Analytic geometry and algebraic geometry of reductive groups

• Applicant: Professorin Dr. Annette Werner
• Subject Area: Mathematics
• Term: from 2013 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 237910448

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.

• DFG Programme: Research Grants