Kristin Shaw (UiO): The Lefschetz (1,1)-theorem in tropical geometry

Abstract: The classical Lefschetz (1,1)-theorem characterizes the cohomology classes of a non-singular projective complex variety which come from Chern classes of line bundles. In joint work with Johannes Rau and Philipp Jell, we proved an analogue of this theorem for polyhedral spaces which is inspired by tropical geometry. For polyhedral spaces which are also non-singular tropical manifolds, we can phrase the theorem in terms of tropical algebraic cycles of codimension one. This leads to a version of the Hodge conjecture in tropical geometry proposed by Mikhalkin and Zharkov. In my talk I will present examples, and also highlight the differences and similarities between the tropical and complex versions of this theorem.