Let $(X, 0)$ be a normal complex surface germ embedded in $(mathbb{C}^n, 0)$, and denote by $mathfrak{m}$ the maximal ideal of the local ring $mathcal{O}_{X,0}$. In this paper, we associate to each $mathfrak{m}$-primary ideal $I$ of $mathcal{O}_{X,0}$ a continuous function $mathcal{I}_I$ defined on the set of positive (suitably normalized) semivaluations of $mathcal{O}_{X,0}$. We prove that the function $mathcal{I}_{mathfrak{m}}$ is determined by the outer Lipschitz geometry of the surface $(X, 0)$. We further demonstrate that for each $mathfrak{m}$-primary ideal $I$, there exists a complex surface germ $(X_I, 0)$ with an isolated singularity whose normalization is isomorphic to $(X, 0)$ and $mathcal{I}_I = mathcal{I}_{mathfrak{m}_I}$, where $mathfrak{m}_I$ is the maximal ideal of $mathcal{O}_{X_I,0}$. Subsequently, we construct an infinite family of complex surface germs with isolated singularities, whose normalizations are isomorphic to $(X,0)$ (in particular, they are homeomorphic to $(X,0)$) but have distinct outer Lipschitz types.
{"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}
We survey old and new results on the existence of moduli spaces of semistable coherent sheaves both in algebraic and in complex geometry.
我们考察了代数几何和复几何中关于半稳相干剪切的模空间存在性的新老结果。
{"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}
We prove that the Dirichlet problem for the complex Hessian equation has the H"older continuous solution provided it has a subsolution with this property. Compared to the previous result of Benali-Zeriahi and Charabati-Zeriahi we remove the assumption on the finite total mass of the measure on the right hand side.
与贝纳利-泽里阿希和查拉巴蒂-泽里阿希之前的结果相比，我们取消了右侧量度总质量有限的假设。
{"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}
Dual pairs of interior and exterior Hardy spaces associated to a simple closed Lipschitz planar curve are considered, leading to a M"obius invariant function bounding the norm of the Cauchy transform $bf{C}$ from below. This function is shown to satisfy strong rigidity properties and is closely connected via the Berezin transform to the square of the Kerzman-Stein operator. Explicit example calculations are presented. For ellipses, a new asymptotically sharp lower bound on the norm of $bf{C}$ is produced.
{"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}
We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.
{"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}
We show that in the category of analytic sheaves on a complex analytic space, the full subcategory of quasi-coherent sheaves is an abelian subcategory.
我们证明，在复解析空间上的解析剪切范畴中，准相干剪切的全子类是一个非相干子类。
{"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}
The boundary of every relatively compact Stein domain in a complex manifold of dimension at least two is connected. No assumptions on the boundary regularity are necessary. The same proofs hold also for $q$-complete domains, and in the context of almost complex manifolds as well.
{"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}
If $U$ is a $C^{infty}$ function with compact support in the plane, we let $u$ be its restriction to the unit circle $mathbb{S}$, and denote by $U_i,,U_e$ the harmonic extensions of $u$ respectively in the interior and the exterior of $mathbb S$ on the Riemann sphere. About a hundred years ago, Douglas has shown that begin{align*} iint_{mathbb{D}}|nabla U_i|^2(z)dxdy&= iint_{bar{mathbb{C}}backslashbar{mathbb{D}}}|nabla U_e|^2(z)dxdy &= frac{1}{2pi}iint_{mathbb Stimesmathbb S}left|frac{u(z_1)-u(z_2)}{z_1-z_2}right|^2|dz_1||dz_2|, end{align*} thus giving three ways to express the Dirichlet norm of $u$. On a rectifiable Jordan curve $Gamma$ we have obvious analogues of these three expressions, which will of course not be equal in general. The main goal of this paper is to show that these $3$ (semi-)norms are equivalent if and only if $Gamma$ is a chord-arc curve.
{"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}
A complete characterization of parabolic self-maps of finite shift is given in terms of their Herglotz's representation. This improves a previous result due to Contreras, D'iaz-Madrigal, and Pommerenke. We also derive some consequences for the rate of convergence of these functions to their Denjoy-Wolff point, improving a related result of Kourou, Theodosiadis, and Zarvalis for the continuous setting.
{"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}
This paper surveys Campana's theory of C-pairs (or "geometric orbifolds") in the complex-analytic setting, to serve as a reference for future work. Written with a view towards applications in hyperbolicity, rational points, and entire curves, it introduces the fundamental definitions of C-pair-theory systematically. In particular, it establishes an appropriate notion of "morphism", which agrees with notions from the literature in the smooth case, but is better behaved in the singular setting and has functorial properties that relate it to minimal model theory.
本文概述了坎帕纳在复解析背景下的 C 对（或 "几何球面"）理论，为今后的工作提供参考。本文着眼于双曲、有理点和全曲线的应用，系统地介绍了 C 对理论的基本定义。特别是，它建立了一个适当的 "态 "概念，这个概念与光滑情况下的文献中的概念一致，但在奇异情况下表现得更好，并且具有与最小模型理论相关的函数特性。
{"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}
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