Cohomology of real-valued differential forms on Berkovich analytic spaces
Final Report Abstract
The area of my project was algebraic geometry, more concretely tropical and nonarchimedean geometry. Most of the project was concerned with two new cohomology theories in these areas. Namely Dolbeault cohomology of Lagerberg forms for tropical varieties and tropical Dolbeault cohomology on Berkovich spaces, defined via CLD-forms, in the sense of Chambert-Loir and Ducros. My DFG Research Project achieved most of the goals that were outlined in the original proposal. At least as far as those were actually achievable. The goals of Subprojects 1 and 2 was construction of tropicalizations for algebraic varieties that were nice in a suitable sense, the technical terms being “smooth” and “fully faithful”. These constructions were achieved for Mumford curves in my paper “Constructing smooth and fully faithful tropicalizations for Mumford curves”. It was also shown that the goal of subproject 2 is actually only achievable for Mumford curves. The goal of subproject 3 was computation of tropical Dolbeault cohomology for algebraic curves. This was already achieved beforehand by myself in the paper “Tropical Hodge numbers of non-archimedean curves”. Here there are some curves for which duality does not hold. It is however exactly shown for which curves it is true and a conceptual explanation why this is not true for other curves is given. In my current ongoing project with Rabinoff the goal is to slightly tweak the theory of CLD-forms in a way that duality is true for all curves. This will be shown in our paper. The goal of subproject 4 was construction of integral classes in tropical Dolbeault cohomology. This will be, among other results, part of my ongoing project “Non-trivial and integral classes in tropical Dolbeault cohomology”. The goal of subproject 5 was to establish a Hodge theory for tropical cohomology. My joint project with Rau and Shaw “Lefschetz (l,l)-theorem in tropical geometry” achieved a first step in this direction. The completion of this project will be the subject of further work. Additionally, the joint project with Scheiderer and Yu “Real tropicalization and an-alvtification of semialgebraic sets” opens a new direction of research. We study tropicalizations over real closed fields. This opens opportunities for the use of tropical and non-archimedean geometry in real (semi)algebraic geometry. In total, the research project achieved most of its goals and opened new and interesting new research possibilities in tropical, non-archimedean and real algebraic geometry.
Publications
- Lefschetz (1, 1)-theorem in tropical geometry, Épijournal Géométrie Algébrique, 2: Art, 2018
P. Jell, J. Rau, and K. Shaw
