代数曲面纤维化的不变量及其应用

在这个项目中，我们主要研究纤维化的不变量的性质以及它们的应用。(1) 我们研究纤维化的模不变量的关系。为此要证明奇异纤维陈数的一些不等式，估计非常模情形下模不变量的下界， 证明亏格3非超椭圆纤维化的相对不变量的Reid-Xiao猜想，寻找三点式纤维化的不变量计算公式。(2)我们研究非半稳定奇异纤维个数的下界的问题。我们要研究最多含三条奇异纤维的纤维化的性质。此外，我们要证明Gurjar-张德琪猜想。(3) 计算叶状结构的陈数，寻找它们的关系，用它们研究叶状结构更精细的分类。

In this research, our main goal is to study the invariants of fibrations and their applications. (1) We study the relations between the modular invariants. For this purpose, we will prove some inequalities for Chern numbers of singular fibers, give the lower bounds of the modular invariants, prove Reid-Xiao conjecture for the relative invariants of nonhyperelliptic fibrations of genus 3 and find the formulas for the invariants of a trigonal fibration. (2) We will study the problem on the lower bound of the number of singular fibers. We study the properties of fibrations with 2 or 3 singular fibers. Moreover, we will prove Gurjar-Zhang's conjecture. (3) Compute the Chern numbers of foliations ,find their relations and give more precise classification by using the Chern numbers.