Tropical approach to Chern-Weil theory for singular metrics

Given a smooth complex manifold, Chern-Weil theory asks for the Chern classes of holomorphic vector bundles to be represented by forms and currents in de Rham cohomology. If the vector bundle is endowed with a smooth (even mildly singular) metric, then Chern Weil theory holds. There are however cases of rich arithmetic interest (e.g. universal abelian varieties) where Chern-Weil theory does not longer hold. The goal of this talk is to propose an infinite version of Chern-Weil theory for line bundles using tropical geometry. The key idea is that the algebraic geometric analogue of the first Chern current of a singular metric should actually be a limit of divisors, a so-called b-divisor. Assuming some toroidal properties of the metric, this limit can be expressed in terms of a function on a tropical variety. Then, top wedge products of Chern currents of singular metrics correspond to a mixture of algebraic and tropical intersection numbers.