Abstract

Metric graphs are piecewise-linear objects that arise as tropicalizations of algebraic curves. For a metric graph, we can define notions such as line bundles, divisors, linear equivalence, and the Picard group, in complete analogy with the algebraic setting. Furthermore, all of these notions behave well with respect to tropicalization.

I will give a naive definition of vector bundles on metric graphs as torsors over the tropical general linear group. A simple dimension count shows that our construction is incomplete and does not fully capture tropicalization of algebraic vector bundles. Nevertheless, tropical vector bundles satisfy a number of results that are analogous to the algebraic setting, such as the Birkhoff—Grothendieck theorem, the Weil—Riemann—Roch theorem, a Narasimhan—Seshadri correspondence, and a version of Atiyah’s classification of vector bundles on an elliptic curve.

Joint work with Andreas Gross and Martin Ulirsch.