Minimal models for fiber spaces

点击翻译按钮获取中文摘要

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.