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Vector bundles on Olsson fans

Luca Battistella, Francesca Carocci, Jonathan Wise

TL;DR

The paper develops a coherent framework for vector bundles on Olsson fans, a relative generalization of Artin fans that interpolates between toric geometry and logarithmic geometry via weakly convex cones and tropical tori. It establishes local models and cohomological tools, showing strict étale charts by Olsson cones and combinatorial control of étale cohomology, and Classifies locally free sheaves on cones via Klyachko–Perling data, including explicit moduli descriptions for toric vector bundles. It proves that, in favorable cases (integral cones and subdivisions), the moduli of vector bundles is an algebraic stack described as a limit of classifying stacks BG on cones, but also presents substantial cautionary examples showing that, in general, vector bundles on Olsson fans can fail to split and their moduli need not be algebraic. The results illuminate how logarithmic structure and tropical geometry interact to govern vector bundles, with implications for logarithmic linear series and related moduli problems in logarithmic Gromov–Witten and DT theory. Overall, the work lays a combinatorial–geometric foundation for vector bundles on Olsson fans, clarifying when classical toric techniques extend and when new obstructions appear.

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.

Vector bundles on Olsson fans

TL;DR

The paper develops a coherent framework for vector bundles on Olsson fans, a relative generalization of Artin fans that interpolates between toric geometry and logarithmic geometry via weakly convex cones and tropical tori. It establishes local models and cohomological tools, showing strict étale charts by Olsson cones and combinatorial control of étale cohomology, and Classifies locally free sheaves on cones via Klyachko–Perling data, including explicit moduli descriptions for toric vector bundles. It proves that, in favorable cases (integral cones and subdivisions), the moduli of vector bundles is an algebraic stack described as a limit of classifying stacks BG on cones, but also presents substantial cautionary examples showing that, in general, vector bundles on Olsson fans can fail to split and their moduli need not be algebraic. The results illuminate how logarithmic structure and tropical geometry interact to govern vector bundles, with implications for logarithmic linear series and related moduli problems in logarithmic Gromov–Witten and DT theory. Overall, the work lays a combinatorial–geometric foundation for vector bundles on Olsson fans, clarifying when classical toric techniques extend and when new obstructions appear.

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.
Paper Structure (26 sections, 57 theorems, 85 equations)

This paper contains 26 sections, 57 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Olsson fans and their étale sheaves
  3. Weakly convex cones and locally free sheaves
  4. History
  5. Logarithmic linear series
  6. Acknowledgments
  7. Monoids and logarithmic structures
  8. Properties of monoid homomorphisms
  9. Logarithmic geometry and algebraic stacks
  10. Olsson cones and Olsson fans
  11. Olsson cones
  12. Weakly convex cones
  13. The star of a subcone
  14. On toric vector bundles
  15. Equivariant sheaves as graded modules
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $S$ be a fine, saturated logarithmic scheme and let $\Sigma$ be an Olsson fan over $S$. Then $\Sigma$ admits a strict étale cover by Olsson cones.

Theorems & Definitions (154)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • proof
  • Definition 2.3: Ogus
  • ...and 144 more