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Exotic surfaces 奇异的表面
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-17 DOI: 10.1007/s00208-024-02916-7
Javier Reyes, Giancarlo Urzúa
{"title":"Exotic surfaces","authors":"Javier Reyes, Giancarlo Urzúa","doi":"10.1007/s00208-024-02916-7","DOIUrl":"https://doi.org/10.1007/s00208-024-02916-7","url":null,"abstract":"<p>Although exotic blow-ups of the projective plane at <i>n</i> points have been constructed for every <span>(n ge 2)</span>, the only examples known by means of rational blowdowns satisfy <span>(n ge 5)</span>. It has been an intriguing problem whether it is possible to decrease <i>n</i>. In this paper, we construct the first exotic <span>({mathbb {C}}{mathbb {P}}^2 # 4 overline{{mathbb {C}}{mathbb {P}}^2})</span> with this technique. We also construct exotic <span>(3{mathbb {C}}{mathbb {P}}^2 # b^- overline{{mathbb {C}}{mathbb {P}}^2})</span> for <span>(b^-=9,8,7)</span>. All of them are minimal and symplectic, as they are produced from projective surfaces <i>W</i> with Wahl singularities and <span>(K_W)</span> big and nef. In more generality, we elaborate on the problem of finding exotic </p><span>$$begin{aligned} (2chi ({mathcal {O}}_W)-1) {mathbb {C}}{mathbb {P}}^2 # (10chi ({mathcal {O}}_W)-K^2_W-1) overline{{mathbb {C}}{mathbb {P}}^2} end{aligned}$$</span><p>from these Kollár–Shepherd-Barron–Alexeev surfaces <i>W</i>, obtaining explicit geometric obstructions on the corresponding configurations of rational curves.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"72 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces 塞雷代数、矩阵因式分解和超曲面的分类托雷里定理
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-17 DOI: 10.1007/s00208-024-02915-8
Xun Lin, Shizhuo Zhang
{"title":"Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces","authors":"Xun Lin, Shizhuo Zhang","doi":"10.1007/s00208-024-02915-8","DOIUrl":"https://doi.org/10.1007/s00208-024-02915-8","url":null,"abstract":"<p>Let <i>X</i> be a smooth Fano variety. We attach a bi-graded associative algebra <span>(textrm{HS}(mathcal {K}u(X))=bigoplus _{i,jin mathbb {Z}} textrm{Hom}(textrm{Id},S_{mathcal {K}u(X)}^{i}[j]))</span> to the Kuznetsov component <span>(mathcal {K}u(X))</span> whenever it is defined. Then we construct a natural sub-algebra of <span>(textrm{HS}(mathcal {K}u(X)))</span> when <i>X</i> is a Fano hypersurface and establish its relation with Jacobian ring <span>(textrm{Jac}(X))</span>. As an application, we prove a categorical Torelli theorem for Fano hypersurface <span>(Xsubset mathbb {P}^n(nge 2))</span> of degree <i>d</i> if <span>(textrm{gcd}((n+1),d)=1.)</span> In addition, we give a new proof of the main theorem [15, Theorem 1.2] using a similar idea.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"24 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Simultaneous extreme values of zeta and L-functions zeta 函数和 L 函数的同时极值
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-14 DOI: 10.1007/s00208-024-02892-y
Winston Heap, Junxian Li
{"title":"Simultaneous extreme values of zeta and L-functions","authors":"Winston Heap, Junxian Li","doi":"10.1007/s00208-024-02892-y","DOIUrl":"https://doi.org/10.1007/s00208-024-02892-y","url":null,"abstract":"<p>We show that distinct primitive <i>L</i>-functions can achieve extreme values <i>simultaneously</i> on the critical line. Our proof uses a modification of the resonance method and can be applied to establish simultaneous extreme central values of <i>L</i>-functions in families.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"69 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Quantization of the energy for the inhomogeneous Allen–Cahn mean curvature 非均质艾伦-卡恩平均曲率的能量量子化
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-14 DOI: 10.1007/s00208-024-02909-6
Huy The Nguyen, Shengwen Wang
{"title":"Quantization of the energy for the inhomogeneous Allen–Cahn mean curvature","authors":"Huy The Nguyen, Shengwen Wang","doi":"10.1007/s00208-024-02909-6","DOIUrl":"https://doi.org/10.1007/s00208-024-02909-6","url":null,"abstract":"<p>We consider the varifold associated to the Allen–Cahn phase transition problem in <span>({mathbb {R}}^{n+1})</span>(or <span>(n+1)</span>-dimensional Riemannian manifolds with bounded curvature) with integral <span>(L^{q_0})</span> bounds on the Allen–Cahn mean curvature (first variation of the Allen–Cahn energy) in this paper. It is shown here that there is an equidistribution of energy between the Dirichlet and Potential energy in the phase field limit and that the associated varifold to the total energy converges to an integer rectifiable varifold with mean curvature in <span>(L^{q_0}, q_0 &gt; n)</span>. The latter is a diffused version of Allard’s convergence theorem for integer rectifiable varifolds.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"113 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 generating families on Lagrangian cobordisms 拉格朗日共线性上族的存在性
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-14 DOI: 10.1007/s00208-024-02913-w
Wenyuan Li
{"title":"Existence of generating families on Lagrangian cobordisms","authors":"Wenyuan Li","doi":"10.1007/s00208-024-02913-w","DOIUrl":"https://doi.org/10.1007/s00208-024-02913-w","url":null,"abstract":"<p>For an embedded exact Lagrangian cobordism between Legendrian submanifolds in the 1-jet bundle, we prove that a generating family linear at infinity on the Legendrian at the negative end extends to a generating family linear at infinity on the Lagrangian cobordism after stabilization if and only if the formal obstructions vanish. In particular, a Lagrangian filling with trivial stable Lagrangian Gauss map admits a generating family linear at infinity.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"41 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Bounded Fatou and Julia components of meromorphic functions 有界法图和非定常函数的 Julia 分量
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-08 DOI: 10.1007/s00208-023-02725-4
David Martí-Pete, Lasse Rempe, James Waterman
{"title":"Bounded Fatou and Julia components of meromorphic functions","authors":"David Martí-Pete, Lasse Rempe, James Waterman","doi":"10.1007/s00208-023-02725-4","DOIUrl":"https://doi.org/10.1007/s00208-023-02725-4","url":null,"abstract":"<p>We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering compacta using approximation theory.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Bias in the distribution of holonomy on compact hyperbolic 3-manifolds 紧凑双曲3-manifolds上整体性分布的偏差
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-06 DOI: 10.1007/s00208-024-02903-y
Lindsay Dever
{"title":"Bias in the distribution of holonomy on compact hyperbolic 3-manifolds","authors":"Lindsay Dever","doi":"10.1007/s00208-024-02903-y","DOIUrl":"https://doi.org/10.1007/s00208-024-02903-y","url":null,"abstract":"<p>Ambient prime geodesic theorems provide an asymptotic count of closed geodesics by their length and holonomy and imply effective equidistribution of holonomy. We show that for a smoothed count of closed geodesics on compact hyperbolic 3-manifolds, there is a persistent bias in the secondary term which is controlled by the number of zero spectral parameters. In addition, we show that a normalized, smoothed bias count is distributed according to a probability distribution, which we explicate when all distinct, non-zero spectral parameters are linearly independent.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"27 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Fractional Sobolev spaces on Riemannian manifolds 黎曼流形上的分数索波列夫空间
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-03 DOI: 10.1007/s00208-024-02894-w
Michele Caselli, Enric Florit-Simon, Joaquim Serra
{"title":"Fractional Sobolev spaces on Riemannian manifolds","authors":"Michele Caselli, Enric Florit-Simon, Joaquim Serra","doi":"10.1007/s00208-024-02894-w","DOIUrl":"https://doi.org/10.1007/s00208-024-02894-w","url":null,"abstract":"<p>This article studies the canonical Hilbert energy <span>(H^{s/2}(M))</span> on a Riemannian manifold for <span>(sin (0,2))</span>, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type <span>({mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+int _M F(v) , dV)</span>, with <span>(Fge 0)</span>, is given, which includes in particular the case of nonlocal <i>s</i>-minimal surfaces. Finally, we prove some estimates for the Caffarelli–Silvestre extension problem, which are of general interest. This work is motivated by Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"35 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Generalized Schauder theory and its application to degenerate/singular parabolic equations 广义绍德理论及其在退化/星状抛物方程中的应用
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-03 DOI: 10.1007/s00208-024-02898-6
Takwon Kim, Ki-Ahm Lee, Hyungsung Yun
{"title":"Generalized Schauder theory and its application to degenerate/singular parabolic equations","authors":"Takwon Kim, Ki-Ahm Lee, Hyungsung Yun","doi":"10.1007/s00208-024-02898-6","DOIUrl":"https://doi.org/10.1007/s00208-024-02898-6","url":null,"abstract":"<p>In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form </p><span>$$begin{aligned} u_t = a^{i'j'}u_{i'j'} + 2 x_n^{gamma /2} a^{i'n} u_{i'n} + x_n^{gamma } a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{gamma /2} b^n u_{n} + c u + f quad (gamma le 1). end{aligned}$$</span><p>When the equation above is singular, it can be derived from Monge–Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution <i>u</i>, which is called <i>s</i>-polynomial. To prove <span>(C_s^{2+alpha })</span>-regularity and higher regularity of the solution <i>u</i>, we establish generalized Schauder theory which approximates coefficients of the operator with <i>s</i>-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap argument to obtain higher regularity.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"53 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Exponential decay for the quintic wave equation with locally distributed damping 具有局部分布阻尼的五次波方程的指数衰减
IF 1.4 2区 数学
Mathematische Annalen Pub Date : 2024-06-03 DOI: 10.1007/s00208-024-02904-x
Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, André Vicente
{"title":"Exponential decay for the quintic wave equation with locally distributed damping","authors":"Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, André Vicente","doi":"10.1007/s00208-024-02904-x","DOIUrl":"https://doi.org/10.1007/s00208-024-02904-x","url":null,"abstract":"<p>We study the stabilization and the well-posedness of solutions of the quintic wave equation with locally distributed damping. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we approximate weak solutions for regular solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"30 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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