Linear Spaces, Buildings and Principal Bundles on Curves in Tropical Geometry / Arne Kuhrs ; Betreuer: Prof. Dr. Martin Ulirsch. Paderborn, 2026
Inhalt
- Introduction
- Embedded Tropical Geometry
- Non-Archimedean Geometry and Berkovich Spaces
- Tropical Linear Spaces and Valuated Matroids
- Tropical Real Algebraic Geometry
- Bruhat–Tits Buildings
- Tropical Reductive Groups
- Tropical Line Bundles on Metric Graphs
- Tropicalization of Line Bundles on Curves
- Contributions of this Thesis
- Buildings, Valuated Matroids, and Tropical Linear Spaces
- The Affine Building and its Compactification
- Analytification and Tropicalization
- Limits of Linear Tropicalizations
- Valuated Matroids and Tropical Linear Spaces
- Essentials of Matroids
- Valuated Matroids
- Tropical Linear Spaces and Matroid Polytopes
- Tropicalized Linear Spaces
- The Trivial Valuation Case
- Faithful Linear Tropicalization
- The Trivially and the Discretely Valued Case
- The Universal Realizable Valuated Matroid
- The Signed Goldman–Iwahori Space and Real Tropical Linear Spaces
- Real Tropical Geometry
- Matroids over the Real Tropical Hyperfield
- Real Analytification and Tropicalization
- The Signed Goldman–Iwahori Space
- The Limit Theorem
- Relation to the Goldman–Iwahori Space
- Towards a Generalization to Tropical Extensions of Hyperfields
- The Universal Realizable Oriented Valuated Matroid
- The Case of Real Numbers with Trivial Valuation
- Real Bergman Fans
- Tropical Reductive Groups and Principal Bundles on Metric Graphs
- Tropical reductive groups
- Root systems and tropical reductive groups
- Type An: the tropical general, special, and projective linear groups
- Type Cn: the tropical symplectic group
- Types Bn and Dn: the tropical orthogonal and special orthogonal groups
- Tropical G2
- Tropical principal bundles
- Degree and stability in the algebraic setting
- Moduli spaces of tropical principal bundles
- Degree and stability in the tropical setting
- Moduli of tropical G-bundles
- Stable bundles on tropical elliptic curves
- Semistable bundles on tropical elliptic curves
- Tropicalization of principal bundles
- Bibliography
- Acknowledgements