Pub Date : 2023-12-15DOI: 10.4310/mrl.2023.v30.n3.a4
Vitalijs Brejevs, Andy Wand
We construct an infinite family of genus one open book decompositions supporting Stein-fillable contact structures and show that their monodromies do not admit positive factorisations. This extends a line of counterexamples in higher genera and establishes that a correspondence between Stein fillings and positive factorisations only exists for planar open book decompositions.
{"title":"Stein-fillable open books of genus one that do not admit positive factorisations","authors":"Vitalijs Brejevs, Andy Wand","doi":"10.4310/mrl.2023.v30.n3.a4","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n3.a4","url":null,"abstract":"We construct an infinite family of genus one open book decompositions supporting Stein-fillable contact structures and show that their monodromies do not admit positive factorisations. This extends a line of counterexamples in higher genera and establishes that a correspondence between Stein fillings and positive factorisations only exists for planar open book decompositions.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138682067","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 : 2023-12-15DOI: 10.4310/mrl.2023.v30.n3.a7
Dmitry Kleinbock, Ioannis Konstantoulas, Florian K. Richter
In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauß map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co‑null set respectively).
{"title":"Zero–one laws for eventually always hitting points in rapidly mixing systems","authors":"Dmitry Kleinbock, Ioannis Konstantoulas, Florian K. Richter","doi":"10.4310/mrl.2023.v30.n3.a7","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n3.a7","url":null,"abstract":"In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauß map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co‑null set respectively).","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"19 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138681922","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 : 2023-12-15DOI: 10.4310/mrl.2023.v30.n3.a12
Ming Xiao
In this paper, we first prove a Huang’s lemma type result. Then we discuss its applications in studying rigidity problems of mappings into indefinite hyperbolic spaces and bounded symmetric domains.
{"title":"A theorem on Hermitian rank and mapping problems","authors":"Ming Xiao","doi":"10.4310/mrl.2023.v30.n3.a12","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n3.a12","url":null,"abstract":"In this paper, we first prove a Huang’s lemma type result. Then we discuss its applications in studying rigidity problems of mappings into indefinite hyperbolic spaces and bounded symmetric domains.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"19 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138681986","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 : 2023-12-15DOI: 10.4310/mrl.2023.v30.n3.a5
Rankeya Datta, Nicholas Switala, Wenliang Zhang
Let $R$ be a polynomial or formal power series ring with coefficients in a DVR $V$ of mixed characteristic with a uniformizer $pi$. We prove that the $R$-module annihilator of any nonzero $mathcal{D}(R,V)$-module is either zero or is generated by a power of $pi$. In contrast to the equicharacteristic case, nonzero annihilators can occur; we give an example of a top local cohomology module of the ring $mathbb{Z}_2 [[x_0, dotsc, x_5]]$ that is annihilated by $2$, thereby answering a question of Hochster in the negative. The same example also provides a counterexample to a conjecture of Lyubeznik and Yildirim.
{"title":"Annihilators of $D$-modules in mixed characteristic","authors":"Rankeya Datta, Nicholas Switala, Wenliang Zhang","doi":"10.4310/mrl.2023.v30.n3.a5","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n3.a5","url":null,"abstract":"Let $R$ be a polynomial or formal power series ring with coefficients in a DVR $V$ of mixed characteristic with a uniformizer $pi$. We prove that the $R$-module annihilator of any nonzero $mathcal{D}(R,V)$-module is either zero or is generated by a power of $pi$. In contrast to the equicharacteristic case, nonzero annihilators can occur; we give an example of a top local cohomology module of the ring $mathbb{Z}_2 [[x_0, dotsc, x_5]]$ that is annihilated by $2$, thereby answering a question of Hochster in the negative. The same example also provides a counterexample to a conjecture of Lyubeznik and Yildirim.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"172 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138681996","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 : 2023-12-15DOI: 10.4310/mrl.2023.v30.n3.a6
Yan Mary He, Hongming Nie
We introduce a Riemannian metric on certain hyperbolic components in the moduli space of degree at least $2$ rational maps in one complex variable. Our metric is constructed by considering the measure-theoretic entropy of a rational map with respect to some equilibrium state.
{"title":"A Riemannian metric on hyperbolic components","authors":"Yan Mary He, Hongming Nie","doi":"10.4310/mrl.2023.v30.n3.a6","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n3.a6","url":null,"abstract":"We introduce a Riemannian metric on certain hyperbolic components in the moduli space of degree at least $2$ rational maps in one complex variable. Our metric is constructed by considering the measure-theoretic entropy of a rational map with respect to some equilibrium state.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"33 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138682000","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 : 2023-12-15DOI: 10.4310/mrl.2023.v30.n3.a10
Katrina Morgan, Jared Wunsch
We obtain estimates on the rate of decay of a solution to the wave equation on a stationary spacetime that tends to Minkowski space at a rate $O ({lvert x rvert}^{-kappa}), kappa in (1,infty) backslash mathbb{N}$. Given suitably smooth and decaying initial data, we show a wave locally enjoys the decay rate $O(t^{-kappa-2+epsilon})$.
我们得到了关于静止时空中波方程解的衰减率的估计，该方程以 $O ({lvert x rvert}^{-kappa}), kappa in (1,infty) backslash mathbb{N}$的速率趋向于闵科夫斯基空间。给定合适的平滑和衰变的初始数据，我们展示了一个波局部享有衰变率 $O(t^{-kappa-2+epsilon})$。
{"title":"Generalized Price’s law on fractional-order asymptotically flat stationary spacetimes","authors":"Katrina Morgan, Jared Wunsch","doi":"10.4310/mrl.2023.v30.n3.a10","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n3.a10","url":null,"abstract":"We obtain estimates on the rate of decay of a solution to the wave equation on a stationary spacetime that tends to Minkowski space at a rate $O ({lvert x rvert}^{-kappa}), kappa in (1,infty) backslash mathbb{N}$. Given suitably smooth and decaying initial data, we show a wave locally enjoys the decay rate $O(t^{-kappa-2+epsilon})$.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"92 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138681991","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 : 2023-09-13DOI: 10.4310/mrl.2023.v30.n2.a9
Kevin J. McGown, Frank Thorne, Amanda Tucker
We prove several results concerning genus numbers of quintic fields: we compute the proportion of quintic fields with genus number one; we prove that a positive proportion of quintic fields have arbitrarily large genus number; and we compute the average genus number of quintic fields. All of these results also hold when restricted to $S_5$-quintic fields only.
{"title":"Counting quintic fields with genus number one","authors":"Kevin J. McGown, Frank Thorne, Amanda Tucker","doi":"10.4310/mrl.2023.v30.n2.a9","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a9","url":null,"abstract":"We prove several results concerning genus numbers of quintic fields: we compute the proportion of quintic fields with genus number one; we prove that a positive proportion of quintic fields have arbitrarily large genus number; and we compute the average genus number of quintic fields. All of these results also hold when restricted to $S_5$-quintic fields only.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"14 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138509869","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 : 2023-06-21DOI: 10.4310/mrl.2023.v30.n1.a6
D. Huybrechts, M. Mauri
We give a simple argument to prove Nagai’s conjecture for type II degenerations of compact hyperkähler manifolds and cohomology classes of middle degree. Under an additional assumption, the techniques yield the conjecture in arbitrary degree. This would complete the proof of Nagai’s conjecture in general, as it was proved already for type I degenerations by Kollár, Laza, Saccà, and Voisin [10] and independently by Soldatenkov [18], while it is immediate for type III degenerations. Our arguments are close in spirit to a recent paper by Harder [8] proving similar results for the restrictive class of good degenerations.
{"title":"On type II degenerations of hyperkähler manifolds","authors":"D. Huybrechts, M. Mauri","doi":"10.4310/mrl.2023.v30.n1.a6","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n1.a6","url":null,"abstract":"We give a simple argument to prove Nagai’s conjecture for type II degenerations of compact hyperkähler manifolds and cohomology classes of middle degree. Under an additional assumption, the techniques yield the conjecture in arbitrary degree. This would complete the proof of Nagai’s conjecture in general, as it was proved already for type I degenerations by Kollár, Laza, Saccà, and Voisin [<b>10</b>] and independently by Soldatenkov [<b>18</b>], while it is immediate for type III degenerations. Our arguments are close in spirit to a recent paper by Harder [<b>8</b>] proving similar results for the restrictive class of good degenerations.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"13 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138509870","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 : 2023-01-01DOI: 10.4310/mrl.2023.v30.n2.a7
Sergei V. Konyagin, Min Sha, Igor E. Shparlinski, Cameron L. Stewart
{"title":"On the distribution of multiplicatively dependent vectors","authors":"Sergei V. Konyagin, Min Sha, Igor E. Shparlinski, Cameron L. Stewart","doi":"10.4310/mrl.2023.v30.n2.a7","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a7","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135403482","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 : 2023-01-01DOI: 10.4310/mrl.2023.v30.n2.a10
Sheng Meng, Guolei Zhong
Let $X$ be a rationally connected smooth projective variety of dimension $n$. We show that $X$ is a toric variety if and only if $X$ admits an int-amplified endomorphism with totally invariant ramification divisor. We also show that $Xcong (mathbb{P}^1)^{times n}$ if and only if $X$ admits a surjective endomorphism $f$ such that the eigenvalues of $f^*|_{text{N}^1(X)}$ (without counting multiplicities) are $n$ distinct real numbers greater than $1$.
{"title":"Rigidity of rationally connected smooth projective varieties from dynamical viewpoints","authors":"Sheng Meng, Guolei Zhong","doi":"10.4310/mrl.2023.v30.n2.a10","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a10","url":null,"abstract":"Let $X$ be a rationally connected smooth projective variety of dimension $n$. We show that $X$ is a toric variety if and only if $X$ admits an int-amplified endomorphism with totally invariant ramification divisor. We also show that $Xcong (mathbb{P}^1)^{times n}$ if and only if $X$ admits a surjective endomorphism $f$ such that the eigenvalues of $f^*|_{text{N}^1(X)}$ (without counting multiplicities) are $n$ distinct real numbers greater than $1$.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"190 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136297298","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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