Pub Date : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a1
Georgios Daskalopoulos, Chikako Mese
We prove uniqueness of equivariant harmonic maps into irreducible symmetric spaces of non-compact type and Bruhat–Tits buildings associated to isometric actions by Zariski dense subgroups.
{"title":"Uniqueness of equivariant harmonic maps to symmetric spaces and buildings","authors":"Georgios Daskalopoulos, Chikako Mese","doi":"10.4310/mrl.2023.v30.n6.a1","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a1","url":null,"abstract":"We prove uniqueness of equivariant harmonic maps into irreducible symmetric spaces of non-compact type and Bruhat–Tits buildings associated to isometric actions by Zariski dense subgroups.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"10 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744902","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a3
Siarhei Finski
We study the behavior of the Quillen metric for families of Riemann surfaces with hyperbolic cusps when the additional cusps are created by degeneration. More precisely, in our previous paper, we’ve shown that the renormalization of the Quillen metric associated with a family of Riemann surfaces with cusps extends continuously over the locus of singular curves. Here we show that modulo some explicit universal constant, this continuous extension coincides with the Quillen metric of the normalization of singular curves. As a consequence, we get an explicit relation in terms of the Bott–Chern classes between the Quillen metric associated with a metric with cusps and the Quillen metric associated with a metric on the compactified Riemann surface. We also prove compatibility between our version of the analytic torsion and the version of Takhtajan–Zograf, defined through lengths of closed geodesics.
{"title":"Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theorem","authors":"Siarhei Finski","doi":"10.4310/mrl.2023.v30.n6.a3","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a3","url":null,"abstract":"We study the behavior of the Quillen metric for families of Riemann surfaces with hyperbolic cusps when the additional cusps are created by degeneration. More precisely, in our previous paper, we’ve shown that the renormalization of the Quillen metric associated with a family of Riemann surfaces with cusps extends continuously over the locus of singular curves. Here we show that modulo some explicit universal constant, this continuous extension coincides with the Quillen metric of the normalization of singular curves. As a consequence, we get an explicit relation in terms of the Bott–Chern classes between the Quillen metric associated with a metric with cusps and the Quillen metric associated with a metric on the compactified Riemann surface. We also prove compatibility between our version of the analytic torsion and the version of Takhtajan–Zograf, defined through lengths of closed geodesics.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745088","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a11
Yuji Yang
We prove a rank-two potential automorphy theorem for $mod l$ representations satisfying an ordinary condition. Combined with an ordinary automorphy lifting theorem from $href{https://doi.org/10.4007/annals.2023.197.3.2}{textrm{[1]}}$, we prove a rank-two, $p neq l$ case of local-global compatibility for regular algebraic cuspidal automorphic representations of arbitrary weight over CM fields that is $iota$-ordinary for some $overline{mathbb{Q}}_l tilde{to} mathbb{C}$.
{"title":"An ordinary rank-two case of local-global compatibility for automorphic representations of arbitrary weight over CM fields","authors":"Yuji Yang","doi":"10.4310/mrl.2023.v30.n6.a11","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a11","url":null,"abstract":"We prove a rank-two potential automorphy theorem for $mod l$ representations satisfying an ordinary condition. Combined with an ordinary automorphy lifting theorem from $href{https://doi.org/10.4007/annals.2023.197.3.2}{textrm{[1]}}$, we prove a rank-two, $p neq l$ case of local-global compatibility for regular algebraic cuspidal automorphic representations of arbitrary weight over CM fields that is $iota$-ordinary for some $overline{mathbb{Q}}_l tilde{to} mathbb{C}$.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"575 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745094","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a10
Jia Shen, Yifei Wu
In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation. Recently, Dodson $href{https://dx.doi.org/10.4171/RMI/1295}{textrm{[16]}}$ studied the global well-posedness in a critical Sobolev space $dot{W}^{11/7,7/6}$. In this paper, we aim to show that if the initial data belongs to $dot{H}^{frac{1}{2}}$ to guarantee the local existence, then some extra weak space which is supercritical, is sufficient to prove the global well-posedness. More precisely, we prove that if the initial data belongs to $dot{H}^{1/2} cap dot{W}^{s,1}$ for $12/13 lt s leqslant 1$, then the corresponding solution exists globally and scatters.
本文研究了三维离焦立方薛定谔方程的全局好拟性和散射问题。最近，Dodson $href{https://dx.doi.org/10.4171/RMI/1295}{textrm{[16]}}$ 研究了临界 Sobolev 空间 $dot{W}^{11/7,7/6}$ 中的全局好摆性。本文旨在证明，如果初始数据属于$dot{H}^{frac{1}{2}}$以保证局部存在，那么一些额外的超临界弱空间就足以证明全局良好性。更准确地说，我们证明了如果初始数据属于 $dot{H}^{1/2} cap dot{W}^{s,1}$ 中的 $12/13 lt s leqslant 1$，那么相应的解在全局上存在并且是分散的。
{"title":"Global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation","authors":"Jia Shen, Yifei Wu","doi":"10.4310/mrl.2023.v30.n6.a10","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a10","url":null,"abstract":"In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation. Recently, Dodson $href{https://dx.doi.org/10.4171/RMI/1295}{textrm{[16]}}$ studied the global well-posedness in a critical Sobolev space $dot{W}^{11/7,7/6}$. In this paper, we aim to show that if the initial data belongs to $dot{H}^{frac{1}{2}}$ to guarantee the local existence, then some extra weak space which is supercritical, is sufficient to prove the global well-posedness. More precisely, we prove that if the initial data belongs to $dot{H}^{1/2} cap dot{W}^{s,1}$ for $12/13 lt s leqslant 1$, then the corresponding solution exists globally and scatters.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"23 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745100","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a6
Daniel Kasprowski, Mark Powell, Arunima Ray
Let $M$ be a compact 4-manifold and let $S$ and $T$ be embedded $2$-spheres in $M$, both with trivial normal bundle. We write $M_{S}$ and $M_T$ for the 4-manifolds obtained by the Gluck twist operation on $M$ along $S$ and $T$ respectively. We show that if $S$ and $T$ are concordant, then $M_S$ and $M_T$ are $s$-cobordant, and so if $pi_1(M)$ is good, then $M_S$ and $M_T$ are homeomorphic. Similarly, if $S$ and $T$ are homotopic then we show that $M_S$ and $M_T$ are simple homotopy equivalent.Under some further assumptions, we deduce th $M_S$ and $M_T$ are homeomorphic. We show that additional assumptions are necessary by giving an example where $S$ and $T$ are homotopic but $M_S$ and $M_T$ are not homeomorphic. We also give an example where $S$ and $T$ are homotopic and $M_S$ and $M_T$ are homeomorphic but not diffeomorphic.
{"title":"Gluck twists on concordant or homotopic spheres","authors":"Daniel Kasprowski, Mark Powell, Arunima Ray","doi":"10.4310/mrl.2023.v30.n6.a6","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a6","url":null,"abstract":"Let $M$ be a compact 4-manifold and let $S$ and $T$ be embedded $2$-spheres in $M$, both with trivial normal bundle. We write $M_{S}$ and $M_T$ for the 4-manifolds obtained by the Gluck twist operation on $M$ along $S$ and $T$ respectively. We show that if $S$ and $T$ are concordant, then $M_S$ and $M_T$ are $s$-cobordant, and so if $pi_1(M)$ is good, then $M_S$ and $M_T$ are homeomorphic. Similarly, if $S$ and $T$ are homotopic then we show that $M_S$ and $M_T$ are simple homotopy equivalent.Under some further assumptions, we deduce th $M_S$ and $M_T$ are homeomorphic. We show that additional assumptions are necessary by giving an example where $S$ and $T$ are homotopic but $M_S$ and $M_T$ are not homeomorphic. We also give an example where $S$ and $T$ are homotopic and $M_S$ and $M_T$ are homeomorphic but not diffeomorphic.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"64 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745091","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a8
Alberto Raffero
We consider the source-free Type IIA flow introduced by Fei–Phong–Picard–Zhang $href{https://dx.doi.org/10.4310/CJM.2021.v9.n3.a3}{textrm{[10]}}$, and we study it in the case where the relevant geometric datum is a symplectic half-flat SU(3)-structure. We show the existence of ancient, immortal and eternal solutions to the flow, provided that the initial symplectic half-flat structure satisfies suitable properties. In particular, we prove that the solution starting at a symplectic half-flat structure with Hermitian Ricci tensor is ancient and evolves self-similarly by scaling the initial datum. These results apply to all known (locally) homogeneous spaces admitting invariant symplectic half-flat SU(3)-structures.
{"title":"Special solutions to the Type IIA flow","authors":"Alberto Raffero","doi":"10.4310/mrl.2023.v30.n6.a8","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a8","url":null,"abstract":"We consider the source-free Type IIA flow introduced by Fei–Phong–Picard–Zhang $href{https://dx.doi.org/10.4310/CJM.2021.v9.n3.a3}{textrm{[10]}}$, and we study it in the case where the relevant geometric datum is a symplectic half-flat SU(3)-structure. We show the existence of ancient, immortal and eternal solutions to the flow, provided that the initial symplectic half-flat structure satisfies suitable properties. In particular, we prove that the solution starting at a symplectic half-flat structure with Hermitian Ricci tensor is ancient and evolves self-similarly by scaling the initial datum. These results apply to all known (locally) homogeneous spaces admitting invariant symplectic half-flat SU(3)-structures.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"15 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745098","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a5
Zhi Jiang, Wenfei Liu, Hang Zhao
$defAutQx{mathrm{Aut}_mathbb{Q}(X)}$ In this paper, we prove that the group $AutQx$ of numerically trivial automorphisms are uniformly bounded for smooth projective threefolds X of general type which either satisfy $q(X) geq 3$ or have a Gorenstein minimal model. If X is furthermore of maximal Albanese dimension, then $lvert AutQx rvert leq 4$, and the equality can be achieved by an unbounded family of threefolds previously constructed by the third author. Along the way we prove a Noether type inequality for log canonical pairs of general type with the coefficients of the boundary divisor from a given subset $mathcal{C} subset (0, 1]$ such that $mathcal{C} cup {1}$ attains the minimum.
{"title":"On numerically trivial automorphisms of threefolds of general type","authors":"Zhi Jiang, Wenfei Liu, Hang Zhao","doi":"10.4310/mrl.2023.v30.n6.a5","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a5","url":null,"abstract":"$defAutQx{mathrm{Aut}_mathbb{Q}(X)}$ In this paper, we prove that the group $AutQx$ of numerically trivial automorphisms are uniformly bounded for smooth projective threefolds X of general type which either satisfy $q(X) geq 3$ or have a Gorenstein minimal model. If X is furthermore of maximal Albanese dimension, then $lvert AutQx rvert leq 4$, and the equality can be achieved by an unbounded family of threefolds previously constructed by the third author. Along the way we prove a Noether type inequality for log canonical pairs of general type with the coefficients of the boundary divisor from a given subset $mathcal{C} subset (0, 1]$ such that $mathcal{C} cup {1}$ attains the minimum.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"76 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745090","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a4
Kazuhiro Ito
We prove $p$-complete arc-descent results for finite projective modules and perfect complexes over integral perfectoid rings.Using our resul,we clarify a reduction argument in the proof of the classification of $p$-divisible groups over integral perfectoid rings given by Scholze–Weinstein.
{"title":"$p$-complete arc-descent for perfect complexes over integral perfectoid rings","authors":"Kazuhiro Ito","doi":"10.4310/mrl.2023.v30.n6.a4","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a4","url":null,"abstract":"We prove $p$-complete arc-descent results for finite projective modules and perfect complexes over integral perfectoid rings.Using our resul,we clarify a reduction argument in the proof of the classification of $p$-divisible groups over integral perfectoid rings given by Scholze–Weinstein.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"18 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745089","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a7
Samir Marouani, Dan Popovici
We introduce two notions of hyperbolicity for not necessarily Kähler $n$-dimensional compact complex manifolds $X$. The first, called balanced hyperbolicity, generalises Gromov’s Kähler hyperbolicity by means of Gauduchon’s balanced metrics. The second, called divisorial hyperbolicity, generalises the Brody hyperbolicity by ruling out the existence of non-degenerate holomorphic maps from $mathbb{C}^{n-1}$ to $X$ that have what we term a subexponential growth. Our main result in the first part of the paper asserts that every balanced hyperbolic $X$ is also divisorially hyperbolic. We provide a certain number of examples and counter-examples and discuss various properties of these manifolds. In the second part of the paper, we introduce the notions of divisorially Kähler and divisorially nef real De Rham cohomology classes of degree $2$ and study their properties. They also apply to $C^infty$, not necessarily holomorphic, complex line bundles and are expected to be implied in certain cases by the hyperbolicity properties introduced in the first part of the work. While motivated by the observation of hyperbolicity properties of certain non-Kähler manifolds, all these four new notions seem to have a role to play even in the Kähler and the projective settings.
{"title":"Balanced hyperbolic and divisorially hyperbolic compact complex manifolds","authors":"Samir Marouani, Dan Popovici","doi":"10.4310/mrl.2023.v30.n6.a7","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a7","url":null,"abstract":"We introduce two notions of hyperbolicity for not necessarily Kähler $n$-dimensional compact complex manifolds $X$. The first, called <i>balanced hyperbolicity</i>, generalises Gromov’s Kähler hyperbolicity by means of Gauduchon’s balanced metrics. The second, called <i>divisorial hyperbolicity</i>, generalises the Brody hyperbolicity by ruling out the existence of non-degenerate holomorphic maps from $mathbb{C}^{n-1}$ to $X$ that have what we term a subexponential growth. Our main result in the first part of the paper asserts that every balanced hyperbolic $X$ is also divisorially hyperbolic. We provide a certain number of examples and counter-examples and discuss various properties of these manifolds. In the second part of the paper, we introduce the notions of <i>divisorially Kähler</i> and <i>divisorially nef</i> real De Rham cohomology classes of degree $2$ and study their properties. They also apply to $C^infty$, not necessarily holomorphic, complex line bundles and are expected to be implied in certain cases by the hyperbolicity properties introduced in the first part of the work. While motivated by the observation of hyperbolicity properties of certain non-Kähler manifolds, all these four new notions seem to have a role to play even in the Kähler and the projective settings.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"47 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745092","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 : 2024-07-17DOI: 10.4310/mrl.2023.v30.n6.a9
Filippo Sarti, Alessio Savini
Let $Gamma$ be a finitely generated group and let $(X,mu_X)$ be an ergodic standard Borel probability $Gamma$-space. Suppose that $mathcal{X}$ is a Hermitian symmetric space not of tube type and assume that $G=operatorname{Isom}(mathcal{X})^{circ}$ is simple. Given a Zariski dense measurable cocycle $sigma:Gamma times X to G$, we define the notion of parametrized Kähler class and we show that it completely determines the cocycle up to cohomology.
让 $Gamma$ 是一个有限生成的群，让 $(X,mu_X)$ 是一个遍历标准伯尔概率 $Gamma$ 空间。假设$(X,mu_X)$ 是一个不属于管型的赫米蒂对称空间，并假设$G=operatorname{Isom}(mathcal{X})^{circ}$ 是简单的。给定一个扎里斯基密集可测环 $sigma:Gamma times X to G$，我们定义了参数化凯勒类的概念，并证明它完全决定了这个环的同调性。
{"title":"Parametrized Kähler class and Zariski dense orbital 1-cohomology","authors":"Filippo Sarti, Alessio Savini","doi":"10.4310/mrl.2023.v30.n6.a9","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n6.a9","url":null,"abstract":"Let $Gamma$ be a finitely generated group and let $(X,mu_X)$ be an ergodic standard Borel probability $Gamma$-space. Suppose that $mathcal{X}$ is a Hermitian symmetric space not of tube type and assume that $G=operatorname{Isom}(mathcal{X})^{circ}$ is simple. Given a Zariski dense measurable cocycle $sigma:Gamma times X to G$, we define the notion of parametrized Kähler class and we show that it completely determines the cocycle up to cohomology.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"18 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745093","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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