Cohomologically Tropical Varieties

This talk asks which tropicalisations of subvarieties of the torus know the cohomology of the original variety. A motivating example are linear embeddings of complements of hyperplane arrangements.

We can prove that the tropicalisation knows the cohomology of the variety in a strong sense if and only if it satisfies local tropical Poincaré duality and the original variety is so-called “wunderschön”. Following the work of Itenberg, Katzarkov, Mikhalkin, and Zharkov, we can obtain information about the mixed Hodge structure of a family of varieties from its tropicalisation when it locally satisfies the two conditions above.