A tropical version of the Lefschetz (1, 1) theorem

In this talk I will present joint work in preparation with Philipp Jell and Johannes Rau on a tropical version of the Lefschetz (1, 1) theorem.

In complex algebraic geometry this theorem completely describes which degree 2 cohomology classes of a smooth projective variety can be represented by (Poincare duals to) algebraic cycles. For tropical varieties, we of course must consider tropical cohomology classes. This is just the cohomology of certain cellular sheaves on a tropical variety and can be computed using the cellularSheaves package for polymake (joint work with Lars Kastner and Anna-Lena Winz).

The tropical analogue of the Lefschetz (1,1) theorem uses a "wave map” on cohomology introduced by Mikhalkin and Zharkov. I will explain how this map has the ability to detect tropical algebraic cycles and how it might also be used in moduli problems, for example in the case of K3 surfaces.