2001 Fiscal Year Annual Research Report

3次元特異複素射影超曲面の位相的,解析的研究

Research Project

Project/Area Number	13640083
Research Institution	Kagoshima University
Principal Investigator	坪井昭二鹿児島大学, 理学部, 教授 (80027375)
Co-Investigator(Kenkyū-buntansha)	大本亨鹿児島大学, 理学部, 助教授 (20264400) 輿倉昭治鹿児島大学, 理学部, 教授 (60182680) 宮嶋公夫鹿児島大学, 理学部, 教授 (40107850) 中島正治鹿児島大学, 理学部, 教授 (40041230) 小櫃邦夫鹿児島大学, 理学部, 助教授 (00325763)
Keywords	3次元特異複素射影超曲面 / 通常特異点 / オイラー数 / 交点理論 / 通常3重点の退化型 / 3次元剛直孤立特異点 / グレブナー基底 / イデアルの自由分解
Research Abstract	本年度の主たる研究成果は次の通りである。 (1)X⊂P^4(C)を4次元複素射影空間内の通常3重点、通常4重点、および停留点以外の通常特異点(通常2重点、尖点)を持った超曲面、D : Xの2重曲面(特異点集合)、C : Xの尖点の集合、n=degX、m=dogD、γ=degC(それぞれX, D, Cの次数)としたとき、Xの非特異正規化モデルYのオイラー数χ(Y)は、次で与えられる:χ(Y)=n(n^3-n^2-4n+8)-(4n^2-3n-8)m-6(n-1)γ+2D・D+2π(C)-2ただし、π(C)は、Cの仮想種数を表わす。 (2)アフィン座標で、(xy)^2+(yz)^2+(zx)^2+xyzw=0で定義される通常3重点の退化型特異点の正規化モデルは、v : P^2(C)→P^5(C)を2次のVeronese埋め込み(ξ_0^2 : ξ_1^2 : ξ_2^2 : ξ_0 : ξ_1 : ξ_0 : ξ_2 : ξ_1 : ξ_2)→(x_0 : x_1 : x_2 : y_0 : y_: y_2)としたとき、S : =v(P^2(C))の上のconeであるが、これが変形でrigidあることを示した。これは、Sの定義イデアルI(S)のグレブナー基底(y_0^2-x_0x_1,y_1^2-x_0x_2,y_2^2-x_1x_2,y_0y_2-x_1y_1,y_1y_0-x_0y_2,y_1y_2-x_2y_0)の自由分解を構成し、H^i(P^5,Θ_P5(-X))=0 (i=1,2)を示し、これより、H^1(S, N_<S/P5>)=0、H^0(P^5,Θ_P5)→H^0(S, N_<S/P5>)が全射であることを導くことによって結論できる。

Research Products

(6 results)

All Other

All Publications (6 results)

[Publications] S.Tsuboi: "The Euler number of the non-singular normalization of an algebraic threefold with ordinary singulaities."Procccdings of the International Symposium on Singularity Theory and its Applications, Beijing University of Chemical Technology. 113-119 (2001)
[Publications] S.Tsuboi: "A Certain Degenerate Ordinarity Singulary of Dimension Three"Finite or infinite Dimensional Complex Analysis, Shandon Science and Technology Press. 223-228 (2001)
[Publications] K.Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities"Proceeings of the Japan-Korea Join Workshop on Complex Analysis and Related Topics. 115-124 (2002)
[Publications] K.Miyajima: "CR descripion of the formal deformations of uasi-homgeneous singularities"Quaderni di Matematica, Series ed.by Dipartimento di Matematica, Seconda Universita di Napoli. (in press). 24 (2002)
[Publications] S.Yokura: "An application of bivariant theory to Milnor classes"Topology and Its Applications. 115. 43-61 (2001)
[Publications] K.Obitsu: "The asymptoic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics"Pub1.RIMS.Kyoto Univ.. 37. 459-478 (2001)

2001 Fiscal Year Annual Research Report

3次元特異複素射影超曲面の位相的,解析的研究

Principal Investigator

坪井 昭二 鹿児島大学, 理学部, 教授 (80027375)

Research Products

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

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

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

[Publications] K.Miyajima: "CR descripion of the formal deformations of uasi-homgeneous singularities"Quaderni di Matematica, Series ed.by Dipartimento di Matematica, Seconda Universita di Napoli. (in press). 24 (2002)

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

[Publications] K.Obitsu: "The asymptoic behavior of Eisenstein series and a comparison of the Weil-Petersson and the Zograf-Takhtajan metrics"Pub1.RIMS.Kyoto Univ.. 37. 459-478 (2001)

坪井昭二鹿児島大学, 理学部, 教授 (80027375)