TGiZ-Seminar: Tropical geometry in Zoom (Third meeting Summer Semester 2022)
Matilde Manzaroli (Universität Tübingen)

Abstract:  The talk is about a work in progress with Emiliano Ambrosi.
Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin and Ilia Zharkov proved in “Tropical homology” that for a smooth proper family of complex varieties over the punctured disk with smooth tropicalisation X the Hodge numbers of the general fiber coincide with the dimensions of the tropical homology groups of X. We explore the possibility of extending this result over more general discrete valued fields of arithmetic interest, such as R((t)) or Qp, the field of p-adique numbers. In the process of doing this, we get an action of the Galois group on the tropical homology groups and we compare this action, in certain cases, with the action defined by Tyler Foster in “Galois actions on analytifications and tropicalisation”.