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Forms on Berkovich spaces based on harmonic tropicalizations

Walter Gubler, Philipp Jell, Joseph Rabinoff

TL;DR

The paper develops a general framework for Berkovich spaces over non-Archimedean fields by replacing smooth tropicalizations with harmonic tropicalizations to define weakly smooth differential forms that enjoy favorable cohomological properties. It introduces tropical skeletons, tropical multiplicities, and a balancing condition for harmonic tropicalizations, enabling robust integration and Dolbeault theory in the non-Archimedean analytic setting. A key achievement is constructing a Dolbeault cohomology H^{p,q}(X) via weakly smooth forms, proving a local Poincaré lemma, and establishing a tropical cycle class map compatible with Liu’s framework; Galois actions are carefully treated, and canonical tropicalizations for abelian varieties are shown to be harmonic, with new examples clarifying subtleties in smoothness. These advances provide a cohesive toolkit for non-Archimedean Arakelov-type theories, with potential implications for positivity, metric theory, and the study of abelian varieties in analytic geometry. The balancing and residue-formalisms connect tropical geometry with analytic, cohomological, and Galois structures in a way that mirrors complex-analytic intuition while leveraging non-Archimedean geometry.

Abstract

We introduce tropical skeletons for Berkovich spaces based on results of Ducros. Then we study harmonic functions on good strictly analytic spaces over a non-trivially valued non-Archimedean field. Chambert-Loir and Ducros introduced bigraded sheaves of smooth real-valued differential forms on Berkovich spaces by pulling back Lagerberg forms with respect to tropicalization maps. We give a new approach in which we allow pullback by more general harmonic tropicalizations to get a larger sheaf of differential forms with essentially the same properties, but with a better cohomological behavior. A crucial ingredient is that tropical varieties arising from harmonic tropicalization maps are balanced.

Forms on Berkovich spaces based on harmonic tropicalizations

TL;DR

The paper develops a general framework for Berkovich spaces over non-Archimedean fields by replacing smooth tropicalizations with harmonic tropicalizations to define weakly smooth differential forms that enjoy favorable cohomological properties. It introduces tropical skeletons, tropical multiplicities, and a balancing condition for harmonic tropicalizations, enabling robust integration and Dolbeault theory in the non-Archimedean analytic setting. A key achievement is constructing a Dolbeault cohomology H^{p,q}(X) via weakly smooth forms, proving a local Poincaré lemma, and establishing a tropical cycle class map compatible with Liu’s framework; Galois actions are carefully treated, and canonical tropicalizations for abelian varieties are shown to be harmonic, with new examples clarifying subtleties in smoothness. These advances provide a cohesive toolkit for non-Archimedean Arakelov-type theories, with potential implications for positivity, metric theory, and the study of abelian varieties in analytic geometry. The balancing and residue-formalisms connect tropical geometry with analytic, cohomological, and Galois structures in a way that mirrors complex-analytic intuition while leveraging non-Archimedean geometry.

Abstract

We introduce tropical skeletons for Berkovich spaces based on results of Ducros. Then we study harmonic functions on good strictly analytic spaces over a non-trivially valued non-Archimedean field. Chambert-Loir and Ducros introduced bigraded sheaves of smooth real-valued differential forms on Berkovich spaces by pulling back Lagerberg forms with respect to tropicalization maps. We give a new approach in which we allow pullback by more general harmonic tropicalizations to get a larger sheaf of differential forms with essentially the same properties, but with a better cohomological behavior. A crucial ingredient is that tropical varieties arising from harmonic tropicalization maps are balanced.

Paper Structure

This paper contains 35 sections, 81 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Tropical Skeletons and Tropical Multiplicities
  3. Formal and Semipositive Metrics
  4. Piecewise Linear and Harmonic Functions
  5. Piecewise Linear and Harmonic Tropicalizations
  6. The Balancing Condition
  7. Weakly Smooth Forms
  8. Integration and Strong Currents
  9. Dolbeault Cohomology and Curves
  10. Canonical Tropicalizations for Abelian Varieties
  11. Acknowledgments
  12. Prerequisites
  13. General Notation and Conventions
  14. Non-Archimedean Fields
  15. Non-Archimedean geometry
  16. ...and 20 more sections

Key Result

Theorem 1

Under the above assumptions, assume that the metrics $\|\space\|_i$ converge uniformly to the metric $\|\space\|$ of $L$. Then the first Chern current $c_1(L,\|\space\|)$ is positive.

Figures (2)

  • Figure 1: The skeleton $S$ of the curve $X$ from Example \ref{['harmonictrop:eg:tropical.skeleton.specific']}, and its tropicalization. The dashed line segment $BD$ is included in $S$ but not in $S_\varphi(X)$. The small numbers indicate tropical multiplicities; see Example \ref{['harmonictrop:eg:tropical.skeleton.multiplicities']}.
  • Figure 2: The graph $\Theta$ and its Jacobian.

Theorems & Definitions (204)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Definition 8
  • Remark 10
  • Remark 11
  • ...and 194 more