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 正则丛公式，并作为应用证明了 Kodaira 维数至多为 3 的代数簇具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 重的有界性。 Mori 还根据方程和修补的坐标，对每个具有两个非 Gorenstein 点的不可约半稳定极值邻域进行了明确的描述。 Mukai 证明了具有最大 Clifford 指数的 9 格正则曲线，是辛的线性空间部分六维格拉斯曼簇。Masahiko Saito 介绍了代数曲面的 Okamoto-Painleve 对及其反规范除数，以代数几何方式表征 Painleve 方程的初始值空间。他还利用其变形理论从该对重建了 Painleve 方程。Nakayama 使用 a-etale 上同调群描述了给定解析空间上的某些椭圆纤维结构直至双亚纯等价。Namikawa 研究了复辛簇之间的双有理映射，并证明了该类比“里德之梦”的命题并不属于复辛簇的范畴。他还构建了复辛簇的双有理 Torelli 问题的反例。Oguiso 将克莱因曲线的表征推广到 K3 曲面的情况，其中指出 Klein-Mukai 曲面是唯一允许忠实作用的 K3 曲面168阶简单群的四次扩张。Takagi推广了Takeuchi的方法，得到了指数为2的Q-Fano 3重分类表，并证实了Fujino 已经证明，每个对数翻转序列都以 4 维规范对终止。