Abstract It is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,ldots , f^{k_0}$ if ${f^{k_1}=cdots =f^{k_m}=0}$ is a nondegenerate complete intersection variety for any $k_1,ldots ,k_min {1,ldots , k_0}$ .
摘要众所周知，（牛顿）非退化多项式函数f的Milnor fibration的微分同胚型是由f的牛顿边界唯一确定的。本文将这一结果推广到某些退化函数，即，我们证明了形式为$f=f^1cdots f^｛k_0｝$的（可能退化的）多项式函数的Milnor fibration的微分同胚型是由$f^1，ldots，f^｛k _0｝$的牛顿边界唯一确定的，如果$｛f^{k_1｝=cdots=f^｛k_m｝=0｝$对于任何$k_1，ldot，k_m in ｛1，ldott，k_0。
{"title":"ON THE MILNOR FIBRATION OF CERTAIN NEWTON DEGENERATE FUNCTIONS","authors":"C. Eyral, M. Oka","doi":"10.1017/nmj.2022.37","DOIUrl":"https://doi.org/10.1017/nmj.2022.37","url":null,"abstract":"Abstract It is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form \u0000$f=f^1cdots f^{k_0}$\u0000 is uniquely determined by the Newton boundaries of \u0000$f^1,ldots , f^{k_0}$\u0000 if \u0000${f^{k_1}=cdots =f^{k_m}=0}$\u0000 is a nondegenerate complete intersection variety for any \u0000$k_1,ldots ,k_min {1,ldots , k_0}$\u0000 .","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45906677","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}
Abstract For a finite-dimensional Lie algebra $mathfrak {L}$ over $mathbb {C}$ with a fixed Levi decomposition $mathfrak {L} = mathfrak {g} ltimes mathfrak {r}$ , where $mathfrak {g}$ is semisimple, we investigate $mathfrak {L}$ -modules which decompose, as $mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $mathfrak {g}$ -modules with finite multiplicities. We call such modules $mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $mathfrak {g} = mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $mathfrak {g}$ -Harish-Chandra modules.
{"title":"LIE ALGEBRA MODULES WHICH ARE LOCALLY FINITE AND WITH FINITE MULTIPLICITIES OVER THE SEMISIMPLE PART","authors":"V. Mazorchuk, Rafael Mrðen","doi":"10.1017/nmj.2021.8","DOIUrl":"https://doi.org/10.1017/nmj.2021.8","url":null,"abstract":"Abstract For a finite-dimensional Lie algebra \u0000$mathfrak {L}$\u0000 over \u0000$mathbb {C}$\u0000 with a fixed Levi decomposition \u0000$mathfrak {L} = mathfrak {g} ltimes mathfrak {r}$\u0000 , where \u0000$mathfrak {g}$\u0000 is semisimple, we investigate \u0000$mathfrak {L}$\u0000 -modules which decompose, as \u0000$mathfrak {g}$\u0000 -modules, into a direct sum of simple finite-dimensional \u0000$mathfrak {g}$\u0000 -modules with finite multiplicities. We call such modules \u0000$mathfrak {g}$\u0000 -Harish-Chandra modules. We give a complete classification of simple \u0000$mathfrak {g}$\u0000 -Harish-Chandra modules for the Takiff Lie algebra associated to \u0000$mathfrak {g} = mathfrak {sl}_2$\u0000 , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple \u0000$mathfrak {g}$\u0000 -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff \u0000$mathfrak {sl}_2$\u0000 and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple \u0000$mathfrak {g}$\u0000 -Harish-Chandra modules.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87837443","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}
Abstract We investigate a novel geometric Iwasawa theory for ${mathbf Z}_p$ -extensions of function fields over a perfect field k of characteristic $p>0$ by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if $cdots to X_2 to X_1 to X_0$ is the tower of curves over k associated with a ${mathbf Z}_p$ -extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of $X_n$ as $nrightarrow infty $ . By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$ -module structure of the space $M_n:=H^0(X_n, Omega ^1_{X_n/k})$ of global regular differential forms as $nrightarrow infty .$ For example, for each tower in a basic class of ${mathbf Z}_p$ -towers, we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + lambda _r n + c_r(n)$ for all n sufficiently large, where $a_r, lambda _r$ are rational constants and $c_r : {mathbf Z}/m_r {mathbf Z} to {mathbf Q}$ is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on ${mathbf Z}_p$ -towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$ .
摘要本文用p-扭转类群方案的研究取代了通常的类群中p-扭转的研究，研究了特征为$p>0$的完美域k上${mathbf Z}_p$ -函数域扩展的一个新的几何Iwasawa理论。也就是说，如果$cdots to X_2 to X_1 to X_0$是k上的曲线塔，与在有限非空位置集合上完全分叉的函数场的${mathbf Z}_p$ -扩展相关联，我们研究了$X_n$为$nrightarrow infty $的雅可比矩阵中p-扭转群格式的增长。根据dieudonn理论，这相当于研究了具有Frobenius和Cartier算子v的自然作用的$X_n$的第一个de Rham上同群。我们制定并测试了一些猜想，这些猜想预测了整体正则微分形式$nrightarrow infty .$的空间$M_n:=H^0(X_n, Omega ^1_{X_n/k})$的$k[V]$ -模块结构中的惊人规律性。例如，对于${mathbf Z}_p$ -塔的基本类中的每个塔，我们推测，对于所有足够大的n, $M_n$上$V^r$核的维数由$a_r p^{2n} + lambda _r n + c_r(n)$给出，其中$a_r, lambda _r$是有理数常数，$c_r : {mathbf Z}/m_r {mathbf Z} to {mathbf Q}$是一个周期函数，取决于r和塔。为了为这些猜想提供证据，我们收集了大量的实验数据，这些数据基于新的和更有效的算法，用于处理${mathbf Z}_p$ -曲线塔上的微分，我们在$p=2$和$r=1$的情况下证明了我们的猜想。
{"title":"IWASAWA THEORY FOR p-TORSION CLASS GROUP SCHEMES IN CHARACTERISTIC p","authors":"J. Booher, Bryden Cais","doi":"10.1017/nmj.2022.30","DOIUrl":"https://doi.org/10.1017/nmj.2022.30","url":null,"abstract":"Abstract We investigate a novel geometric Iwasawa theory for \u0000${mathbf Z}_p$\u0000 -extensions of function fields over a perfect field k of characteristic \u0000$p>0$\u0000 by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if \u0000$cdots to X_2 to X_1 to X_0$\u0000 is the tower of curves over k associated with a \u0000${mathbf Z}_p$\u0000 -extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of \u0000$X_n$\u0000 as \u0000$nrightarrow infty $\u0000 . By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of \u0000$X_n$\u0000 equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the \u0000$k[V]$\u0000 -module structure of the space \u0000$M_n:=H^0(X_n, Omega ^1_{X_n/k})$\u0000 of global regular differential forms as \u0000$nrightarrow infty .$\u0000 For example, for each tower in a basic class of \u0000${mathbf Z}_p$\u0000 -towers, we conjecture that the dimension of the kernel of \u0000$V^r$\u0000 on \u0000$M_n$\u0000 is given by \u0000$a_r p^{2n} + lambda _r n + c_r(n)$\u0000 for all n sufficiently large, where \u0000$a_r, lambda _r$\u0000 are rational constants and \u0000$c_r : {mathbf Z}/m_r {mathbf Z} to {mathbf Q}$\u0000 is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on \u0000${mathbf Z}_p$\u0000 -towers of curves, and we prove our conjectures in the case \u0000$p=2$\u0000 and \u0000$r=1$\u0000 .","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43187510","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}
Abstract We construct a convex and strongly pseudoconvex Kobayashi positive Finsler metric on a vector bundle E under the assumption that the symmetric power of the dual $S^kE^*$ has a Griffiths negative $L^2$ -metric for some k. The proof relies on the negativity of direct image bundles and the Minkowski inequality for norms. As a corollary, we show that given a strongly pseudoconvex Kobayashi positive Finsler metric, one can upgrade to a convex Finsler metric with the same property. We also give an extremal characterization of Kobayashi curvature for Finsler metrics.
{"title":"POSITIVELY CURVED FINSLER METRICS ON VECTOR BUNDLES","authors":"Kuang-Ru Wu","doi":"10.1017/nmj.2022.2","DOIUrl":"https://doi.org/10.1017/nmj.2022.2","url":null,"abstract":"Abstract We construct a convex and strongly pseudoconvex Kobayashi positive Finsler metric on a vector bundle E under the assumption that the symmetric power of the dual \u0000$S^kE^*$\u0000 has a Griffiths negative \u0000$L^2$\u0000 -metric for some k. The proof relies on the negativity of direct image bundles and the Minkowski inequality for norms. As a corollary, we show that given a strongly pseudoconvex Kobayashi positive Finsler metric, one can upgrade to a convex Finsler metric with the same property. We also give an extremal characterization of Kobayashi curvature for Finsler metrics.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47001755","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}
Abstract By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the $(n-1,n)$ th mild $partial overline {partial }$ -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of $(p,p)$ -Bott–Chern numbers.
{"title":"POWER SERIES PROOFS FOR LOCAL STABILITIES OF KÄHLER AND BALANCED STRUCTURES WITH MILD \u0000$partial overline {partial }$\u0000 -LEMMA","authors":"S. Rao, Xueyuan Wan, Quanting Zhao","doi":"10.1017/nmj.2021.4","DOIUrl":"https://doi.org/10.1017/nmj.2021.4","url":null,"abstract":"Abstract By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an n-dimensional balanced manifold with the \u0000$(n-1,n)$\u0000 th mild \u0000$partial overline {partial }$\u0000 -lemma by power series method and the other one on p-Kähler structures with the deformation invariance of \u0000$(p,p)$\u0000 -Bott–Chern numbers.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/nmj.2021.4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48068824","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}
Pub Date : 2021-06-01DOI: 10.1017/s0027763020000240
{"title":"NMJ volume 242 Cover and Back matter","authors":"","doi":"10.1017/s0027763020000240","DOIUrl":"https://doi.org/10.1017/s0027763020000240","url":null,"abstract":"","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/s0027763020000240","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41359745","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}
Pub Date : 2021-06-01DOI: 10.1017/s0027763020000239
{"title":"NMJ volume 242 Cover and Front matter","authors":"","doi":"10.1017/s0027763020000239","DOIUrl":"https://doi.org/10.1017/s0027763020000239","url":null,"abstract":"","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/s0027763020000239","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45796606","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}
Abstract We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.
摘要本文建立了任意余维对数正则中心的完全一般的附结和附结反演。
{"title":"ADJUNCTION AND INVERSION OF ADJUNCTION","authors":"O. Fujino, K. Hashizume","doi":"10.1017/nmj.2022.24","DOIUrl":"https://doi.org/10.1017/nmj.2022.24","url":null,"abstract":"Abstract We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43057417","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}
Abstract Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in $mathsf {ZF}$ , i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.
{"title":"CONSTRUCTING MAXIMAL COFINITARY GROUPS","authors":"David Schrittesser","doi":"10.1017/nmj.2022.46","DOIUrl":"https://doi.org/10.1017/nmj.2022.46","url":null,"abstract":"Abstract Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in \u0000$mathsf {ZF}$\u0000 , i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44324189","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}
Abstract We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U with Lie algebra $mathfrak {u}$ extends holomorphically to an action of the complexified group $U^{mathbb {C}}$ and that the U-action on Z is Hamiltonian. If $Gsubset U^{mathbb {C}}$ is compatible, there exists a gradient map $mu _{mathfrak p}:X longrightarrow mathfrak p$ where $mathfrak g=mathfrak k oplus mathfrak p$ is a Cartan decomposition of $mathfrak g$ . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map $mu _{mathfrak p}$ .
{"title":"COMPACT ORBITS OF PARABOLIC SUBGROUPS","authors":"L. Biliotti, O. J. Windare","doi":"10.1017/nmj.2021.14","DOIUrl":"https://doi.org/10.1017/nmj.2021.14","url":null,"abstract":"Abstract We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U with Lie algebra \u0000$mathfrak {u}$\u0000 extends holomorphically to an action of the complexified group \u0000$U^{mathbb {C}}$\u0000 and that the U-action on Z is Hamiltonian. If \u0000$Gsubset U^{mathbb {C}}$\u0000 is compatible, there exists a gradient map \u0000$mu _{mathfrak p}:X longrightarrow mathfrak p$\u0000 where \u0000$mathfrak g=mathfrak k oplus mathfrak p$\u0000 is a Cartan decomposition of \u0000$mathfrak g$\u0000 . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map \u0000$mu _{mathfrak p}$\u0000 .","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43646940","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}
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