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.