{"title":"Topological classification of circle-valued simple Morse-Bott functions on closed orientable surfaces","authors":"E. B. Batista, J. Costa, I. S. Meza-Sarmiento","doi":"10.5427/jsing.2018.17q","DOIUrl":"https://doi.org/10.5427/jsing.2018.17q","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"150 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79453223","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 discuss examples of smooth quasi-projective manifolds with non-reduced Alexander modules, giving a non-semisimple Alexander module in one variable case and prove a result giving sufficient conditions for semisimplicity.
{"title":"Remarks on semi-simplicity of Alexander modules","authors":"A. Libgober","doi":"10.5427/jsing.2022.25n","DOIUrl":"https://doi.org/10.5427/jsing.2022.25n","url":null,"abstract":"We discuss examples of smooth quasi-projective manifolds with non-reduced Alexander modules, giving a non-semisimple Alexander module in one variable case and prove a result giving sufficient conditions for semisimplicity.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"25 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88453466","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}
{"title":"Monster towers from differential and algebraic viewpoints","authors":"P. Mormul","doi":"10.5427/jsing.2022.25o","DOIUrl":"https://doi.org/10.5427/jsing.2022.25o","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"56 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88522220","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}
{"title":"Invariants and classification of simple function germs with respect to Lipschitz A-equivalence","authors":"Nhan Nguyen, S. Trivedi","doi":"10.5427/jsing.2022.25p","DOIUrl":"https://doi.org/10.5427/jsing.2022.25p","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74429213","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}
J. Brasselet, Nivaldo G. Grulha Jr., Thuy Thi Bích Nguyên
{"title":"Local Euler obstruction, old and new, III","authors":"J. Brasselet, Nivaldo G. Grulha Jr., Thuy Thi Bích Nguyên","doi":"10.5427/jsing.2022.25e","DOIUrl":"https://doi.org/10.5427/jsing.2022.25e","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"14 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80880909","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}
Raimundo Nonato Araújo dos Santos, O. Saeki, T. O. Souza
{"title":"Algebraic knots associated with Milnor fibrations","authors":"Raimundo Nonato Araújo dos Santos, O. Saeki, T. O. Souza","doi":"10.5427/jsing.2022.25b","DOIUrl":"https://doi.org/10.5427/jsing.2022.25b","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"108 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88594503","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}
{"title":"On bi-Lipschitz invariance and the uniqueness of tangent cones","authors":"J. E. Sampaio, E. D. da Silva","doi":"10.5427/jsing.2022.25s","DOIUrl":"https://doi.org/10.5427/jsing.2022.25s","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"132 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75418880","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 the sum of the local cohomological dimension and the rectified $mathbb Q$-homological depth of a closed analytic subspace of a complex manifold coincide with the dimension of the ambient manifold. The local cohomological dimension is then calculated using the cohomology of the links of the analytic space. In the algebraic case the first assertion is equivalent to the coincidence of the rectified $mathbb Q$-homological depth with the de Rham depth studied by Ogus, and follows essentially from his work. As a corollary we show that the local cohomological dimension of a quasi-projective variety is determined by that of its general hyperplane section together with the link cohomology at 0-dimensional strata of a complex analytic Whitney stratification.
{"title":"Topological calculation of local cohomological dimension","authors":"Thomas Reichelt, M. Saito, U. Walther","doi":"10.5427/jsing.2023.26b","DOIUrl":"https://doi.org/10.5427/jsing.2023.26b","url":null,"abstract":"We show that the sum of the local cohomological dimension and the rectified $mathbb Q$-homological depth of a closed analytic subspace of a complex manifold coincide with the dimension of the ambient manifold. The local cohomological dimension is then calculated using the cohomology of the links of the analytic space. In the algebraic case the first assertion is equivalent to the coincidence of the rectified $mathbb Q$-homological depth with the de Rham depth studied by Ogus, and follows essentially from his work. As a corollary we show that the local cohomological dimension of a quasi-projective variety is determined by that of its general hyperplane section together with the link cohomology at 0-dimensional strata of a complex analytic Whitney stratification.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"28 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80902712","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}
From a procedure to calculate the $C_5$-cone of a reduced complex analytic curve $X subset mathbb{C}^n$ at a singular point $0 in X$, we extract a collection of integers that we call {it auxiliary multiplicities} and we prove they characterize the Lipschitz type of complex curve singularities. We then use them to improve the known bounds for the number of irreducible components of the $C_5$-cone. We finish by giving an example showing that in a Lipschitz equisingular family of curves the number of planes in the $C_5$-cone may not be constant.
{"title":"On the fifth Whitney cone of a complex analytic curve","authors":"A. G. Flores, O. N. Silva, J. Snoussi","doi":"10.5427/jsing.2022.24c","DOIUrl":"https://doi.org/10.5427/jsing.2022.24c","url":null,"abstract":"From a procedure to calculate the $C_5$-cone of a reduced complex analytic curve $X subset mathbb{C}^n$ at a singular point $0 in X$, we extract a collection of integers that we call {it auxiliary multiplicities} and we prove they characterize the Lipschitz type of complex curve singularities. We then use them to improve the known bounds for the number of irreducible components of the $C_5$-cone. We finish by giving an example showing that in a Lipschitz equisingular family of curves the number of planes in the $C_5$-cone may not be constant.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"34 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75640045","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 smooth rigidity inequalitiy provides an explicit lower bound for the (d+1)st derivatives of a smooth function f , which holds, if f exhibits certain patterns, forbidden for polynomials of degree d. The main goal of the present paper is twofold: first, we provide an overview of some recent results and questions related to smooth rigidity, which recently were obtained in Singularity Theory, in Approximation Theory, and in Whitney smooth extensions. Second, we prove some new results, specifically, a new Remez-type inequality, and on this base we obtain a new rigidity inequality. In both parts of the paper we stress the topology of the level sets, as the input information. Here are the main new results of the paper: Let B be the unit n-dimensional ball. For a given integer d let Z ⊂ B be a smooth compact hypersurface with N = (d − 1) + 1 connected components Zj . Let μj be the n-volume of the interior of Zj, and put μ = minμj , j = 1, . . . , N . Then for each polynomial P of degree d on R we have maxBn |P | max Z |P | ≤ ( 4n μ ). As a consequence, we provide an explicit lower bound for the (d+1)-st derivatives of any smooth function f , which vanishes on Z, while being of order 1 on B (smooth rigidity): ||f || ≥ 1 (d+ 1)! ( 4n μ ). We also provide an interpretation, in terms of smooth rigidity, of one of the simplest versions of the results in [8].
光滑刚性不等式为光滑函数f的(d+1)st阶导数提供了一个显式下界，如果f表现出某些模式，则对d次多项式是禁止的。本文的主要目标有两个:首先，我们概述了最近在奇点理论、近似理论和惠特尼光滑扩展中获得的与光滑刚性有关的一些结果和问题。其次，我们证明了一些新的结果，特别是一个新的remez型不等式，并在此基础上得到了一个新的刚性不等式。在本文的两部分中，我们都强调水平集的拓扑结构作为输入信息。以下是本文的主要新结果:设B为单位n维球。对于给定的整数d，设Z∧B是一个光滑紧超曲面，具有N = (d−1)+ 1个连通分量Zj。设μj为Zj内部的n体积，设μ = minμj, j = 1，…。，名词;那么对于R上的每个d次多项式P，我们有maxBn |P | max Z |P |≤(4n μ)。因此，我们为任意光滑函数f的(d+1)-st阶导数提供了一个显式下界，该函数在Z上消失，而在B上是1阶(光滑刚性):||f ||≥1 (d+1) !(4n μ)。我们还对[8]中结果的一个最简单版本提供了平滑刚性的解释。
{"title":"Smooth rigidity and Remez inequalities via Topology of level sets","authors":"Y. Yomdin","doi":"10.5427/jsing.2022.25v","DOIUrl":"https://doi.org/10.5427/jsing.2022.25v","url":null,"abstract":"A smooth rigidity inequalitiy provides an explicit lower bound for the (d+1)st derivatives of a smooth function f , which holds, if f exhibits certain patterns, forbidden for polynomials of degree d. The main goal of the present paper is twofold: first, we provide an overview of some recent results and questions related to smooth rigidity, which recently were obtained in Singularity Theory, in Approximation Theory, and in Whitney smooth extensions. Second, we prove some new results, specifically, a new Remez-type inequality, and on this base we obtain a new rigidity inequality. In both parts of the paper we stress the topology of the level sets, as the input information. Here are the main new results of the paper: Let B be the unit n-dimensional ball. For a given integer d let Z ⊂ B be a smooth compact hypersurface with N = (d − 1) + 1 connected components Zj . Let μj be the n-volume of the interior of Zj, and put μ = minμj , j = 1, . . . , N . Then for each polynomial P of degree d on R we have maxBn |P | max Z |P | ≤ ( 4n μ ). As a consequence, we provide an explicit lower bound for the (d+1)-st derivatives of any smooth function f , which vanishes on Z, while being of order 1 on B (smooth rigidity): ||f || ≥ 1 (d+ 1)! ( 4n μ ). We also provide an interpretation, in terms of smooth rigidity, of one of the simplest versions of the results in [8].","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":"12 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77938837","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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