Analytic geometry and algebraic geometry of reductive groups

“解析几何与代数约化群的解析几何”项目旨在用解析几何的新方法解决代数问题。我们方法的新特点是解析几何在任意地场上的应用。这些可以配备平凡的绝对值，这使得它们可以用于Berkovich发展的解析空间理论。在这个框架中，在所选的地面场上的每一个代数变化都会产生一个解析变化。该空间比品种的几何点包含更多的点，并且具有更好的连通性。这个项目是基于与Bertrand rsammy和Amaury Thuillier的联合论文，我们在有限域上确定了Drinfeld上半平面的自同构群。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变体。确定其自同构群是有限域上两族代数几何中的一个问题。我们通过研究相应的Berkovich解析空间的一个子集，用解析几何的方法来求解它，这个子集反映了一个合适的紧化的几何形式。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值地面场的约化群及其齐次空间的理论。我们的目标是在两个方向上取得进展。其中一个目标是发展向量建筑在解析空间中的等变嵌入理论。另一方面是利用解析几何研究有限域上其他变异的自同构群。

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.