A Lefschetz (1,1) theorem in tropical geometry

The classical Lefschetz theorem characterizes the cohomology classes of complex projective varieties which arise as Chern classes of line bundles. It asserts that the these classes are precisely the integral classes contained in the (1, 1) part of the Hodge decomposition. In this talk I will explain an analogous result for polyhedral spaces. In joint work with Philipp Jell and Johannes Rau, we show that the tropical cohomology classes arising from tropical line bundles are precisely the integral classes contained in the kernel of the so-called wave map of Mikhalkin and Zharkov. Combining this result with Poincaré duality, given a tropical manifold of dimension n we can completely describe the tropical homology classes of type (n-1, n-1) that can be represented by so-called straight cycles. These are the candidates for the tropicalization of algebraic cycles. I will explain the proof of these facts along with corollaries and examples of our results.