Geometry Seminar

The Euler characteristic is a linear map from the K-ring of vector bundles on a smooth, proper variety to the integers, and when that variety happens to be the toric variety associated to a complete unimodular fan, the Euler characteristic can be interpreted combinatorially in terms of lattice-point counting in polytopes.  On the other hand, recent work of Larson, Li, Payne, and Proudfoot constructed an Euler characteristic on Bergman fans of matroids, despite the fact that these fans are incomplete.  Motivated by the structural properties of lattice-point counts, we introduce a new class of fans whose K-rings admit a canonical linear map to the integers, which includes complete unimodular fans and Bergman fans of matroids as special cases, and thus creates a general framework for studying Euler characteristics of matroids and smooth proper toric varieties together.  This is joint work with Melody Chan, Caroline Klivans, and Dusty Ross.