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From amoebas to pluripotential theory on hybrid analytic spaces

Sebastien Boucksom

TL;DR

This work develops a comprehensive hybrid framework assembling complex degenerations with non-Archimedean Berkovich geometry to study degenerations of Calabi–Yau and related varieties. It builds a bridge between complex pluripotential theory and its non-Archimedean counterpart through hybrid spaces, model metrics, and dual complexes, proving convergence of volume forms and potentials in the hybrid topology and establishing stability results for Monge–Ampère equations in degenerations. Key contributions include a detailed analysis of convergence of measures and potentials in hybrids, a robust theory of hybrid model functions and log maps, and a maximal hybrid extension principle for non-Archimedean metrics, with applications to essential skeletons and Calabi–Yau degenerations. The results provide foundational tools for understanding how complex degenerations collapse to non-Archimedean limits, enabling new insights into the geometry of degenerations and the behavior of Monge–Ampère dynamics across the Archimedean/non-Archimedean divide.

Abstract

These lecture notes are an introduction to the use of non-Archimedean geometry in the study of meromorphic degenerations of complex algebraic varieties. They provide a self-contained discussion of hybrid spaces, which fill in one-parameter degenerations with the associated non-Archimedean Berkovich space as a central fiber. The main focus is on the interplay between complex and non-Archimedean pluripotential theory, and on the relation between convergence of psh metrics and the associated Monge-Ampere measures in the hybrid space, following work of the author with M.Jonsson, and recent breakthrough work by Y.Li.

From amoebas to pluripotential theory on hybrid analytic spaces

TL;DR

This work develops a comprehensive hybrid framework assembling complex degenerations with non-Archimedean Berkovich geometry to study degenerations of Calabi–Yau and related varieties. It builds a bridge between complex pluripotential theory and its non-Archimedean counterpart through hybrid spaces, model metrics, and dual complexes, proving convergence of volume forms and potentials in the hybrid topology and establishing stability results for Monge–Ampère equations in degenerations. Key contributions include a detailed analysis of convergence of measures and potentials in hybrids, a robust theory of hybrid model functions and log maps, and a maximal hybrid extension principle for non-Archimedean metrics, with applications to essential skeletons and Calabi–Yau degenerations. The results provide foundational tools for understanding how complex degenerations collapse to non-Archimedean limits, enabling new insights into the geometry of degenerations and the behavior of Monge–Ampère dynamics across the Archimedean/non-Archimedean divide.

Abstract

These lecture notes are an introduction to the use of non-Archimedean geometry in the study of meromorphic degenerations of complex algebraic varieties. They provide a self-contained discussion of hybrid spaces, which fill in one-parameter degenerations with the associated non-Archimedean Berkovich space as a central fiber. The main focus is on the interplay between complex and non-Archimedean pluripotential theory, and on the relation between convergence of psh metrics and the associated Monge-Ampere measures in the hybrid space, following work of the author with M.Jonsson, and recent breakthrough work by Y.Li.

Paper Structure

This paper contains 51 sections, 60 theorems, 297 equations, 1 figure.

Table of Contents

  1. Amoebas, valuations and tropicalization
  2. The asymptotic cone of an amoeba
  3. Semivaluations
  4. The non-Archimedean amoeba
  5. Tropicalization
  6. Degenerations of amoebas
  7. Toric degenerations
  8. Toric degenerations
  9. Berkovich analytification
  10. The Berkovich spectrum
  11. General Berkovich analytification
  12. Tropicalization, reprise
  13. Base change
  14. Continuous metrics
  15. Hybrid spaces
  16. ...and 36 more sections

Key Result

Lemma 1.3

Pick $v\in X^\mathrm{an}$ and a finite set $(f_i)$ in $\mathcal{O}(X)$. If $v(\sum_i f_i)>\min_i v(f_i)$, then the minimum is achieved at least twice.

Figures (1)

  • Figure 1: Amoeba and tropicalization of $Z=(z_1+z_2+1=0)\subset(\mathbb{C}^\times)^2$

Theorems & Definitions (151)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • Definition 1.6
  • Example 1.7
  • Definition 2.1
  • Remark 2.2
  • ...and 141 more