Tropical varieties are polyhedral complexes that are combinatorial shadows of degenerations of algebraic varieties embedded into a projective space. Mikhalkin and Zharkov defined bigraded (co)homology groups for tropical varieties that under a combinatorially defined condition of tropical smoothness can be related to the limiting mixed Hodge structure on the cohomology of the nearby fibre. On the other hand, the properties of the tropical homology groups of smooth tropical varieties are tightly connected to combinatorics of matroids and to the recent work of Adiprasito, Huh and Katz on combinatorial Lefschetz-type theorems. This summer school aims to introduce interested PhD students and postdocs, both from tropical geometry proper and from the adjacent subjects areas, to this topic.
The meeting will be held over Zoom. In order to receive Zoom links please fill out the registration form.
Eric Katz (Ohio State University): Toric schemes, semistable degenerations, and tropicalization
We give an introduction to tropical geometry focusing on the operation of tropicalization which assigns a polyhedral complex to a subvariety of an algebraic torus. Under a smoothness hypothesis, the algebraic variety will have a stratification (in the constant coefficient case) or a semistable degeneration (in the valued field case) modeled on a toric variety or a toric scheme, respectively. Time permitting, we will also discuss how to leverage this description into a computation of cohomology.
notes: KU-1.pdf KU-2.pdf KU-3.pdf
Kris Shaw (University of Oslo): Tropical homology and real algebraic geometry
notes: Lecture 1, Lecture 2
Omid Amini (École Polytechnique, CNRS), Matthieu Piquerez (École Polytechnique): Hodge theory for tropical varieties
Notes
Times are given in Central European Summer Time zone (Paris, Brussels, Berlin). To get time in London and UK substract 1, New York and East coast 6, Moscow is +1, Hong Kong +6. Lectures are 1 hour long.