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A finiteness property of postcritically finite unicritical polynomials 后临界有限单临界多项式的有限性质
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n2.a1
Robert L. Benedetto, Su-Ion Ih
{"title":"A finiteness property of postcritically finite unicritical polynomials","authors":"Robert L. Benedetto, Su-Ion Ih","doi":"10.4310/mrl.2023.v30.n2.a1","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a1","url":null,"abstract":"Let $k$ be a number field with algebraic closure $bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $dgeq 2$ and $alpha in bar{k}$ such that the map $zmapsto z^d+alpha$ is not postcritically finite. Assuming a technical hypothesis on $alpha$, we prove that there are only finitely many parameters $cinbar{k}$ for which $zmapsto z^d+c$ is postcritically finite and for which $c$ is $S$-integral relative to $(alpha)$. That is, in the moduli space of unicritical polynomials of degree d, there are only finitely many PCF $bar{k}$-rational points that are $((alpha),S)$-integral. We conjecture that the same statement is true without the technical hypothesis.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"4 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":"135784442","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}
引用次数: 1
On Kakeya maps with regularity assumptions 在有规律性假设的Kakeya地图上
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n1.a4
Yuqiu Fu, Shengwen Gan
{"title":"On Kakeya maps with regularity assumptions","authors":"Yuqiu Fu, Shengwen Gan","doi":"10.4310/mrl.2023.v30.n1.a4","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n1.a4","url":null,"abstract":"For a $n-$dimensional Kakeya set $(ngeq 3)$ we may define a Kakeya map associated to it which parametrizes the Kakeya set by $[0,1]times S^{n-1},$ where $S^{n-1}$ is thought of as the space of unit directions. We show that if the Kakeya map is either $alpha-$H\"{o}lder continuous with $alpha>frac{(n-2)n}{(n-1)^2},$ or continuous and in the Sobolev space $ H^{s} $ for some $s>(n-1)/2,$ then the Kakeya set has positive Lebesgue measure.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"17 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":"135784626","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}
引用次数: 0
Hodge symmetry for rigid varieties via $log$ hard Lefschetz 通过$log$ hard Lefschetz的刚性变量的Hodge对称
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n1.a1
Piotr Achinger
{"title":"Hodge symmetry for rigid varieties via $log$ hard Lefschetz","authors":"Piotr Achinger","doi":"10.4310/mrl.2023.v30.n1.a1","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n1.a1","url":null,"abstract":"Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base non-archimedean field $K$ is of residue characteristic zero, (2) $K$ is $p$-adic and $X$ has good ordinary reduction, (3) $K$ is $p$-adic and $X$ has combinatorial reduction.' We also reprove a version of their result, Hodge symmetry for $H^1$, without the use of moduli spaces of semistable sheaves. All of this relies on cases of Kato's log hard Lefschetz conjecture, which we prove for $H^1$ and for log schemes of combinatorial type.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"78 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":"135784811","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}
引用次数: 0
Higher $operatorname{Ext}$-groups in the triple product case 更高的$operatorname{Ext}$-组在三重积的情况下
IF 1 3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n1.a2
L. Cai, Yangyu Fan
{"title":"Higher $operatorname{Ext}$-groups in the triple product case","authors":"L. Cai, Yangyu Fan","doi":"10.4310/mrl.2023.v30.n1.a2","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n1.a2","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516788","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}
引用次数: 0
Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics 加权Mabuchi泛函的凸性与加权极值度量的唯一性
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n2.a8
Abdellah Lahdili
{"title":"Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics","authors":"Abdellah Lahdili","doi":"10.4310/mrl.2023.v30.n2.a8","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a8","url":null,"abstract":"We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal Kahler metrics on a compact Kahler manifold introduced in our previous work. This extends a result by Berman--Berndtsson and Chen--Paun--Zeng in the extremal Kahler case. Furthermore, we show that a weighted extremal Kahler metric is a global minimum of a suitable weighted version of the modified Mabuchi energy. This implies a suitable notion of weighted K-semistability of a Kahler manifold admitting a weighted extremal Kahler metric.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"242 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":"136092726","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}
引用次数: 12
Mather classes of Schubert varieties via small resolutions 通过小分辨率的舒伯特变种的马瑟类
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n2.a6
Minyoung Jeon
{"title":"Mather classes of Schubert varieties via small resolutions","authors":"Minyoung Jeon","doi":"10.4310/mrl.2023.v30.n2.a6","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a6","url":null,"abstract":"We express a Schubert expansion of the Chern-Mather class for Schubert varieties in the even orthogonal Grassmannian via integrals involving Pfaffians and pushforward of the small resolutions in the sense of Intersection Cohomology (IH) constructed by Sankaran and Vanchinathan, instead of the Nash blowup. The equivariant localization is employed to show the way of computing the integral. As a byproduct, we present the computations. For analogy and the completion of the method in ordinary Grassmannians, we also suggest Kazhdan-Lusztig classes associated to Schubert varieties in the Lagrangian and odd orthogonal Grassmannian.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"238 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":"135403493","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}
引用次数: 0
Existence of an exotic plane in an acylindrical 3-manifold 非圆柱形三流形中奇异平面的存在性
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n2.a11
Yongquan Zhang
{"title":"Existence of an exotic plane in an acylindrical 3-manifold","authors":"Yongquan Zhang","doi":"10.4310/mrl.2023.v30.n2.a11","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a11","url":null,"abstract":"Let $P$ be a geodesic plane in a convex cocompact, acylindrical hyperbolic 3-manifold $M$. Assume that $P^*=M^*cap P$ is nonempty, where $M^*$ is the interior of the convex core of $M$. Does this condition imply that $P$ is either closed or dense in $M$? A positive answer would furnish an analogue of Ratner's theorem in the infinite volume setting. In arXiv:1802.03853 it is shown that $P^*$ is either closed or dense in $M^*$. Moreover, there are at most countably many planes with $P^*$ closed, and in all previously known examples, $P$ was also closed in $M$. In this note we show more exotic behavior can occur: namely, we give an explicit example of a pair $(M,P)$ such that $P^*$ is closed in $M^*$ but $P$ is not closed in $M$. In particular, the answer to the question above is no. Thus Ratner's theorem fails to generalize to planes in acylindrical 3-manifolds, without additional restrictions.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"6 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":"135403481","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}
引用次数: 5
Metrics of constant negative scalar-Weyl curvature 常数负标量-魏尔曲率的度量
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n2.a2
Giovanni Catino
{"title":"Metrics of constant negative scalar-Weyl curvature","authors":"Giovanni Catino","doi":"10.4310/mrl.2023.v30.n2.a2","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a2","url":null,"abstract":"Extending Aubin's construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, tinmathbb{R}$. In particular, there are no topological obstructions for metrics with $varepsilon$-pinched Weyl curvature and negative scalar curvature.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"70 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":"135784133","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}
引用次数: 1
Positive currents on non-kählerian surfaces non-kählerian表面的正电流
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n2.a4
Ionuţ Chiose, Matei Toma
{"title":"Positive currents on non-kählerian surfaces","authors":"Ionuţ Chiose, Matei Toma","doi":"10.4310/mrl.2023.v30.n2.a4","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a4","url":null,"abstract":"We propose a classification of non-k\"ahlerian surfaces from a dynamical point of view and show how the known non-k\"ahlerian surfaces fit into it.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"238 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":"135784604","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}
引用次数: 0
Affine symmetries in quantum cohomology: corrections and new results 量子上同调中的仿射对称性:修正和新结果
3区 数学
Mathematical Research Letters Pub Date : 2023-01-01 DOI: 10.4310/mrl.2023.v30.n2.a3
Pierre-Emmanuel Chaput, Nicolas Perrin
{"title":"Affine symmetries in quantum cohomology: corrections and new results","authors":"Pierre-Emmanuel Chaput, Nicolas Perrin","doi":"10.4310/mrl.2023.v30.n2.a3","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n2.a3","url":null,"abstract":"In a previous paper Affine symmetries of the equivariant quantum cohomology of rational homogeneous spaces, a general formula was given for the multiplication by some special Schubert classes in the quantum cohomology of any homogeneous space. Although this formula is correct in the non equivariant setting, the stated equivariant version was wrong. We provide corrections for the equivariant formula, thus giving a correct argument for the non equivariant formula. We also give new formulas in the equivariant homology of the affine grassmannian that could lead to Pieri type formulas.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"3 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":"135784451","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}
引用次数: 0
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