Rationally Connected Varieties

本提案的主要目的是探讨几何中的几个问题。第一部分从多个角度对最简单的代数变种——理性连通变种进行了研究。对于非固有变量，有各种不同的有理连通性定义。提议者将研究它们之间的关系。他还将继续研究理性连接型的算术性质。其他部分主要讨论了变异或对的有界性问题。特别是，当对是log一般类型时，PI研究者试图建立其体积的均匀性质。代数变量大致是多项式的解所描述的图形。代数几何是一门旨在对所有代数变体进行分类的学科。在双域意义上，代数变量是由一些特定类型的变量构成的。合理连接的品种和一般类型品种是这样的基本组成部分。理解它们将在很大程度上提高我们对所有代数变体的理解。

The main aim of this proposal is to investigate a few questions in birational geometry. The first parts intends to study rationally connected varieties, which are the simplest algebraic varieties from many points of view. For non-proper varieties, there are various definitions of rational connectedness.The proposer will study their relations. He will also continue his study on the arithmetic property of rationally connected varieties. The other parts focus on the boundedness type questions of varieties or pairs. Especially, when the pair is of log general type, the PI investigator is trying to establish the uniform properties of their volumes.Algebraic varieties are roughly speaking the figures described by the solutions of polynomials. Algebraic geometry is a subject which aims to classify all algebraic varieties. In the birational sense, algebraic varieties are built up by some specific types of varieties. Rationally connected varieties and general type varieties are such fundamental building blocks. To understand them will largely improve our understanding of all algebraic varieties.