Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M setminus L$ which can be realized as the singular point set of a generic map $M to mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $mathbb{R}^2$.
{"title":"Linking between singular locus and regular fibers","authors":"O. Saeki","doi":"10.5427/jsing.2020.21n","DOIUrl":"https://doi.org/10.5427/jsing.2020.21n","url":null,"abstract":"Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M setminus L$ which can be realized as the singular point set of a generic map $M to mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $mathbb{R}^2$.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78563538","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}
In a joint work [9] with Kazuya Kato and Chikara Nakayama, log higher Albanese manifolds was constructed as an application of log mixed Hodge theory with group action. In this framework, we describe a work of Deligne in [3] on some nilpotent quotients of the fundamental group of the projective line minus three points, where polylogarithms appear. As a result, we have $q$-expansions of higher Albanese maps at boundary points, i.e., log higher Albanese maps over the boundary.
{"title":"A description of a result of Deligne by log higher Albanese map","authors":"S. Usui","doi":"10.5427/jsing.2020.21q","DOIUrl":"https://doi.org/10.5427/jsing.2020.21q","url":null,"abstract":"In a joint work [9] with Kazuya Kato and Chikara Nakayama, log higher Albanese manifolds was constructed as an application of log mixed Hodge theory with group action. In this framework, we describe a work of Deligne in [3] on some nilpotent quotients of the fundamental group of the projective line minus three points, where polylogarithms appear. As a result, we have $q$-expansions of higher Albanese maps at boundary points, i.e., log higher Albanese maps over the boundary.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79937596","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}
In this work we prove a Brunella-Khanedani-Suwa variational type residue theorem for currents invariant by holomorphic foliations. As a consequence, we give conditions for the leaves of a singular holomorphic foliation to accumulate in the intersection of the singular set of the foliation with the support of an invariant current.
{"title":"Brunella-Khanedani-Suwa variational residues for invariant currents","authors":"M. Corrêa, A. Fern'andez-P'erez, Marcio G. Soares","doi":"10.5427/JSING.2021.23F","DOIUrl":"https://doi.org/10.5427/JSING.2021.23F","url":null,"abstract":"In this work we prove a Brunella-Khanedani-Suwa variational type residue theorem for currents invariant by holomorphic foliations. As a consequence, we give conditions for the leaves of a singular holomorphic foliation to accumulate in the intersection of the singular set of the foliation with the support of an invariant current.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72418278","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}
M. Evangelista-Alvarado, P. Su'arez-Serrato, J. T. Orozco, R. Vera
We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.
{"title":"On Bott-Morse Foliations and their Poisson structures in dimension three","authors":"M. Evangelista-Alvarado, P. Su'arez-Serrato, J. T. Orozco, R. Vera","doi":"10.5427/jsing.2019.19b","DOIUrl":"https://doi.org/10.5427/jsing.2019.19b","url":null,"abstract":"We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83210679","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 consider a finite analytic morphism $phi = (f,g) : (X,p)to (C^2,0)$ where $(X,p)$ is a complex analytic normal surface germ and $f$ and $g$ are complex analytic function germs. Let $pi : (Y,E_{Y})to (X,p)$ be a good resolution of $phi$ with exceptional divisor $E_{Y}=pi ^{-1}(p)$. We denote $G(Y)$ the dual graph of the resolution $pi $. We study the behaviour of the Hironaka quotients of $(f,g)$ associated to the vertices of $G(Y)$. We show that there exists maximal oriented arcs in $G(Y)$ along which the Hironaka quotients of $(f,g)$ strictly increase and they are constant on the connected components of the closure of the complement of the union of the maximal oriented arcs.
{"title":"On the growth behaviour of Hironaka quotients","authors":"H. Maugendre, F. Michel","doi":"10.5427/jsing.2020.20b","DOIUrl":"https://doi.org/10.5427/jsing.2020.20b","url":null,"abstract":"We consider a finite analytic morphism $phi = (f,g) : (X,p)to (C^2,0)$ where $(X,p)$ is a complex analytic normal surface germ and $f$ and $g$ are complex analytic function germs. Let $pi : (Y,E_{Y})to (X,p)$ be a good resolution of $phi$ with exceptional divisor $E_{Y}=pi ^{-1}(p)$. We denote $G(Y)$ the dual graph of the resolution $pi $. We study the behaviour of the Hironaka quotients of $(f,g)$ associated to the vertices of $G(Y)$. We show that there exists maximal oriented arcs in $G(Y)$ along which the Hironaka quotients of $(f,g)$ strictly increase and they are constant on the connected components of the closure of the complement of the union of the maximal oriented arcs.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76541181","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 develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem that the projections $K^{n} times mathbb{P}^{m}(K) to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, Holder continuity of definable functions on closed bounded subsets of $K^{n}$, the existence of definable retractions onto closed definable subsets of $K^{n}$, and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent preprint, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with some applications.
{"title":"A closedness theorem and applications in geometry of rational points over Henselian valued fields","authors":"K. Nowak","doi":"10.5427/jsing.2020.21m","DOIUrl":"https://doi.org/10.5427/jsing.2020.21m","url":null,"abstract":"We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem that the projections $K^{n} times mathbb{P}^{m}(K) to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, Holder continuity of definable functions on closed bounded subsets of $K^{n}$, the existence of definable retractions onto closed definable subsets of $K^{n}$, and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent preprint, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with some applications.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73818470","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}
Let $(R, mathfrak m)$ be a Noetherian local ring and $I$ a $mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay $R$ of dimension at most two. We also discuss relations to various topics: classical results on rings with minimal multiplicity and rational singularities, the recent work on $p_g$ ideals by Okuma-Watanabe-Yoshida, multiplicity of the fiber cone, and the $h$-vector of the associated graded ring.
设$(R， mathfrak m)$是一个诺瑟局部环，$I$ a $mathfrak m$-初级理想。本文研究了一类不等式，它涉及生成子数、Loewy长度和$I$的多重性。有强有力的证据表明，当且仅当$R$具有足够好的奇点时，该不等式对所有有限长度的整闭理想都成立。我们验证了所有维上正则局部环的不等式，验证了$2维上的有理奇点，以及$3维上的cDV奇点。此外，对于最大维数为2的Cohen-Macaulay $R$，当不等式总是成立时，我们可以进行分类。我们还讨论了与各种主题的关系:最小多重性和有理奇点环的经典结果，Okuma-Watanabe-Yoshida关于p_g$理想的最新工作，光纤锥的多重性，以及相关的梯度环的$h$-向量。
{"title":"The multiplicity and the number of generators of an integrally closed ideal","authors":"Hailong Dao, I. Smirnov","doi":"10.5427/JSING.2019.19E","DOIUrl":"https://doi.org/10.5427/JSING.2019.19E","url":null,"abstract":"Let $(R, mathfrak m)$ be a Noetherian local ring and $I$ a $mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay $R$ of dimension at most two. We also discuss relations to various topics: classical results on rings with minimal multiplicity and rational singularities, the recent work on $p_g$ ideals by Okuma-Watanabe-Yoshida, multiplicity of the fiber cone, and the $h$-vector of the associated graded ring.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75628504","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 a $C^{infty}$ equivalence between germs holomorphic foliations at $({mathbb C}^2,0)$ establishes a bijection between the sets of formal separatrices preserving equisingularity classes. As a consequence, if one of the foliations is of second type, so is the other and they are equisingular.
{"title":"Differentiable equisingularity of holomorphic foliations","authors":"Rogério Mol, R. Rosas","doi":"10.5427/JSING.2019.19F","DOIUrl":"https://doi.org/10.5427/JSING.2019.19F","url":null,"abstract":"We prove that a $C^{infty}$ equivalence between germs holomorphic foliations at $({mathbb C}^2,0)$ establishes a bijection between the sets of formal separatrices preserving equisingularity classes. As a consequence, if one of the foliations is of second type, so is the other and they are equisingular.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2016-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72688848","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 purpose of this paper is to investigate properties of the minimal free resolution of the modules of multi-logarithmic forms along a reduced equidimensional subspace. We first consider a notion of freeness for reduced complete intersections, and more generally for reduced equidimensional subspaces embedded in a smooth manifold, which generalizes the notion of Saito free divisors. The first main result is a characterization of freeness in terms of the projective dimension of the module of multi-logarithmic k -forms, where k is the codimension. We also prove that there is a perfect pairing between the module of multi-logarithmic differential k -forms and the module of multi-logarithmic k -vector fields which generalizes the duality between the corresponding modules in the hypersurface case. We deduce from this perfect pairing a duality between the Jacobian ideal and the module of multi-residues of multi-logarithmic k -forms. In the last part of this paper, we investigate logarithmic modules along some examples of free singularities. The main result in this section is an explicit computation of the minimal free resolution of the module of multi-logarithmic forms and multi-residues for quasi-homogeneous complete intersection curves which uses our first main theorem.
{"title":"Characterizations of freeness for equidimensional subspaces","authors":"Delphine Pol","doi":"10.5427/jsing.2020.20a","DOIUrl":"https://doi.org/10.5427/jsing.2020.20a","url":null,"abstract":"The purpose of this paper is to investigate properties of the minimal free resolution of the modules of multi-logarithmic forms along a reduced equidimensional subspace. We first consider a notion of freeness for reduced complete intersections, and more generally for reduced equidimensional subspaces embedded in a smooth manifold, which generalizes the notion of Saito free divisors. The first main result is a characterization of freeness in terms of the projective dimension of the module of multi-logarithmic k -forms, where k is the codimension. We also prove that there is a perfect pairing between the module of multi-logarithmic differential k -forms and the module of multi-logarithmic k -vector fields which generalizes the duality between the corresponding modules in the hypersurface case. We deduce from this perfect pairing a duality between the Jacobian ideal and the module of multi-residues of multi-logarithmic k -forms. In the last part of this paper, we investigate logarithmic modules along some examples of free singularities. The main result in this section is an explicit computation of the minimal free resolution of the module of multi-logarithmic forms and multi-residues for quasi-homogeneous complete intersection curves which uses our first main theorem.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2015-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86649208","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}
Stratied-algebra ic vector bundles on real algebraic varieties have many desirable features of algebraic vector bundles but are more exible. We give a characterization of the compact real algebraic varieties X having the following property: There exists a positive integer r such that for any topological vector bundle on X, the direct sum of r copies of is isomorphic to a stratied- algebraic vector bundle. In particular, each compact real algebraic variety of dimension at most 8 has this property. Our results are expressed in terms of K-theory.
{"title":"Comparison of stratified-algebraic and topological K-theory","authors":"W. Kucharz, K. Kurdyka","doi":"10.5427/jsing.2020.22t","DOIUrl":"https://doi.org/10.5427/jsing.2020.22t","url":null,"abstract":"Stratied-algebra ic vector bundles on real algebraic varieties have many desirable features of algebraic vector bundles but are more exible. We give a characterization of the compact real algebraic varieties X having the following property: There exists a positive integer r such that for any topological vector bundle on X, the direct sum of r copies of is isomorphic to a stratied- algebraic vector bundle. In particular, each compact real algebraic variety of dimension at most 8 has this property. Our results are expressed in terms of K-theory.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2015-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76667081","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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