Minimal models for fiber spaces

纤维空间的极小模型理论是射影空间极小模型理论的相对版本。翻转猜想仍然存在，这是证明最小模型存在性的一个障碍。本研究的目的是明确地研究椭圆型颤振的最小模型，避免翻转。与椭圆曲面的情况相反，我们不能假设高维基底的振动是最小的。然而，通过尝试和错误，我们成功地构建了一个最小模型，局部在基础之上，在局部单性是单性的，并且在除数上没有多个纤维存在的情况下。这是我们的起点。这里需要霍奇结构变分理论和环面嵌入理论。利用上述最小模型，我们可以对在固定法向交叉除数外光滑的射影椭圆颤振进行双亚纯分类。这是我们的第二阶段。第三阶段是给出每个双纯类的最小模型的显式构造。在下一阶段，我们想要描述所有的最小模型，并研究一个精炼版本的规范束公式，这将对许多问题有用。在本研究中，我们完成了第二阶段，来到了第三阶段的最后一步。研究结果同样适用于复杂环面振动的情况。

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.