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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的正则丛公式，并作为应用证明了Kodaira维数至多为3的代数簇有有限生成的正则环。Mori，Miyaoka和Takagi与Kollar一起证明了仅具有正则奇点的Fano 3-折叠的有界性。Mori还给出了每一个具有两个非Gorenstein点的不可约半稳定极值邻域的显式描述，用方程和面片的坐标表示。Mukai证明了每条具有最大Clifford指数的亏格9的标准曲线都是6维辛格拉斯曼变种的线性空间截面。他还利用Painleve对的形变理论重建了Painleve方程。Nakayama用a-a-上同调群描述了给定解析空间上的某些椭圆纤维结构，直到双同调等价。纳米川研究了复辛簇之间的双态映射，证明了类似的“Reid‘s Dream”不适用于复辛簇范畴。Oguiso将Klein曲线的刻画推广到K3曲面的情形，指出Klein-Mukai曲面是唯一允许168阶单群的四次扩张的忠实作用的K3曲面。Takagi推广了Takeuchi的方法，得到了指数为2的q-Fano 3-折叠的分类列表，并在几种情况下证明了它的存在性。Fujino证明了对于4维正则对，每个对数翻转序列都是终止的。