Analytic geometry and algebraic geometry of reductive groups

该项目“解析几何和代数几何的还原群”的目的是使用解析几何在一个新的方式来解决代数问题。我们的方法的新特点是在任意地面场的解析几何的应用。这些可以配备平凡的绝对值，使他们接近理论的分析空间开发的Berkovich。在这个框架下，每一个代数簇在选定的地面领域产生了一个分析簇。这个空间包含的点比该簇的几何点多得多，并且具有更好的连通性。这个项目是基于一个联合文件与贝特朗雷米和Amaury Thuillier在那里我们确定自同构群德林费尔德的上半平面在有限域。有限域上的Drinfeld上半平面定义为从射影空间中删除所有有理超平面而得到的仿射簇。确定它的自同构群是有限域上双有理代数几何中的一个问题。我们解决它与解析几何通过研究相关的Berkovich解析空间的一个子集，它反映了一个合适的紧化的正常交叉因子的几何。这个项目的目标是进一步应用解析几何在有限和其他平凡价值的地面领域的理论约化群及其齐性空间。我们的目标是在两个方向取得进展。一个目标是发展理论的等变嵌入向量建筑物到分析空间。另一个方面是借助解析几何研究有限域上其他簇的自同构群。

The project "Analytic geometry and algebraic geometry of reductive groups" aims at using analytic geometry in a new way for solving algebraic problems. The new feature in our approach is the application of analytic geometry over arbitrary ground fields. These can be equipped with the trivial absolute value which makes them accessible for the theory of analytic spaces developed by Berkovich. In this framework every algebraic variety over the chosen ground field gives rise to an analytic variety. This space contains many more points than the geometric points of the variety and has better connectivity properties. This project is based on a joint paper with Bertrand Rémy and Amaury Thuillier where we determine the automorphism group of Drinfeld's upper half plane over a finite field. Drinfeld's upper half plane over a finite field is defined as the affine variety obtained by deleting all rational hyperplanes from projective space. Determining its automorphism group is a problem in birational algebraic geometry over finite fields. We solve it with analytic geometry by studying a subset of the associated Berkovich analytic space which reflects the geometry of a suitable compactification by a normal crossing divisor. The goals of this project are further applications of analytic geometry over finite and other trivially valued ground fields to the theory of reductive groups and their homogeneous spaces. We aim at progress in two directions. One goal is the development of a theory of equivariant embeddings of vectorial buildings into analytic spaces. Another aspect is the study of automorphism group of other varieties over finite fields with the help of analytic geometry.