Pub Date : 2023-01-01DOI: 10.4310/mrl.2023.v30.n2.a1
Robert L. Benedetto, Su-Ion Ih
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.
设$k$为一个代数闭包为$bar{k}$的数域，设$S$为$k$中包含所有阿基米德数的有限位集。修复$dgeq 2$和$alpha in bar{k}$，使映射$zmapsto z^d+alpha$不是后临界有限的。假设$alpha$上的一个技术假设，我们证明只有有限多个参数$cinbar{k}$，其中$zmapsto z^d+c$是后批判有限的，并且$c$相对于$(alpha)$是$S$ -积分。即在d次单临界多项式的模空间中，只有有限多个PCF $bar{k}$ -有理点$((alpha),S)$ -积分。我们推测，没有技术假设，同样的陈述也是正确的。
{"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}
Pub Date : 2023-01-01DOI: 10.4310/mrl.2023.v30.n1.a4
Yuqiu Fu, Shengwen Gan
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.
{"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}
Pub Date : 2023-01-01DOI: 10.4310/mrl.2023.v30.n1.a1
Piotr Achinger
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.
在Hansen和Li的一个问题的激励下，我们证明了具有投影约简的光滑固有刚性解析空间$X$在下列情况下满足Hodge对称:(1)基非阿基米德域$K$具有残差特征为零，(2)$K$为$p$-进，$X$具有良好的普通约简，(3)$K$为$p$-进，$X$具有组合约简。我们还在不使用半稳定轴的模空间的情况下，证明了他们的结果H^1的Hodge对称的一个版本。所有这些都依赖于加藤的log hard Lefschetz猜想，我们对H^1和组合型的log格式证明了这个猜想。
{"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}
Pub Date : 2023-01-01DOI: 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}
Pub Date : 2023-01-01DOI: 10.4310/mrl.2023.v30.n2.a8
Abdellah Lahdili
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.
{"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}
Pub Date : 2023-01-01DOI: 10.4310/mrl.2023.v30.n2.a6
Minyoung Jeon
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.
{"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}
Pub Date : 2023-01-01DOI: 10.4310/mrl.2023.v30.n2.a11
Yongquan Zhang
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.
{"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}
Pub Date : 2023-01-01DOI: 10.4310/mrl.2023.v30.n2.a2
Giovanni Catino
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.
{"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}
{"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}
Pub Date : 2023-01-01DOI: 10.4310/mrl.2023.v30.n2.a3
Pierre-Emmanuel Chaput, Nicolas Perrin
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.
{"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}
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