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Classification problems in Higher Dimensional Birational Geometry

高维双有理几何中的分类问题

基本信息

批准号：
12440005
负责人：
MORI Shigefumi
金额：
$ 4.16万
依托单位：
KYOTO 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 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440005/
关键词：
flip flop Fano 3-fold canonical divisor symplectic variety extremal neighborhood elliptic structure terminal singularity symplectic singularity invariants monodromy derived category K3 surface complex symplectic structure Painl

项目摘要

Mori and Fujino have generalized Kodaira's canonical bundle formula and proved, as an application, that algebraic varieties with Kodaira dimension at most three have finitely generated canonical rings. Mori, Miyaoka and Takagi, together with Kollar, have proved the boundedness of Fano 3-folds with only canonical singularities. Mori also gave an explicit description of every irreducible semistable extremal neighborhood with two non-Gorenstein points, in terms of coordinates with equations and patching.Mukai proved that every canonical curve of genus 9 with maximal Clifford index, is a linear space section of the symplectic Grassmannian variety of dimension six.Masahiko Saito have introduced the Okamoto-Painleve pair of an algebraic surface and its anti-canonical divisor which algebro-geometrically characterizes the initial value space of the Painleve equation. He also reconstructed the Painleve equation from the pair using its deformation theory.Nakayama described certain elliptic fiber structures over a given analytic space upto bimeromorphic equivalence using the a-etale cohomology group.Namikawa have studied birational maps between complex symplectic varieties and proved that the analogue of "Reid's dream" does not hold in the category of complex symplectic varieties. He also constructed a counterexample to birational Torelli problem for complex symplectic varieties.Oguiso has generalized the characterization of the Klein curve to the case of K3 surfaces, which states that the Klein-Mukai surface is the only K3 surface which admits a faithful action of the quartic extension of the simple group of order 168.Takagi has generalized Takeuchi's method, obtained a classification list of Q-Fano 3-folds with index two and confirmed the existence in several cases.Fujino has proved that every sequence of log flips terminates for 4 dimensional canonical pairs.

Mori和Fujino推广了Kodaira的正则束公式，并作为一个应用证明了Kodaira维数最多为3的代数变元能够有限生成正则环。Mori， Miyaoka和Takagi与Kollar一起证明了只有正则奇点的Fano 3-fold的有界性。Mori还给出了具有两个非gorenstein点的每一个不可约半稳定极值邻域的方程坐标和补点的明确描述。Mukai证明了每条具有极大Clifford指数的9属正则曲线都是六维辛格拉斯曼变的线性空间剖面。Masahiko Saito引入了代数曲面的Okamoto-Painleve对及其反正则除数，该除数在代数几何上表征了Painleve方程的初值空间。他还利用变形理论重构了Painleve方程。Nakayama利用a- ettale上同调群描述了给定解析空间上的某些椭圆型纤维结构直至双纯等价。Namikawa研究了复辛变之间的双域映射，证明了在复辛变的范畴中，“Reid的梦”的类比不成立。他还构造了复辛变异体双民族Torelli问题的一个反例。Oguiso将Klein曲线的性质推广到K3曲面，指出Klein- mukai曲面是唯一允许有168阶单群四次扩张的忠实作用的K3曲面。Takagi推广了Takeuchi的方法，得到了一个指标为2的Q-Fano 3-fold分类表，并在若干情况下证实了其存在性。Fujino证明了对于四维正则对，每一个对数翻转序列都终止。