2001 Fiscal Year Annual Research Report

孤立特異点の境界構造のモジュライに関する研究

Research Project

Project/Area Number	12640080
Research Institution	Kagoshima University
Principal Investigator	宮嶋公夫鹿児島大学, 理学部, 教授 (40107850)
Co-Investigator(Kenkyū-buntansha)	愛甲正鹿児島大学, 理学部, 教授 (00192831) 與倉昭治鹿児島大学, 理学部, 教授 (60182680) 坪井昭二鹿児島大学, 理学部, 教授 (80027375) 小櫃邦夫鹿児島大学, 理学部, 助教授 (00325763) 大本亨鹿児島大学, 理学部, 助教授 (20264400)
Keywords	CR多様体 / 孤立特異点 / モジュライ / 変形 / 超曲面特異点の境界 / 錐特異点の境界 / 準斉次特異点の境界 / 商特異点の境界
Research Abstract	前年度の研究を継続して、正規孤立特異点の強擬凸境界上のCR構造のstably embeddable変形の変形空間の記述を進めた.特に、次のような典型的な場合にそれを記述する方法の確立を目指した:(1)準斉次特異点、(2)2次元巡回商特異点.得られた結果は次の通り.(1)準斉次特異点:stably embemable変形を統御する2重複体にU(1)-作用から得られる次数付けを導入し、変形空間の次数付けを得た.更に、それは特異点の定義方程式の変形の次数付けを統御していることを明らかにした.また、準斉次特異点が錐特異点の構造をも持つ場合は、その次数付けは昨年度の研究で得られた次数付けに一致し、代数幾何におけるH. Pinkhamの結果やJ. Wahlの結果のCR記述を与えることも明らかになった.(2)2次元巡回商特異点(この特異点は、有限群の表現、ツイスター空間、超ケーラー構造、quiver等、種々の興味深い幾何学の源泉となった特異点であり、しかも一般的な場合には完備変形空間の詳細な解析が完全には行われていない特異点である):球調和関数を使った解析により、1次Ext-群の記述を完全に一般的な状況下で行った.更に、巡回次数が(n,1)(nは4以下)の場合については、変形空間の完全な記述を得た.巡回次数が(n,1)(nは5以上)の場合も、個々の場合に変形空間を記述するアルゴリズムは得られたが、一般的な形での記述を得るには至らなかった.また、同時特異点解消の完全な記述を行った.特に、巡回次数(2,1)の場合には(Brieskorn resolutionとして知られている)パラメータ空間の2次拡大が現れる非自明な場合のCR-記述を完全に行った.

Research Products

(14 results)

All Other

All Publications (14 results)

[Publications] 宮嶋公夫: "強擬凸CR多様体と正規孤立特異点の変形"数学. 53(2). 172-184 (2001)
[Publications] K.Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities""Complex Analysis and Related Topics", Proc. of the Japan-Korea Joint Workshop in Mathematics 2001. 115-124 (2002)
[Publications] K.Miyajima: "CR description of the formal deformations of quasi-homogeneous singularities"in "Selected Topics in Cauchy-Riemann Geometry", (Ed. by S.Dragomir). (印刷中). 24 (2002)
[Publications] K.Miyajima: "Strongly pseudoconvex CR manifolds and deformation of normal isolated singularities"Sugaku Expositions, A.M.S.. (印刷中). 19 (2002)
[Publications] S.Tsuboi: "A Certain Degenerate Ordinary Singularity of Dimension Three"Finite or infinite Dimensinal Complex Analysis, Shandon Science and Technology Press. 223-228 (2001)
[Publications] S.Tsuboi: "The Euler number of the non-singular normalization of an algebraic threefold with ordinary singulaities"Proceeings of the International Symposium on Singularity Theory and its Applications. 113-119 (2001)
[Publications] S.Yokura: "An application of bivariant theory to Milnor classes"Topology and Its Applications. 115. 43-61 (2001)
[Publications] L.Ernstrom, S.Yokora: "On bivariant Chem-Schwartz-MacPherson classes with values in Chow groups"Selecta Mathematica. 7(印刷中). 25 (2001)
[Publications] S.Yokura: "Bivariant theories of constructible functions and Grothendieck transformations"Topology and Its Applications. (印刷中). 14 (2001)
[Publications] S.Yokura: "Verdier-Riemann-Roch for Chem class and Milnor class"Asian J. Mathematics. (印刷中). 23 (2002)
[Publications] S.Yokura: "Remarks on Ginzburg's bivariant Chern classes"Proc. of A.M.S.. (印刷中). 8 (2002)
[Publications] T.Aikou: "Applications of Bott connection to Finsler geometry""Steps in Differential Geometry", The Institute of Mathematics, Univ. of Debrecen. 1-13 (2001)
[Publications] K.Obitsu: "Differential geometry of Kaehler fibrations and its application to Finsler geometry"Far East Journal Mathematical Science. (印刷中).
[Publications] K.Obitsu: "The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics"Publ. RIMS., Kyoto Univ.. 37. 49-478 (2001)

2001 Fiscal Year Annual Research Report

孤立特異点の境界構造のモジュライに関する研究

Principal Investigator

宮嶋 公夫 鹿児島大学, 理学部, 教授 (40107850)

Research Products

[Publications] 宮嶋公夫: "強擬凸CR多様体と正規孤立特異点の変形"数学. 53(2). 172-184 (2001)

[Publications] K.Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities""Complex Analysis and Related Topics", Proc. of the Japan-Korea Joint Workshop in Mathematics 2001. 115-124 (2002)

[Publications] K.Miyajima: "CR description of the formal deformations of quasi-homogeneous singularities"in "Selected Topics in Cauchy-Riemann Geometry", (Ed. by S.Dragomir). (印刷中). 24 (2002)

[Publications] K.Miyajima: "Strongly pseudoconvex CR manifolds and deformation of normal isolated singularities"Sugaku Expositions, A.M.S.. (印刷中). 19 (2002)

[Publications] S.Tsuboi: "A Certain Degenerate Ordinary Singularity of Dimension Three"Finite or infinite Dimensinal Complex Analysis, Shandon Science and Technology Press. 223-228 (2001)

[Publications] S.Tsuboi: "The Euler number of the non-singular normalization of an algebraic threefold with ordinary singulaities"Proceeings of the International Symposium on Singularity Theory and its Applications. 113-119 (2001)

[Publications] S.Yokura: "An application of bivariant theory to Milnor classes"Topology and Its Applications. 115. 43-61 (2001)

[Publications] L.Ernstrom, S.Yokora: "On bivariant Chem-Schwartz-MacPherson classes with values in Chow groups"Selecta Mathematica. 7(印刷中). 25 (2001)

[Publications] S.Yokura: "Bivariant theories of constructible functions and Grothendieck transformations"Topology and Its Applications. (印刷中). 14 (2001)

[Publications] S.Yokura: "Verdier-Riemann-Roch for Chem class and Milnor class"Asian J. Mathematics. (印刷中). 23 (2002)

[Publications] S.Yokura: "Remarks on Ginzburg's bivariant Chern classes"Proc. of A.M.S.. (印刷中). 8 (2002)

[Publications] T.Aikou: "Applications of Bott connection to Finsler geometry""Steps in Differential Geometry", The Institute of Mathematics, Univ. of Debrecen. 1-13 (2001)

[Publications] K.Obitsu: "Differential geometry of Kaehler fibrations and its application to Finsler geometry"Far East Journal Mathematical Science. (印刷中).

[Publications] K.Obitsu: "The asymptotic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics"Publ. RIMS., Kyoto Univ.. 37. 49-478 (2001)

宮嶋公夫鹿児島大学, 理学部, 教授 (40107850)