Tropical Dolbeault cohomology of non-archimedean analytic spaces

Series: 
Algebra Seminar
Monday, November 20, 2017 - 15:05
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Georgia Tech
Organizer: 
Real-valued smooth differential forms on Berkovich analytic spaces were introduced by Chambert-Loir and Ducros. They show many fundamental properties analogous to smooth real differential forms on complex manifolds, which are used for example in Arakelov geometry. In particular, these forms define a real valued bigraded cohomology theory for Berkovich analytic space, called tropical Dolbeault cohomology.  I will explain the definition and properties of these forms and their link to tropical geometry. I will then talk about results regarding the tropical Dolbeault cohomology of varietes and in particular curves. In particular, I will look at finite dimensionality and Poincar\'e duality.