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Complex symplectic manifolds and related topics

复辛流形及相关主题

基本信息

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

项目摘要

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辛流形的全纯映射，在其中他获得了一系列关于射影空间和光滑超二次曲面的数值特征的基本结果。他的一个结果断言，一个光滑的Fano n-折叠X同构于一个超二次曲面的射影n-空间当且仅当“X的长度“是n + 1或n，其中长度被定义为（C，-Kx）的最小值，C贯穿X上的曲线集。使我们能够证明以下关于来自复流形的态射的结构定理：·设Y是维数为2n的射影复辛流形，π：Y → Y^^^是正规簇上的双有理态射。设E为任意不可约分支，设B = π（E）.则B是维数为2 m [小于或等于] 2n的复辛簇，并且对于一般点b ∈ B，逆像π^<-1>（b）<$E是维数为n + m的射影空间。设f：Y → X是2n维本原复辛流形上的非平凡纤维空间结构.如果f容许一个全纯截面，则X是射影n-空间，f的纤维是拉格朗日子簇。Mori和H. Takagi，证明了Fano 3-folds的有界性。