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Skeletons and fans of logarithmic structures. (English) Zbl 1364.14047
Baker, Matthew (ed.) et al., Nonarchimedean and tropical geometry. Based on two Simons symposia, Island of St. John, March 31 – April 6, 2013 and Puerto Rico, February 1–7, 2015. Cham: Springer (ISBN 978-3-319-30944-6/hbk; 978-3-319-30945-3/ebook). Simons Symposia, 287-336 (2016).
This is an expository paper that reviews different generalizations of fans of toric varieties including Kato fans, Artin fans, polyhedral cone complexes, and skeletons. It also investigates the relations among these generalizations and provides several applications.
Given a logarithmically regular scheme \(X\), K. Kato [Am. J. Math. 116, No. 5, 1073–1099 (1994; Zbl 0832.14002)] associates a monoidal space \(F_X\), the so called Kato fan, which encodes the combinatorial structure of \(X\). By considering sheaves on the category of Kato fans the authors explain that Kato fans and their generalizations can be constructed for more general logarithmic structures.
Artin fan \(A_X\) of a logarithmic scheme \(X\) is an alternative generalization of fans of toric varieties, however, Artin fans are not functorial with respect to general morphisms of logarithmic schemes. Following M. C. Olsson’s ideas in [Math. Ann. 333, No. 4, 859–931 (2005; Zbl 1095.14016)], the authors work with stacks of diagrams of logarithmic structures to cope with this difficulty. Concerning the relations of these generalizations, they provide a fully faithful functor from the category of Kato fans into the category of Artin fans.
The paper discusses some applications of the theory of Artin fans in Gromov-Witten theory and boundedness of logarithmic stable maps.
For the entire collection see [Zbl 1354.14004].

MSC:
14T05 Tropical geometry (MSC2010)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14G22 Rigid analytic geometry
14A20 Generalizations (algebraic spaces, stacks)
14D20 Algebraic moduli problems, moduli of vector bundles
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