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Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction

Philip Engel, Olivier de Gaay Fortman, Stefan Schreieder

Abstract

We survey the Mumford construction of degenerating abelian varieties, with a focus on the analytic version of the construction, and its relation to toric geometry. Moreover, we study the geometry and Hodge theory of multivariable degenerations of abelian varieties associated to regular matroids, and extend some fundamental results of Clemens on 1-parameter semistable degenerations to the multivariable setting.

Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction

Abstract

We survey the Mumford construction of degenerating abelian varieties, with a focus on the analytic version of the construction, and its relation to toric geometry. Moreover, we study the geometry and Hodge theory of multivariable degenerations of abelian varieties associated to regular matroids, and extend some fundamental results of Clemens on 1-parameter semistable degenerations to the multivariable setting.

Paper Structure

This paper contains 30 sections, 36 theorems, 161 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Contents
  3. Index of constructions
  4. Acknowledgments
  5. Preliminary material
  6. Algebraic and analytic stacks
  7. Principally polarized abelian varieties
  8. Degenerations of PPAVs
  9. Toric varieties
  10. Toroidal extensions
  11. The Mumford construction
  12. Mumford construction, fan version
  13. Weight filtration and dual complex
  14. Mumford construction, polytope version
  15. Comparison of polytope and fan constructions
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $f^*\colon X^*\to (\Delta^*)^k$ be a smooth family of PPAVs of dimension $g$, whose monodromy cone is matroidal (Defs. monodromy-cone, matroidal-cone). There is a flat, $K$-trivial extension $f\colon X\to \Delta^k$ which is a nodal morphism (Def. D-nodal), and $f$ may be assumed strictly nodal

Figures (23)

  • Figure 1: Nearby fiber $C_t$ to the universal deformation $\pi \colon C\to \Delta^3$ of a nodal genus $2$ curve $C_0$ with three nodes. Vanishing cycles $\{\gamma_1,\gamma_2,\gamma_3\}\in {\rm gr}^W_{-2}V_{\mathbb Z}$ shown in red, and dual graph $\Gamma(C_0)$ of the central fiber, shown in blue.
  • Figure 2: Mumford polytope construction of the Tate curve, $\Theta_{0/1}(z,u)$ in blue, $\Theta_{0/2}(z,u)$ in red, $\Theta_{1/2}(z,u)$ in green.
  • Figure 3: Normal fan of the Tate curve.
  • Figure 4: Bending complex of the Tate curve in ${\mathbb R}/{\mathbb Z}$. The integer $1$ indicates the bending parameter, see Definition \ref{['bendinglocus']}.
  • Figure 5: Top left: Universal cover of the Tate curve. Inverse image of $\Delta^*_u$ depicted in red, with embedding into $({\mathbb C}^*)^2$, in grey. In the toroidal extension $Y({\rm Cone}({\mathcal{T}}))$, the fiber over $0\in \Delta_u$ in blue is an infinite ${\mathbb Z}$-periodic quilt of toric varieties, given by gluing an infinite chain of ${\mathbb P}^1$s. A ${\mathbb Z}$-orbit of co-characters passing through $(1,1)\in ({\mathbb C}^*)^2$ and forming sections over ${\mathbb C}_u$ is depicted in green. Top right: The Tate curve, with general fiber ${\mathbb C}^*/u^{\mathbb Z}$ in red, central nodal fiber in blue, and section in green.
  • ...and 18 more figures

Theorems & Definitions (144)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • ...and 134 more