325.353","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.353","url":null,"abstract":"In arXiv1312.7267, the first non-trivial example of a Poisson manifold of strong compact type is given. The construction uses the theory of K3 surfaces and results in a Poisson manifold with leaf space $S^1$. We modify the construction to obtain a new class of examples. Specifically, we obtain for each strongly integral affine 2-torus a Poisson manifold of strong compact type with said torus as leaf space.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"243 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775118","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 : 2023-11-03DOI: 10.2140/pjm.2023.325.239
Jason DeVito
A double disk bundle is any smooth closed manifold obtained as the union of the total spaces of two disk bundles, glued together along their common boundary. The Double Soul Conjecture asserts that a closed simply connected manifold admitting a metric of non-negative sectional curvature is necessarily a double disk bundle. We study a generalization of this conjecture by dropping the requirement that the manifold be simply connected. Previously, a unique counterexample was known to this generalization, the Poincar'e dodecahedral space $S^3/I^ast$. We find infinitely many $3$-dimensional counterexamples, as well as another infinite family of flat counterexamples whose dimensions grow without bound.
{"title":"Counterexamples to the nonsimply connected double soul conjecture","authors":"Jason DeVito","doi":"10.2140/pjm.2023.325.239","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.239","url":null,"abstract":"A double disk bundle is any smooth closed manifold obtained as the union of the total spaces of two disk bundles, glued together along their common boundary. The Double Soul Conjecture asserts that a closed simply connected manifold admitting a metric of non-negative sectional curvature is necessarily a double disk bundle. We study a generalization of this conjecture by dropping the requirement that the manifold be simply connected. Previously, a unique counterexample was known to this generalization, the Poincar'e dodecahedral space $S^3/I^ast$. We find infinitely many $3$-dimensional counterexamples, as well as another infinite family of flat counterexamples whose dimensions grow without bound.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"26 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819357","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 : 2023-11-03DOI: 10.2140/pjm.2023.325.299
Yat-Hin Suen
It is well-known that toric line bundles on a toric variety correspond to piecewise linear functions on the fan. For toric vector bundles, Payne constructed a branched covering over the fan and a piecewise linear function on the domain. We think of these objects as the tropicalization of Lagrangian multi-sections and therefore, deserve the name tropical Lagrangian multi-sections. In this paper, we study the reconstruction problem of toric vector bundles from a given tropical Lagrangian multi-section. Those tropical Lagrangian multi-sections that arise from toric vector bundles are called unobstructed. We reformulate Kaneyama's classification of toric vector bundles in terms of the language of tropical Lagrangian multi-sections. We also provide a ``SYZ-type approach to construct toric vector bundles from tropical Lagrangian multi-sections. In dimension 2, such ``mirror-symmetric approach provides us a combinatorial condition for checking which rank 2 tropical Lagrangian multi-section is unobstructed.
{"title":"Tropical Lagrangian multisections and toric vector bundles","authors":"Yat-Hin Suen","doi":"10.2140/pjm.2023.325.299","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.299","url":null,"abstract":"It is well-known that toric line bundles on a toric variety correspond to piecewise linear functions on the fan. For toric vector bundles, Payne constructed a branched covering over the fan and a piecewise linear function on the domain. We think of these objects as the tropicalization of Lagrangian multi-sections and therefore, deserve the name tropical Lagrangian multi-sections. In this paper, we study the reconstruction problem of toric vector bundles from a given tropical Lagrangian multi-section. Those tropical Lagrangian multi-sections that arise from toric vector bundles are called unobstructed. We reformulate Kaneyama's classification of toric vector bundles in terms of the language of tropical Lagrangian multi-sections. We also provide a ``SYZ-type approach to construct toric vector bundles from tropical Lagrangian multi-sections. In dimension 2, such ``mirror-symmetric approach provides us a combinatorial condition for checking which rank 2 tropical Lagrangian multi-section is unobstructed.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"220 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775000","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 : 2023-11-03DOI: 10.2140/pjm.2023.325.255
Payman Eskandari, V. Kumar Murty
{"title":"The fundamental group of an extension in a Tannakian category and the unipotent radical of the Mumford–Tate group of an open curve","authors":"Payman Eskandari, V. Kumar Murty","doi":"10.2140/pjm.2023.325.255","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.255","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"26 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135819356","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 : 2023-11-03DOI: 10.2140/pjm.2023.325.331
Matteo Verzobio
Let $P$ be a non-torsion point on an elliptic curve defined over a number field $K$ and consider the sequence ${B_n}_{nin mathbb{N}}$ of the denominators of $x(nP)$. We prove that every term of the sequence of the $B_n$ has a primitive divisor for $n$ greater than an effectively computable constant that we will explicitly compute. This constant will depend only on the model defining the curve.
{"title":"Some effectivity results for primitive divisors of elliptic divisibility sequences","authors":"Matteo Verzobio","doi":"10.2140/pjm.2023.325.331","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.331","url":null,"abstract":"Let $P$ be a non-torsion point on an elliptic curve defined over a number field $K$ and consider the sequence ${B_n}_{nin mathbb{N}}$ of the denominators of $x(nP)$. We prove that every term of the sequence of the $B_n$ has a primitive divisor for $n$ greater than an effectively computable constant that we will explicitly compute. This constant will depend only on the model defining the curve.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"25 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135820225","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 : 2023-11-03DOI: 10.2140/pjm.2023.325.209
Anthony Conway, Diarmuid Crowley, Mark Powell
We show that for every odd prime $q$, there exists an infinite family ${M_i}_{i=1}^{infty}$ of topological 4-manifolds that are all stably homeomorphic to one another, all the manifolds $M_i$ have isometric rank one equivariant intersection pairings and boundary $L(2q, 1) # (S^1 times S^2)$, but they are pairwise not homotopy equivalent via any homotopy equivalence that restricts to a homotopy equivalence of the boundary.
{"title":"Infinite homotopy stable class for 4-manifolds with boundary","authors":"Anthony Conway, Diarmuid Crowley, Mark Powell","doi":"10.2140/pjm.2023.325.209","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.209","url":null,"abstract":"We show that for every odd prime $q$, there exists an infinite family ${M_i}_{i=1}^{infty}$ of topological 4-manifolds that are all stably homeomorphic to one another, all the manifolds $M_i$ have isometric rank one equivariant intersection pairings and boundary $L(2q, 1) # (S^1 times S^2)$, but they are pairwise not homotopy equivalent via any homotopy equivalence that restricts to a homotopy equivalence of the boundary.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"26 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135820395","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}
Rondinelle Batista, Barnab'e Lima, Paulo Sousa, Bruno Vieira
. We explore the fourth-order Steklov problem of a compact embedded hyper-surface (cid:54) n with free boundary in a ( n + 1 ) -dimensional compact manifold M n + 1 which has nonnegative Ricci curvature and strictly convex boundary. If (cid:54) is minimal we establish a lower bound for the first eigenvalue of this problem. When M = B n + 1 is the unit ball in (cid:82) n + 1 , if (cid:54) has constant mean curvature H (cid:54) we prove that the first eigenvalue satisfies σ 1 ≤ n + | H (cid:54) | . In the minimal case ( H (cid:54) = 0), we prove that σ 1 = n .
{"title":"Estimate for the first fourth Steklov eigenvalue of a minimal hypersurface with free boundary","authors":"Rondinelle Batista, Barnab'e Lima, Paulo Sousa, Bruno Vieira","doi":"10.2140/pjm.2023.325.1","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.1","url":null,"abstract":". We explore the fourth-order Steklov problem of a compact embedded hyper-surface (cid:54) n with free boundary in a ( n + 1 ) -dimensional compact manifold M n + 1 which has nonnegative Ricci curvature and strictly convex boundary. If (cid:54) is minimal we establish a lower bound for the first eigenvalue of this problem. When M = B n + 1 is the unit ball in (cid:82) n + 1 , if (cid:54) has constant mean curvature H (cid:54) we prove that the first eigenvalue satisfies σ 1 ≤ n + | H (cid:54) | . In the minimal case ( H (cid:54) = 0), we prove that σ 1 = n .","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45478580","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 : 2023-09-03DOI: 10.2140/pjm.2023.325.105
J. Hebda
This paper introduces the stable systoles of higher rank of a Riemannian manifold as a generalization of the usual stable systoles. Several inequalities involving these higher rank systoles are proved.
本文将黎曼流形的高阶稳定系统作为一般稳定系统的推广。证明了涉及这些高阶系统的几个不等式。
{"title":"Stable systoles of higher rank in Riemannian manifolds","authors":"J. Hebda","doi":"10.2140/pjm.2023.325.105","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.105","url":null,"abstract":"This paper introduces the stable systoles of higher rank of a Riemannian manifold as a generalization of the usual stable systoles. Several inequalities involving these higher rank systoles are proved.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45678182","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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