Speaker: Martin Ulirsch (Bonn)

Title: Logarithmic structures, Artin fans, and tropical compactifications

Abstract: Artin fans are logarithmic algebraic stacks that are logarithmically \'etale over a trivially valued base field $k$. They have made their first appearance in the work of Abramovich-Chen-Marcus-Wise, but can implicitly be traced back to the work of Olsson on classifying stack of logarithmic structures. Despite their seemingly abstract definition their geometry can be described completely in terms of a combinatorial objects, so called Kato stacks, a stack-theoretic generalization of K. Kato's notion of a fan. In fact, expanding on Olsson's work, one can show that an Artin fan is the classifying stack of logarithmic structures of a certain combinatorial type. In particular, every logarithmic stack admits a strict tautological morphism $\phi_\calX:\mathcal{X}\rightarrow\mathcal{A}_\mathcal{X}$ to an associated Artin fan. In earlier work I have studied the tropicalization map of logarithmic schemes expanding on work of Thuillier on the non-Archimedean geometry of toroidal embeddings. The main result I am going to present in this talk is that this tropicalization map is nothing but the non-Archimedean analytic map $\phi^\beth$ associated to $\phi_\calX$ by applying Thuillier's generic fiber functor $(.)^\beth$. In the special case of toric varieties this amounts to showing that the Kajiwara-Payne tropicalization map of a toric variety can be identified with a certain non-Archimedean analytic stack quotient. Finally, I am going to explain how Artin fans can be used to give a reinterpretation of Tevelev's theory of tropical compactifications in terms of logarithmically flat compactifications of subvarieties in a surrounding logarithmically smooth scheme.