Study of unramified extensions, with emphasis on Jacobian problem

无分支扩张的研究，重点是雅可比问题

• 批准号: 14540031
• 负责人: TSUCHIMOTO Yoshifumi
• 金额: $ 1.66万
• 依托单位: Kochi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540031/
• 关键词: unramified extension; Jacobian problem; Dixmier conjecture; p-curvature; algebraic variet; polarized variety; delta genus; arithmetic genus; 代数多根体; デルタ極数; Jacobian; 単純拡大; conductor; Dixmire予想; Weyl代数の自己準同型; Weyl代数の中心; Sympletic構造

(1)We show that a Weyl algebra of positive characteristics is related to a matrix bundle on a affine space, and that we may obtain a process which is a 'inversion' of the 'geometric quantization' for this object.(2)Using ultra filter on Spec(Z), we construct a field Qu of characteristic 0. We showed that an algebra endomorphism of Weyl algebra over Qu corresponds to an symplectic morphism of affine space.(3)We show Dixmier conjecture may be deduced to Jacobian conjecture.(4)Ultra filter limit of Cartier operator gives a new integration theory. It gives an important step to the Jacobian problem.(5)We give a definition of the i-th sectional geometric genus and i-th delta genus of a generalized polarized manifold.(6)We observe that the i-th sectional geometric genus and i-th delta genus have analogous properties to those of sectional genus and delta genus when the complete linear system |L| of L has no base point. We studied further what happens when |L| has some base points.(7)We give a definition of the i-th sectional H-arithmetic genus _X^H_i(X,L) of a polarized manifold(X,L).(8)Expecting that g_2(X,L) and _X^H_2(X,L)(resp.) are analogous to geometric genus and arithmetic genus _X(O)(resp.), we propose some problems on polarized manifold as an analogy to the known results on the theory of surface.

(1)我们证明了正特征的Weyl代数与仿射空间上的矩阵束相关，并且我们可以得到该对象的“几何量化”的“反转”过程。(2)利用Spec(Z)上的超滤波器，构造了特征为0的域Qu。我们证明了Qu上Weyl代数的代数自同态对应于仿射空间的辛态。(3)证明Dixmier猜想可以推导为Jacobian猜想。(4)卡地亚算子的超滤波极限给出了一种新的积分理论。它为雅可比矩阵问题迈出了重要的一步。(5)给出了广义极化流形的第i个截面几何格和第i个δ格的定义。(6)我们观察到，当L的完全线性系统|L|没有基点时，第i条截面几何格和第i条δ格具有与截面格和δ格相似的性质。我们进一步研究了当|L|有一些基点时会发生什么。(7)给出了极化流形（X，L）的第i段h算术格_X^H_i（X，L）的定义。(8)假定g_2（X，L）和_X^H_2(X，L)（resp.）与几何格和算术格_X(O)（resp.）是相似的，我们提出了关于极化流形的一些问题，作为对曲面理论中已知结果的类比。

项目成果

Y.Fukuma: "On the sectional geometric genus of quasi-polarized varieties, II."manuscripta mathematica. 113巻2号. 211-237 (2004)

Y.Fukuma：“关于准极化簇的截面几何属，II”，数学手稿，第 113 卷，第 211-237 期（2004 年）

Y.Fukuma: "On the c_r-sectional geometric genus of generalized polarized manifolds."Japanese Journal of Mathematics. 29巻2号. 335-355 (2003)

Y.Fukuma：“关于广义极化流形的 c_r 截面几何亏格”，《日本数学杂志》，第 29 卷，第 2 期，335-355（2003 年）。

Y.Tsuchimoto: "Preliminaries on Dixmier conjecture"Mem.Fac.Sci.Kochi Univ.Ser.A Math.. 24. 43-59 (2003)

Y.Tsuchimoto：“迪克斯米尔猜想的初步”Mem.Fac.Sci.Kochi Univ.Ser.A Math.. 24. 43-59 (2003)

Y.Fukuma: "On the sectional geometric genus of quasi-polarized varieties, II."manuscripta mathematica. 113(2). 211-237 (2004)

Y.Fukuma：“关于准极化品种的截面几何属，II。”数学手稿。

Y.Fukuma: "On the c_r-sectional geometric genus of generalized polarized manifolds."Japanese Journal of Mathematics. 29(2). 335-355 (2003)

Y.Fukuma：“关于广义极化流形的 c_r 截面几何亏格。”日本数学杂志。