The authors prove that pointed matroids over perfect idylls form a combinatorial proto-exact category with duality and exact direct sum, and that tropical toric reflexive sheaves (modular ones in particular) are proto-exact and proto-abelian, with Khan-Maclagan Harder-Narasimhan filtrations realized
Tropical Vector Bundles
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abstract
The goal of this paper is to introduce a construction of a vector bundle on a tropical variety. When the base is a tropical toric variety these tropicalize toric vector bundles, and are described by the data of a valuated matroid and some flats in the lattice of flats of the underlying matroid. The fibers are tropical linear spaces. We define global sections for tropical toric vector bundles, stability, and Jordan-H\"older and Harder-Narasimhan filtrations. Many of these require additional modularity assumptions on the defining matroid.
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2025 1verdicts
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Proto-Exact Categories of Matroids over Idylls and Tropical Toric Reflexive Sheaves
The authors prove that pointed matroids over perfect idylls form a combinatorial proto-exact category with duality and exact direct sum, and that tropical toric reflexive sheaves (modular ones in particular) are proto-exact and proto-abelian, with Khan-Maclagan Harder-Narasimhan filtrations realized