Dissertation Defense: Suhyon Chong

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Thesis titled "On Tropical Vector Bundles".

Tropical vector bundles are combinatorial analogues of toric vector bundles in which vector spaces and their subspaces are replaced by matroids and their flats. In this thesis, we study combinatorial invariants of tropical vector bundles and extend their theory to valuated matroids over toric schemes defined over a discrete valuation ring. We associate a lattice convex chain introduced by Khovanskii-Pukhlikov to a tropical vector bundle and show that its values at lattice points are the corresponding Euler characteristics. Using the theory of convex chains, we obtain a combinatorial Hirzebruch-Riemann-Roch formula for tropical vector bundles. We also study the tautological tropical vector bundle of a matroid and show that its Euler characteristic equals the rank of its global sections. In the second part, we develop a theory of tropical vector bundles associated with valuated matroids over toric schemes defined over a discrete valuation ring. We describe these bundles in terms of graded piecewise linear maps to extended Bergman fans and compatible tropical lattices. We also study their restrictions to the irreducible components of the special fiber and construct the corresponding equivariant characteristic classes. In particular, we construct the tautological tropical vector bundle of a valuated matroid and show that the support of its top equivariant Chern class recovers the associated Bergman complex. Finally, motivated by specialization in toric geometry, we introduce a recession map on Minkowski weights and study its kernel and cokernel as combinatorial invariants of the valuated matroid.

Advisor: Dr.Kiumars Kaveh