Vector bundles on Olsson fans
Luca Battistella, Francesca Carocci, and Jonathan Wise

TL;DR
This paper explores generalized Olsson fans, including tropical tori and base logarithmic schemes, analyzing their geometric properties and vector bundles, with a focus on combinatorial determination and moduli stack structures.
Contribution
It introduces weakly convex Olsson fans over base schemes, extending Artin fans, and studies their geometric and vector bundle properties, including moduli and algebraicity issues.
Findings
Olsson fans can be well-behaved under certain conditions
Moduli stacks of vector bundles are described for subdivided cones
Failures of algebraicity occur in the general case
Abstract
Artin fans are algebro-geometric incarnations of cone complexes. We study weakly convex Olsson fans, generalising Artin fans in two ways: first, they admit lineality spaces, thus including tropical tori as well; second, they are defined over a base logarithmic scheme, thus providing a relative version of equivariant toric geometries. We determine conditions under which Olsson fans are well-behaved, in the sense that their geometry is determined combinatorially. We undertake the study of quasicoherent sheaves, and in particular vector bundles, on Olsson fans: we describe their moduli stack in the case of a (subdivided) cone, and some failures of algebraicity in the general case; Weyl convexity shows up naturally in this context.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory
