Geometry of twistor spaces

扭量空间的几何

• 批准号: 12440019
• 负责人: FUJIKI Akira
• 金额: $ 5.82万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440019/
• 关键词: twistor space; self-dual manifold; complex manifold; algebraic dimension; scalar curvature; group action; Doady space; hyperkahler manifold; Einstein-Weyl幾何; モジュライ空間

1. We have solved a conjecture of Joyce to the effect that (S^1 × S^1)-invariant self-dual metrics are essentially those constructed by Joyce, by determining the structure of the associated twistor space in detail.2. For a twistor space Z with algebraic dimension a(Z) = 2 we have shown that either M 〜 mP^2 or M 〜 (S^1 × S^3)#mP^2 for some of its finite unramified covering M^^〜. A similar result is obtained also for the case a(Z) = 1 and of +-type.3. For a twistor space Z with a(Z) = 2 corresponding to 4P^2 we have done a detailed study of the structure of its algebraic reduction. Honda obtained significant results on the structure and existence of twistor spaces admitting an S^1-action (not necessarily semifree) in the case M = 3P^2, 4P^2.4. For any m【greater than or equal】5, by considering the deformations of Joyce twistor spaces we have obtained the first examples of a twistor space with a = 2. When m = 4 the construction gives also examples which are different from those constructed by Campana-Kreussler.5. The Case M = (S^1 × S^3)#mP^2 of +-type: When m = 0 we have completely determined the structure of algebraic reduction of Z. Whe m > 0, we have solved affirmatively the a(Z) = 1-conjecture by LeBrun for the LeBrun families which are the unique explicit example in this case. Meantime, we get a detailed description of the structure of Z.

1. 通过详细确定相关扭转空间的结构，我们解决了Joyce的一个猜想，即（S^1 × S^1）不变自对偶度量本质上是由Joyce构造的。对于代数维数为a(Z) = 2的扭曲空间Z，我们证明了M ~ mP^2或M ~ (S^1 × S^3)#mP^2对于它的有限未分叉覆盖M^^ ~。对于A (Z) = 1且+-type.3的情况，也得到了类似的结果。对于a(Z) = 2对应4P^2的扭转空间Z，我们详细研究了其代数约简的结构。Honda在M = 3P^2, 4P^2.4的情况下，得到了具有S^1作用（不一定是半自由）的扭转空间的结构和存在性的重要结果。对于任意m【大于或等于】5，通过考虑乔伊斯扭曲空间的变形，我们得到了a = 2的扭曲空间的第一个例子。当m = 4时，该结构也给出了不同于campana - kreussler构造的例子。当M = 0时，我们完全确定了Z的代数约简结构。当M = 0时，我们肯定地解决了LeBrun族的a(Z) = 1猜想，这是这种情况下唯一的显式例子。同时，对Z的结构进行了详细的描述。

项目成果

A.Fujiki: "Topology of compact self-dual manifolds whose twistar space is of positive algebraic dimension"J.Math.Soc. Japan. 54. 587-608 (2002)

A.Fujiki：“扭星空间具有正代数维数的紧致自对偶流形的拓扑”J.Math.Soc。

Y.Namikawa: "Projectivity criterion of Moishezon spaces and density of projective symplectic varieties"International J.Math.. 13. 125-135 (2002)

Y.Namikawa：“Moishezon 空间的射影准则和射影辛簇的密度”International J.Math.. 13. 125-135 (2002)

Y.Namikawa: "Deformation theory of singular symplectic n-folds"Mathematisches Annalen. 319. 597-623 (2001)

Y.Namikawa：“奇异辛 n 折叠的变形理论”数学年鉴。

Y.Namikawa: "Stratified local moduli of Calabi-Yau threefold"Topology. (to appear).

Y.Namikawa：“卡拉比-丘三重分层局部模量”拓扑。

Y.Ohama: "Classical Solutions of Schlesinger equations and Twistor Theory"CRM Proceeding and Lecture Notes. 31. 51-58 (2002)

Y.Ohama：“施莱辛格方程和扭转理论的经典解”CRM 论文集和讲义。