Vector bundles on Olsson fans

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.