Tropical vector bundles and matroids
| Autor: | Kaveh, Kiumars, Manon, Christopher |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We introduce a notion of tropical vector bundle on a tropical toric variety which is a tropical analogue of a torus equivariant vector bundle on a toric variety. Alternatively it can be called a toric matroid bundle. We define equivariant $K$-theory and characteristic classes of these bundles. As a particular case, we show that any matroid comes with tautological tropical toric vector bundles over the permutahedral toric variety and the corresponding equivariant $K$-classes and Chern classes recover the tautological classes of matroids constructed in the recent work of Berger-Eur-Spink-Tseng. In analogy with toric vector bundles, we define sheaf of sections and Euler characteristic as well as positivity notions such as global generation, ampleness and nefness for tropical toric vector bundles. Moreover, we prove a vanishing of higher cohomologies result. Finally, we study the splitting of our tropical toric vector bundles and, in particular, an analogue of Grothendieck's theorem on splitting of vector bundles on projective line. Comment: Title changed. We now call the main combinatorial objects introduced in the paper "tropical toric vector bundles" (previously we called them "toric matroid bundles"). New material added, in particular a theorem about vanishing of higher cohomologies (after tensoring with a high power of an ample line bundle) was added. 38 pages, 3 figures |
| Databáze: | arXiv |
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