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Lipschitz geometry of complex surface germs via inner rates of primary ideals 通过主理想的内率实现复曲面胚芽的 Lipschitz 几何学
arXiv - MATH - Complex Variables Pub Date : 2024-07-19 DOI: arxiv-2407.14265
Yenni Cherik
{"title":"Lipschitz geometry of complex surface germs via inner rates of primary ideals","authors":"Yenni Cherik","doi":"arxiv-2407.14265","DOIUrl":"https://doi.org/arxiv-2407.14265","url":null,"abstract":"Let $(X, 0)$ be a normal complex surface germ embedded in $(mathbb{C}^n,\u00000)$, and denote by $mathfrak{m}$ the maximal ideal of the local ring\u0000$mathcal{O}_{X,0}$. In this paper, we associate to each $mathfrak{m}$-primary\u0000ideal $I$ of $mathcal{O}_{X,0}$ a continuous function $mathcal{I}_I$ defined\u0000on the set of positive (suitably normalized) semivaluations of\u0000$mathcal{O}_{X,0}$. We prove that the function $mathcal{I}_{mathfrak{m}}$ is\u0000determined by the outer Lipschitz geometry of the surface $(X, 0)$. We further\u0000demonstrate that for each $mathfrak{m}$-primary ideal $I$, there exists a\u0000complex surface germ $(X_I, 0)$ with an isolated singularity whose\u0000normalization is isomorphic to $(X, 0)$ and $mathcal{I}_I =\u0000mathcal{I}_{mathfrak{m}_I}$, where $mathfrak{m}_I$ is the maximal ideal of\u0000$mathcal{O}_{X_I,0}$. Subsequently, we construct an infinite family of complex\u0000surface germs with isolated singularities, whose normalizations are isomorphic\u0000to $(X,0)$ (in particular, they are homeomorphic to $(X,0)$) but have distinct\u0000outer Lipschitz types.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Slope-semistability and moduli of coherent sheaves: a survey 相干剪切的斜向可分性与模态：概览
arXiv - MATH - Complex Variables Pub Date : 2024-07-18 DOI: arxiv-2407.13485
Mihai Pavel, Matei Toma
{"title":"Slope-semistability and moduli of coherent sheaves: a survey","authors":"Mihai Pavel, Matei Toma","doi":"arxiv-2407.13485","DOIUrl":"https://doi.org/arxiv-2407.13485","url":null,"abstract":"We survey old and new results on the existence of moduli spaces of semistable\u0000coherent sheaves both in algebraic and in complex geometry.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A remark on the Hölder regularity of solutions to the complex Hessian equation 关于复赫西恩方程解的赫尔德正则性的评论
arXiv - MATH - Complex Variables Pub Date : 2024-07-18 DOI: arxiv-2407.13130
Slawomir Kolodziej, Ngoc Cuong Nguyen
{"title":"A remark on the Hölder regularity of solutions to the complex Hessian equation","authors":"Slawomir Kolodziej, Ngoc Cuong Nguyen","doi":"arxiv-2407.13130","DOIUrl":"https://doi.org/arxiv-2407.13130","url":null,"abstract":"We prove that the Dirichlet problem for the complex Hessian equation has the\u0000H\"older continuous solution provided it has a subsolution with this property.\u0000Compared to the previous result of Benali-Zeriahi and Charabati-Zeriahi we\u0000remove the assumption on the finite total mass of the measure on the right hand\u0000side.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cauchy transforms and Szegő projections in dual Hardy spaces: inequalities and Möbius invariance 对偶哈代空间中的考奇变换和斯格ő投影：不等式和莫比乌斯不变性
arXiv - MATH - Complex Variables Pub Date : 2024-07-17 DOI: arxiv-2407.13033
David E. Barrett, Luke D. Edholm
{"title":"Cauchy transforms and Szegő projections in dual Hardy spaces: inequalities and Möbius invariance","authors":"David E. Barrett, Luke D. Edholm","doi":"arxiv-2407.13033","DOIUrl":"https://doi.org/arxiv-2407.13033","url":null,"abstract":"Dual pairs of interior and exterior Hardy spaces associated to a simple\u0000closed Lipschitz planar curve are considered, leading to a M\"obius invariant\u0000function bounding the norm of the Cauchy transform $bf{C}$ from below. This\u0000function is shown to satisfy strong rigidity properties and is closely\u0000connected via the Berezin transform to the square of the Kerzman-Stein\u0000operator. Explicit example calculations are presented. For ellipses, a new\u0000asymptotically sharp lower bound on the norm of $bf{C}$ is produced.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A sharp bound on the number of self-intersections of a trigonometric curve 三角曲线自交数的锐界
arXiv - MATH - Complex Variables Pub Date : 2024-07-17 DOI: arxiv-2407.12572
Sergei Kalmykov, Leonid V. Kovalev
{"title":"A sharp bound on the number of self-intersections of a trigonometric curve","authors":"Sergei Kalmykov, Leonid V. Kovalev","doi":"arxiv-2407.12572","DOIUrl":"https://doi.org/arxiv-2407.12572","url":null,"abstract":"We obtain a sharp bound on the number of self-intersections of a closed\u0000planar curve with trigonometric parameterization. Moreover, we show that a\u0000generic curve of this form is normal in the sense of Whitney.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quasi-coherent sheaves on complex analytic spaces 复解析空间上的准相干剪切
arXiv - MATH - Complex Variables Pub Date : 2024-07-16 DOI: arxiv-2407.11656
Haohao Liu
{"title":"Quasi-coherent sheaves on complex analytic spaces","authors":"Haohao Liu","doi":"arxiv-2407.11656","DOIUrl":"https://doi.org/arxiv-2407.11656","url":null,"abstract":"We show that in the category of analytic sheaves on a complex analytic space,\u0000the full subcategory of quasi-coherent sheaves is an abelian subcategory.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the connectedness of the boundary of $q$-complete domains 论$q$完整域边界的连通性
arXiv - MATH - Complex Variables Pub Date : 2024-07-16 DOI: arxiv-2407.11897
Rafael B. Andrist
{"title":"On the connectedness of the boundary of $q$-complete domains","authors":"Rafael B. Andrist","doi":"arxiv-2407.11897","DOIUrl":"https://doi.org/arxiv-2407.11897","url":null,"abstract":"The boundary of every relatively compact Stein domain in a complex manifold\u0000of dimension at least two is connected. No assumptions on the boundary\u0000regularity are necessary. The same proofs hold also for $q$-complete domains,\u0000and in the context of almost complex manifolds as well.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dirichlet spaces over chord-arc domains 弦弧域上的德里赫特空间
arXiv - MATH - Complex Variables Pub Date : 2024-07-16 DOI: arxiv-2407.11577
Huaying Wei, Michel Zinsmeister
{"title":"Dirichlet spaces over chord-arc domains","authors":"Huaying Wei, Michel Zinsmeister","doi":"arxiv-2407.11577","DOIUrl":"https://doi.org/arxiv-2407.11577","url":null,"abstract":"If $U$ is a $C^{infty}$ function with compact support in the plane, we let\u0000$u$ be its restriction to the unit circle $mathbb{S}$, and denote by\u0000$U_i,,U_e$ the harmonic extensions of $u$ respectively in the interior and the\u0000exterior of $mathbb S$ on the Riemann sphere. About a hundred years ago,\u0000Douglas has shown that begin{align*} iint_{mathbb{D}}|nabla U_i|^2(z)dxdy&=\u0000iint_{bar{mathbb{C}}backslashbar{mathbb{D}}}|nabla U_e|^2(z)dxdy &= frac{1}{2pi}iint_{mathbb Stimesmathbb\u0000S}left|frac{u(z_1)-u(z_2)}{z_1-z_2}right|^2|dz_1||dz_2|, end{align*} thus\u0000giving three ways to express the Dirichlet norm of $u$. On a rectifiable Jordan\u0000curve $Gamma$ we have obvious analogues of these three expressions, which will\u0000of course not be equal in general. The main goal of this paper is to show that\u0000these $3$ (semi-)norms are equivalent if and only if $Gamma$ is a chord-arc\u0000curve.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Characterization of finite shift via Herglotz's representation 通过赫格洛茨表示法确定有限位移的特征
arXiv - MATH - Complex Variables Pub Date : 2024-07-15 DOI: arxiv-2407.10664
Francisco J. Cruz-Zamorano
{"title":"Characterization of finite shift via Herglotz's representation","authors":"Francisco J. Cruz-Zamorano","doi":"arxiv-2407.10664","DOIUrl":"https://doi.org/arxiv-2407.10664","url":null,"abstract":"A complete characterization of parabolic self-maps of finite shift is given\u0000in terms of their Herglotz's representation. This improves a previous result\u0000due to Contreras, D'iaz-Madrigal, and Pommerenke. We also derive some\u0000consequences for the rate of convergence of these functions to their\u0000Denjoy-Wolff point, improving a related result of Kourou, Theodosiadis, and\u0000Zarvalis for the continuous setting.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
C-pairs and their morphisms C 对及其态式
arXiv - MATH - Complex Variables Pub Date : 2024-07-15 DOI: arxiv-2407.10668
Stefan Kebekus, Erwan Rousseau
{"title":"C-pairs and their morphisms","authors":"Stefan Kebekus, Erwan Rousseau","doi":"arxiv-2407.10668","DOIUrl":"https://doi.org/arxiv-2407.10668","url":null,"abstract":"This paper surveys Campana's theory of C-pairs (or \"geometric orbifolds\") in\u0000the complex-analytic setting, to serve as a reference for future work. Written\u0000with a view towards applications in hyperbolicity, rational points, and entire\u0000curves, it introduces the fundamental definitions of C-pair-theory\u0000systematically. In particular, it establishes an appropriate notion of\u0000\"morphism\", which agrees with notions from the literature in the smooth case,\u0000but is better behaved in the singular setting and has functorial properties\u0000that relate it to minimal model theory.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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