2003 Fiscal Year Annual Research Report

孤立特異点のモジュライへの境界解析/幾何の応用

Research Project

Project/Area Number	14540087
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) 赤堀隆夫姫路工業大学, 大学院・理学研究科, 教授 (40117560) 大本亨北海道大学, 大学院・理学研究科, 助教授 (20264400)
Keywords	CR構造 / 変形 / モジュライ / 孤立特異点 / コーシー・リーマン構造
Research Abstract	本研究は孤立特異点のモジュライに関して,その境界構造を通じて解析することを主要な目的としている.一般的に,正規孤立特異点には次の3種類のアプローチが可能である. (1)特異点を定義する方程式を通じたアプローチ,(2)特異点解消で得られる強擬凸複素多様体を通じたアプローチ,(3)特異点の境界構造を通じたアプローチ. 昨年度の研究を引き継いで,今年度の研究では(2)のアプローチを完成させることを目指した.(2)で扱う変形は(3)で扱う境界構造を通じた変形とは違いが生じるものの,それ独自で表現論やゲージ理論等との豊かな結びつきを持っており,最近では弦理論との関わりも指摘されている重要な対象である.昨年度の研究で確立した複素構造の安定変形の変形複体に基づいて特異点解消の変形空間を構成した,収束性を保証する部分には新しい解析的考察が必要とされるが,領域内部ではGardingの不等式で保証される統御性に,境界付近ではP.Greiner-E.M.SteinによるNeumann作用素のoptimal評価に還元するテクニックを確立した.更に,このテクニックにより,正規孤立特異点の完備族をその非特異部分の複素構造の安定変形によって構成することにも成功した.この研究により,正規孤立特異点の完備族構成をコーシー・リーマン理論に基づいて構成する方法が(幾何学的にも解析学的にも)確立できたということが言える.

Research Products
(12 results)

All Other

All Publications (12 results)

[Publications] K.Miyajima: "Analytic approach to deformation of resolution of normal isolated singularities : Formal deformations"J.of Korean Mathematical Society. 40(4). 709-725 (2003)
[Publications] K.Miyajima: "Strongly pseudoconvex CR manifolds and deformation of normal isolated singularities"SUGAKU EXPOSITIONS, American Mathematical Society. 16(2). 191-206 (2003)
[Publications] K.Miyajima: "CR description of the formal deformations of quasi-homogeneous singularities""Selected Topics in Cauchy-Riemann Geometry (ed.S.Dragomir)", quademi di matematica. 9. 249-280 (2003)
[Publications] S.Tsuboi: "The Euler number of the normalization of an algebraic threefold with ordinary singularities""Geometric Singularity Theory", Banach Center Publications Polish Academy of Sciences. (印刷中). (2004)
[Publications] S.Tsuboi: "The Euler number of the normalization of an algebraic threefold with ordinary singularities"proccedings of the "Rencontre Singularites Franco-Japonaise (CIRM, Luminy, Marseille, France)". (印刷中)(to appear). 20 (2004)
[Publications] S.Yokura: "On Ginzburg's bivariant Chern classes"Transactions of American Mathematical Society. 355. 2501-2521 (2003)
[Publications] S.Yokura: "On Ginzburg's bivariant Chern classes, II"Geometricae Dedicata. 101. 185-201 (2003)
[Publications] T.Aikou: "A note on Riemannian manifolds with semi-parallel vector fields"Reports of Faculty of Science, Kagoshima University. 36. 1-10 (2003)
[Publications] T.Aikou: "Projective Flatness of Complex Finsler Metrics"Publ.Math.Debrecen. 63. 343-362 (2003)
[Publications] T.Ohmoto: "Self-intersection class for singularities and its application to fold maps"Transactions of American Mathematical Society. 355. 3825-3838 (2003)
[Publications] T.Ohmoto: "A elementary remark on the integral with respect to Euler characteristics on projective hyperplane sections"Reports of Faculty of Science, Kagoshima University. 36. 37-41 (2003)
[Publications] T.Akahori: "Homogeneous polynomial hypersurface isolated sinularities"J. of Korean Mathematical Society. 40(4). 667-680 (2003)

2003 Fiscal Year Annual Research Report

孤立特異点のモジュライへの境界解析/幾何の応用

Principal Investigator

宮嶋 公夫 鹿児島大学, 理学部, 教授 (40107850)

Research Products

[Publications] K.Miyajima: "Analytic approach to deformation of resolution of normal isolated singularities : Formal deformations"J.of Korean Mathematical Society. 40(4). 709-725 (2003)

[Publications] K.Miyajima: "Strongly pseudoconvex CR manifolds and deformation of normal isolated singularities"SUGAKU EXPOSITIONS, American Mathematical Society. 16(2). 191-206 (2003)

[Publications] K.Miyajima: "CR description of the formal deformations of quasi-homogeneous singularities""Selected Topics in Cauchy-Riemann Geometry (ed.S.Dragomir)", quademi di matematica. 9. 249-280 (2003)

[Publications] S.Tsuboi: "The Euler number of the normalization of an algebraic threefold with ordinary singularities""Geometric Singularity Theory", Banach Center Publications Polish Academy of Sciences. (印刷中). (2004)

[Publications] S.Tsuboi: "The Euler number of the normalization of an algebraic threefold with ordinary singularities"proccedings of the "Rencontre Singularites Franco-Japonaise (CIRM, Luminy, Marseille, France)". (印刷中)(to appear). 20 (2004)

[Publications] S.Yokura: "On Ginzburg's bivariant Chern classes"Transactions of American Mathematical Society. 355. 2501-2521 (2003)

[Publications] S.Yokura: "On Ginzburg's bivariant Chern classes, II"Geometricae Dedicata. 101. 185-201 (2003)

[Publications] T.Aikou: "A note on Riemannian manifolds with semi-parallel vector fields"Reports of Faculty of Science, Kagoshima University. 36. 1-10 (2003)

[Publications] T.Aikou: "Projective Flatness of Complex Finsler Metrics"Publ.Math.Debrecen. 63. 343-362 (2003)

[Publications] T.Ohmoto: "Self-intersection class for singularities and its application to fold maps"Transactions of American Mathematical Society. 355. 3825-3838 (2003)

[Publications] T.Ohmoto: "A elementary remark on the integral with respect to Euler characteristics on projective hyperplane sections"Reports of Faculty of Science, Kagoshima University. 36. 37-41 (2003)

[Publications] T.Akahori: "Homogeneous polynomial hypersurface isolated sinularities"J. of Korean Mathematical Society. 40(4). 667-680 (2003)

宮嶋公夫鹿児島大学, 理学部, 教授 (40107850)