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.

莫里（Mori）和富吉诺（Fujino）已将Kodaira的规范束公式概括，并证明，作为一种应用，具有Kodaira尺寸的代数品种，最多三个具有有限生成的规范环。莫里（Mori），米亚卡（Miyaoka）和高吉（Takagi）与科拉尔（Kollar）一起证明了只有规范性奇异的3倍Fano的界限。莫里（Mori）还在与方程式和补丁的坐标方面对每个不可还原的半超值邻里进行了明确的描述。Mukai证明，9属9属的每个规范曲线都具有最大Clifford index的最大clifford index，是Symplectic Grassmann verters of Simeens of sifeens of sifemahikiko sait的线性空间。代数表面及其反典型的分裂，代数形成量表表征了painleve方程的初始值空间。他还使用其变形理论从两对重建了painleve方程。 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在几种情况下，有三倍的索引二，并确认存在。