Problems in Higher Dimensional Algebraic Geometry

作为连接代数和几何的纽带，代数几何的主要目标是理解在几何上被称为代数簇的某些对象(即，多变量多项式的公共零集)。这些非常自然和基本的代数对象经常出现在数学和科学的许多分支中。因此，代数几何在数学的其他部分，如数论或复杂几何，以及其他科学或工程学科，如物理、编码理论、机器人和计算生物学，都有许多应用。特别是，这些代数对象的任何一般结构理论在许多研究领域都非常有用。致力于这一结构理论的代数几何部分被称为高维代数几何，它是本研究项目的主要主题。该项目旨在进一步发展高维代数几何理论。对于特征为零和特征为正的区域上的品种，有非常不同的工具可用。前者的例子是Hodge理论和相关的消失定理，而后者的例子是Frobenius提升技术或p-闭叶商法。特别是，上面的二分法给出了与大多数高维代数几何猜想截然不同的两种情况。这个项目的独特之处在于，它涉及特征为零和正特征的高维代数几何的标准主题和猜想，如高维模空间、极小模型程序、Kodaira维度的次可加性和Shafarevich类型双曲性猜想。

As a link between algebra and geometry, the main goal of algebraic geometry is to understand geometrically certain objects called algebraic varieties (i.e., common zero sets of multivariable polynomials). These very natural and fundamental algebraic objects appear frequently in many branches of mathematics and science. Hence, algebraic geometry sees many applications in other parts of mathematics such as number theory or complex geometry, and also in other scientific or engineering disciplines such as physics, coding theory, robotics, and computational biology. In particular, any general structure theory of these algebraic objects can be very useful in many fields of study. The part of algebraic geometry devoted to this structure theory is called higher dimensional algebraic geometry and it is the main subject of this research project. The project aims to further develop the theory of higher dimensional algebraic geometry. For studying varieties over a field of characteristic zero and of positive characteristic, there are very different tools available. Examples of the former are Hodge theory and the related vanishing theorems, while examples for the latter are Frobenius lifting techniques or quotients by p-closed foliations. In particular, the above dichotomy yields two remarkably different cases to most conjectures of higher dimensional algebraic geometry. The particular flavor of this project is that it concerns standard topics and conjectures of higher dimensional algebraic geometry both in characteristic zero and in positive characteristic, such as higher dimensional moduli spaces, minimal model program, subadditivity of Kodaira dimension, and Shafarevich type hyperbolicity conjectures.