任意特征代数闭域上的不规则代数簇的分类

代数簇的分类是代数几何的重要研究内容，不规则代数簇是非常重要的一大类代数簇。本项目拟研究复数域上和正特征代数闭域上的不规则代数簇的分类。一方面，对于复数域上的不规则代数簇，我们将借助典范层或者多典范层的Fourier-Mukai变换以及上同调支集来研究它们的多典范映射和几何结构，拟得到一个下界N使得N-典范映射和Iitaka纤维化一致，并刻画一类欧拉特征为零的代数簇。另一方面，我们将研究正特征代数闭域上的三维不规则代数簇的极小模型理论中的一个重要问题——abundance猜想。通过改进现有的关于相对典范层的正性和一般消失定理的结论，我们可以利用Albanese映射诱导的纤维化来研究三维不规则代数簇的Kodaira维数，进而对特殊的不规则代数簇证明abundance猜想。 本项目的结论将使我们对复数域上的一些代数簇有更精细的了解，并且能够完善正特征代数闭域上的代数簇的分类理论。

Classification of algebraic varieties is an important task of algebraic geometry. Irregular varieties form a large class of varieties. This project aims to study irregular varieties over both the field of complex numbers and algebraically closed field of positive characteristic. On one hand, for irregular varieties over the field of complex numbers, we will apply the Fourier-Mukai transform and the cohomological locus of (pluri)canonical sheaves to study their pluricanonical maps and their geometry, and aim to get a lower bound N such that N-canonical map coincides with the Iitaka fibration and describe a class of special varieties with vanishing Euler characteristic. On the other hand, we will study the abundance conjecture of three dimensional irregular varieties over an algebraically closed field of positive characteristic, which is an important problem in minimal model program. By improving the results of the semipositivity of relative canonical sheaves and generic vanishing theorem, we can take advantage of the fibration arising from the Stein factorization of Albanese map to study the Kodaira dimension of three dimensional irregular varieties, and in turn prove abundance conjecture for some special irregular varieties. The results of this project help us to know better about some varieties over the field of complex numbers, and consumate the classification theory of varieties over an algebraically closed field of positive characteristic.