Date/Time
Date(s) - 05/05/2026
3:00 pm - 4:00 pm

Speaker: Kiumars Kaveh (University of Pittsburgh)

Location: Hamilton Hall, Room 312

Title: Tropical vector bundles

Abstract: We review Klyachko’s classification of torus equivariant vector bundles on toric varieties (toric vector bundles) in terms of combinatorial and linear algebra data. We then give a tropical reformulation of this classification which allows one to give a definition of a combinatorial object called “tropical vecor bundle” on a toric variety. It involves the notion of a matroid which I will briefly introduce. This is a joint work with Christopher Manon. We note that, independently and around the same time, Khan and Maclagan also have arrived at (basically) the same notion (of a tropical vector bundle).

Somethings which we will not have time to discuss: one defines equivariant K-theory and characteristic classes of these bundles. As a particular case, we can see that matroids come 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 work of Berget-Eur-Spink-Tseng. In analogy with toric vector bundles, one can define sheaf of sections and Euler characteristic as well as positivity notions such as global generation, ampleness and nefness for tropical toric vector bundles and prove a kind of vanishing of higher cohomologies result.