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Moduli of hybrid curves I: Variations of canonical measures

Omid Amini, Noema Nicolussi

Abstract

The present paper is the first in a series devoted to the study of asymptotic geometry of Riemann surfaces and their moduli spaces. We introduce the moduli space of hybrid curves as a new compactification of the moduli space of curves, refining the one obtained by Deligne and Mumford. This is the moduli space for multiscale geometric objects which mix complex and higher rank tropical and non-Archimedean geometries, reflecting both discrete and continuous features. We define canonical measures on hybrid curves which combine and generalize Arakelov-Bergman measures on Riemann surfaces and Zhang measures on metric graphs. We then show that the universal family of canonically measured hybrid curves over this moduli space varies continuously. This provides a precise link between the non-Archimedean Zhang measure and variations of Arakelov-Bergman measures in families of Riemann surfaces, answering a question which has been open since the pioneering work of Zhang on admissible pairing in the nineties.

Moduli of hybrid curves I: Variations of canonical measures

Abstract

The present paper is the first in a series devoted to the study of asymptotic geometry of Riemann surfaces and their moduli spaces. We introduce the moduli space of hybrid curves as a new compactification of the moduli space of curves, refining the one obtained by Deligne and Mumford. This is the moduli space for multiscale geometric objects which mix complex and higher rank tropical and non-Archimedean geometries, reflecting both discrete and continuous features. We define canonical measures on hybrid curves which combine and generalize Arakelov-Bergman measures on Riemann surfaces and Zhang measures on metric graphs. We then show that the universal family of canonically measured hybrid curves over this moduli space varies continuously. This provides a precise link between the non-Archimedean Zhang measure and variations of Arakelov-Bergman measures in families of Riemann surfaces, answering a question which has been open since the pioneering work of Zhang on admissible pairing in the nineties.

Paper Structure

This paper contains 76 sections, 38 theorems, 256 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hybrid curves
  3. Moduli space $\mathscr M \sb{{^{\scaleto{{\mathrm{hyb}}}{4.4pt}}}} \sp{}$ of hybrid curves of genus $g$
  4. Hybrid spaces of higher rank
  5. Definition of the hybrid moduli space
  6. Hybrid canonical measures
  7. Continuity of the canonical measures over the hybrid moduli space
  8. Applications and outlook
  9. Notes and references to related work
  10. Organization of the paper
  11. Basic notation
  12. Preliminaries
  13. Graphs
  14. Ordered partitions
  15. Layered graphs and their graded minors
  16. ...and 61 more sections

Key Result

Theorem 1.2

For each hybrid étale chart $B^{{^{\scaleto{{\mathrm{hyb}}}{4.4pt}}}}$, the family of canonically measured hybrid curves $(\mathcal{S}^{^{\scaleto{{\mathrm{hyb}}}{4.4pt}}},\mu^{{\scaleto{\mathrm{can}}{2.5pt}}})$ forms a continuous family of measured spaces over $B^{{^{\scaleto{{\mathrm{hyb}}}{4.4pt}

Figures (4)

  • Figure 1: An example of a hybrid curve of rank three. The graph of the underlying stable Riemann surface has five vertices and seven edges. Its edges are partitioned into three sets $\pi_1,\pi_2,$ and $\pi_3$.
  • Figure 2: A layered graph with ordered partition $\pi=({\color{red}\pi_1}, {\color{blue}\pi_2}, \pi_3)$, with three layers, and its graded minors $\mathrm{gr}_\pi^1(G)$, $\mathrm{gr}_\pi^2(G)$ and $\mathrm{gr}_\pi^3(G)$, from left to right.
  • Figure 3: A layered graph $(G, \pi)$ with two layers and its four spanning trees $T_1, T_2, T_3,$ and $T_4$. The underlying non-layered graph $G$ has five spanning trees, those of $(G, \pi)$ and the spanning tree $T$ depicted in the figure.
  • Figure 4: Example of two hybrid curves with the same underlying stable Riemann surface. The canonical measures have the same Archimedean parts. The non-Archimedean parts are however different. Note that changing the order of $\pi_1, \pi_2$ in the first example would drastically change the non-Archimedean part.

Theorems & Definitions (71)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Shivaprasad Shiva
  • Example 2.1
  • Proposition 2.2: Genus formula
  • Proposition 2.3
  • proof
  • proof : Proof of Proposition \ref{['prop:genusformula']}
  • Proposition 2.4
  • Definition 2.5: Spanning trees of layered graphs
  • ...and 61 more