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Minimal models for fiber spaces

纤维空间的最小模型

基本信息

批准号：
14540024
负责人：
NAKAYAMA Noboru
金额：
$ 0.77万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540024/
关键词：
minimal model elliptic fibration torus embedding

项目摘要

The minimal model theory for fiber spaces is regarded as a relative version of that for projective varieties. There still remains the flip conjecture, which is an obstruction in showing the existence of minimal models. The purpose of this research is to study the minimal models for elliptic fibrations explicitly, avoiding flips.Contrary to the case of elliptic surfaces, we can not assume the fibration to be minimal in the case of higher dimensional base. However, by trial and error, we succeeded in constructing a minimal model, locally over the base, in the case where local monodromies are unipotent and no multiple fibers exist over divisors. This is our starting point. Here, the theory of variation of Hodge structure and that of torus embeddings are required.By using the minimal model above, we can classify bimeromorphically the projective elliptic fibrations that are smooth outside a fixed normal crossing divisor. This is our second stage. The third stage is to give an explicit construction of a minimal model for each bimeromorphic class. In the next stage, we want to describe all the minimal models and to study a refined version of canonical bundle formula which will be useful for many problems.In this research, we finish the second stage and come to almost the final step of the third stage. The results in the research are also effective in the case of fibrations of complex tori.

纤维空间的极小模型理论被认为是投射簇极小模型理论的相对版本。仍然存在翻转猜想，这是一个障碍，在显示存在的最小模型。本研究的目的是明确地研究椭圆纤维化的极小模型，避免翻转，与椭圆表面的情况相反，我们不能假设纤维化在高维基的情况下是极小的。然而，通过反复试验，我们成功地在局部单值是幂单的并且约数上不存在多个纤维的情况下，在基数上局部构建了一个最小模型。这是我们的出发点。本文利用Hodge结构的变分理论和环面嵌入的变分理论，利用上述极小模型，对在固定法交因子外光滑的射影椭圆纤维化进行了双纯分类。这是我们的第二阶段。第三阶段是给出每个双纯类的极小模型的显式构造。在下一个阶段，我们要描述所有的极小模型，并研究一个精化的规范丛公式，这将有助于许多问题的解决，在这个研究中，我们完成了第二阶段，并几乎进入了第三阶段的最后一步。研究结果对复杂环面的纤维化问题也是有效的。