Kris Shaw, University of Oslo
The cohomology of the complement of a hyperplane arrangement can be recovered from its tropicalisation. This talk asks what which other subvarieties of the torus does this hold. We show 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 what we call “wunderschön”. Following the work of Itenberg, Katzarkov, Mikhalkin, and Zharkov, we have that tropicalisations of families of varieties which locally satisfy these two conditions recover information about the mixed Hodge structures of the family.
This is joint work with Edvard Aksnes, Omid Amini, and Matthieu Piquerez.