一般型复代数曲面与三维代数簇的分类和Bloch猜想的研究

This research project is about the classification of algebraic surfaces and 3-folds of general type over the complex number field and about the study of the Bloch conjecture. It contains three topics. The first topic is aimed to construct new surfaces of general type with geometric genus zero, to study their automorphism groups, deformations, Gieseker moduli spaces and the compactification of the moduli spaces. The second topic is to study the inequalities of Noether type for the algebraic 3-folds of general type , to construct and to classify 3-folds satisfying the equalities. The third topic is about the study of the Bloch conjecture. These three topics are related via the study of the automorphism groups of surfaces with small invariants and the study of the compactification of their Gieseker moduli spaces.

本项目旨在研究复数域上一般型代数曲面与三维代数簇的分类以及研究Bloch猜想。主要研究内容包括三个方面。其一、研究几何亏格为零的一般型代数曲面的构造、自同构群、形变、Gieseker模空间以及Gieseker模空间的紧化。其二、研究一般型三维代数簇的Noether类型不等式，构造并分类满足等式的三维代数簇。其三，对著名的Bloch猜想做探索性的研究。三个方面的内容通过研究小不变量曲面的自同构群和模空间的紧化联系起来。

本项目研究复数域上一般型代数曲面与三维代数簇的分类以及研究Bloch猜想。内容包括：研究几何亏格为零的一般型代数曲面的构造、自同构群、形变、Gieseker模空间以及Gieseker模空间的紧化；研究一般型三维代数簇的Noether类型不等式，构造并分类满足等式的三维代数簇等等。所取得的重要成果：对第一陈类自交数为7、几何亏格为0且双典范映射非双有理的一般型复代数曲面做出部分分类。