Complex symplectic manifolds and related topics

复辛流形及相关主题

• 批准号: 12440006
• 负责人: MIYAOKA Yoichi
• 金额: $ 4.03万
• 依托单位: The University of Tokyo (2001-2002)Kyoto University (2000)
• 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-12440006/
• 关键词: complex symplectic; birational morphism; fiber space structure; projective space; hyperquadrics; Fano 3-folds; 複素シンプレクティック多様体; 高次元多様体; 正則写像; 変形同値類; 有利曲線族; ファノ多様体; 2次超曲面; 双有理写像; 完全可積分ハミルトン系; 大域剛性; モディライ空間; 複素トーラス; Q-ファノ多様体

The head investigator studied holomorphic maps from complex symplectic manifolds, in which he obtained a series of fundamental results on numerical characterisations of projective space and smooth hyperquadrics. One of his results asserts that a smooth Fano n-fold X is isomorphic to projective n-space of a hyperquadric if and only if the"length of X"is n + 1 or n, where the length is defined to be the minimum of (C, -Kx), C running through the set of curves on X.Our new characterisations are strong enough to be applied to complex manifolds, enabling us to prove the following structure theorem on morphisms from complex manifolds :・ Let Y be a projective complex symplectic manifod of dimension 2n and π : Y → Y^^^ a birational morphism onto a normal variety. Let E denote an arbitrary irreducible component and put B = π(E). Then B is a complex symplectic variety of dimension 2m 【less than or equal】 2n and, for a general point b ∈ B, the inverse image π^<-1>(b) ∩ E is projective space of dimension n + m.・ Let f : Y → X be a nontrivial fiber space structure on a primitive ; complex symplectic manifold of dimension 2n. If f admits a holomorphic section, then X is projective n-space and the fibers of f are Lagrangian subvarieties.Another product of his research is a joint work with J. Kollar, S. Mori and H. Takagi, which proved the boundedness of Fano 3-folds with only canonical, singularities.

首席研究员研究了复杂的对称歧管的全态图，其中他就投射空间和光滑的超质量的数值特征获得了一系列基本结果。他的结果之一断言，光滑的fano n倍X与超期的投影n空间是同构的。歧管：・让y成为维度2n和π的投射复杂对称宣言：y→y→y ^^在正常品种上是十亿个某些形态。令E表示一个任意的不可约组件，然后将B =π（E）放置。然后b是尺寸2M的复杂对称品种[小于或相等] 2n，对于一般点b∈B，逆图像π^<-1>（b）e是尺寸n + m。f：y y→y→x的射击空间。维度2n的复杂对称歧管。如果f接受了一个全体形态部分，则x是射影的n空间，而f的纤维是拉格朗日亚群。他的研究的另一个产品是与J. Kollar，S。Mori和H. Takagi的联合作品，这被证明是3倍的Fano的界限，只有规范，奇异性。

项目成果

S.Mori, Y.Miyaoka: "Higher-dimensional Birational Geometry"Math. Soc. of Japan. 295 (2002)

S.Mori，Y.Miyaoka：“高维双有理几何”数学。

S.Mori, Y.Miyaoka: "Higher-Dimensional Birational Geometry"日本数学会. 295 (2002)

S.Mori，Y.Miyaoka：“高维双有理几何”日本数学会295（2002）。

NAKAYAMA,Noboru: "Local structure of an elliptic fibration"to appear in Adv.Stud.Pure Math..

NAKAYAMA,Noboru：“椭圆纤维振动的局部结构”出现在 Adv.Stud.Pure Math..

K.Cho, Y.Miyaoka, N.Shepherd-Barron: "Characterizations of projective space and applications to complex symplectic manifolds"Advanced Studies in Pure Mathematics. 35. 1-89 (2002)

K.Cho、Y.Miyaoka、N.Shepherd-Barron：“射影空间的表征及其在复辛流形中的应用”纯数学高级研究。

O.Fujino, S.Mori: "A canonical bundle formula"J. Differ. Geom.. 56. 167-188 (2000)

O.Fujino，S.Mori：“规范束公式”J。