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Semi-homogeneous vector bundles on abelian varieties: moduli spaces and their tropicalization

Andreas Gross, Inder Kaur, Martin Ulirsch, Annette Werner

Abstract

We construct a moduli space of semi-homogeneous vector bundles with a fixed Néron-Severi class $H$ on an abelian variety $A$ over an algebraically closed field of characteristic zero. When $A$ has totally degenerate reduction over a non-Archimedean field, we describe our moduli space from the perspective of non-Archimedean uniformization and show that the essential skeleton may be identified with a tropical analogue of this moduli space. For $H=0$ our moduli space may be identified with the moduli space $M_{0,r}(A)$ of semistable vector bundles with vanishing Chern classes on $A$. In this case we construct a surjective analytic morphism from the character variety of the analytic fundamental group of $A$ onto $M_{0,r}(A)$, which naturally tropicalizes. One may view this construction as a non-Archimedean uniformization of $M_{0,r}(A)$.

Semi-homogeneous vector bundles on abelian varieties: moduli spaces and their tropicalization

Abstract

We construct a moduli space of semi-homogeneous vector bundles with a fixed Néron-Severi class on an abelian variety over an algebraically closed field of characteristic zero. When has totally degenerate reduction over a non-Archimedean field, we describe our moduli space from the perspective of non-Archimedean uniformization and show that the essential skeleton may be identified with a tropical analogue of this moduli space. For our moduli space may be identified with the moduli space of semistable vector bundles with vanishing Chern classes on . In this case we construct a surjective analytic morphism from the character variety of the analytic fundamental group of onto , which naturally tropicalizes. One may view this construction as a non-Archimedean uniformization of .
Paper Structure (25 sections, 42 theorems, 132 equations)

This paper contains 25 sections, 42 theorems, 132 equations.

Table of Contents

  1. Semi-homogeneous bundles and their moduli spaces
  2. Semi-homogeneous vector bundles on abelian varieties
  3. Semi-homogeneous bundles and Fourier-Mukai transforms
  4. Moduli spaces of semi-homogeneous bundles
  5. Tropical and non-Archimedean Appell--Humbert
  6. Integral affine manifolds
  7. Tropical line bundles
  8. Factors of automorphy on real tori
  9. Line bundles on $A^\mathrm{an}$
  10. Semi-homogeneous tropical bundles on real tori
  11. Tropical vector bundles on integral affine manifolds
  12. Semi-homogeneous tropical vector bundles
  13. A parameter space for semi-homogeneous tropical bundles
  14. Uniformization and semi-homogeneous bundles
  15. Semi-homogeneous vector bundles and free covers
  16. ...and 10 more sections

Key Result

Theorem 1

Let $A$ be an abelian variety over an algebraically closed field $K$ of characteristic zero and fix a class $H\in \mathop{\mathrm{NS}}\nolimits(A)_{\mathbb{Q}}$. The moduli functor $\mathcal{M}_{H,1}(A)$ is represented by a fine moduli space $M_{H,1}(A)$. For $k\geq 1$, the moduli functor $\mathcal{

Theorems & Definitions (111)

  • Theorem 1: see Theorem \ref{['thm_modulispace']}
  • Theorem 2: see Theorem \ref{['thm:skeleton in general case']}
  • Theorem 3: see Corollary \ref{['cor_tropicalcorrespondence']} and Proposition \ref{['prop_compatibilityrep->bundle']}
  • Definition 1.1
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Mukai
  • Definition 1.6
  • Theorem 1.7: Mukai
  • Definition 1.8
  • ...and 101 more