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Degeneration of the archimedean height pairing of algebraically trivial cycles

Zhelun Chen

TL;DR

The paper investigates the limiting behavior of the archimedean height pairing for homologically trivial cycles in a degenerating family and proves that, for algebraically trivial cycles, the limit is controlled by the geometric height pairing on the generic fiber provided Griffiths's incidence equivalence conjecture holds. It builds a bridge between archimedean heights and non-archimedean geometric height pairings via a network of regular homomorphisms, higher Picard varieties, and relative algebraic intermediate Jacobians, culminating in an isomorphism between the relevant height line bundles and Lear extensions. A key outcome is a precise comparison between geometric Beilinson–Bloch height pairings and their Néron–Tate/Néron-model counterparts, formalized through Poincaré biextensions and $G_m$-biextensions, and conditional on Griffiths's conjecture, establishing the desired equivalence of line bundles on the base. The results provide a geometric viewpoint on Beilinson–Bloch height positivity and connect to the positivity results of Brosnan–Pearlstein, with implications for the study of height pairings over one-variable complex function fields.

Abstract

We consider the limiting behaviour of the archimedean height pairing for homologically trivial algebraic cycles in a degenerating one-parameter family of smooth projective complex varieties. We conjecture that the limit is controlled by the non-archimedean geometric height pairing of the cycles on the generic fiber and verify this for algebraically trivial cycles, assuming a conjecture of Griffiths on incidence equivalence. Our work offers a more geometric understanding of a related asymptotic result of Brosnan--Pearlstein and suggests a new perspective on the positivity of the Beilinson--Bloch height pairing over a one-variable complex function field.

Degeneration of the archimedean height pairing of algebraically trivial cycles

TL;DR

The paper investigates the limiting behavior of the archimedean height pairing for homologically trivial cycles in a degenerating family and proves that, for algebraically trivial cycles, the limit is controlled by the geometric height pairing on the generic fiber provided Griffiths's incidence equivalence conjecture holds. It builds a bridge between archimedean heights and non-archimedean geometric height pairings via a network of regular homomorphisms, higher Picard varieties, and relative algebraic intermediate Jacobians, culminating in an isomorphism between the relevant height line bundles and Lear extensions. A key outcome is a precise comparison between geometric Beilinson–Bloch height pairings and their Néron–Tate/Néron-model counterparts, formalized through Poincaré biextensions and -biextensions, and conditional on Griffiths's conjecture, establishing the desired equivalence of line bundles on the base. The results provide a geometric viewpoint on Beilinson–Bloch height positivity and connect to the positivity results of Brosnan–Pearlstein, with implications for the study of height pairings over one-variable complex function fields.

Abstract

We consider the limiting behaviour of the archimedean height pairing for homologically trivial algebraic cycles in a degenerating one-parameter family of smooth projective complex varieties. We conjecture that the limit is controlled by the non-archimedean geometric height pairing of the cycles on the generic fiber and verify this for algebraically trivial cycles, assuming a conjecture of Griffiths on incidence equivalence. Our work offers a more geometric understanding of a related asymptotic result of Brosnan--Pearlstein and suggests a new perspective on the positivity of the Beilinson--Bloch height pairing over a one-variable complex function field.
Paper Structure (18 sections, 39 theorems, 172 equations)

This paper contains 18 sections, 39 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. Künnemann's theorem
  4. Higher Picard varieties and algebraic intermediate Jacobians
  5. Connection to the positivity of the height pairing
  6. Conventions
  7. Algebraically trivial cycles revisit
  8. Regular homomorphisms and higher Picard varieties
  9. Relative setting
  10. Classical setting
  11. Higher Picard varieties
  12. Relative algebraic intermediate Jacobians
  13. The geometric height pairing à la Kahn
  14. The geometric Néron--Tate height pairing
  15. Comparison of the line bundles
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $\overline{X}/\overline{S}$ be a semistable degeneration of $X/S$ as in standard setup. Let $\Delta\subset \mathbb{C}$ be a (small) analytic neighbourhood around $s_0\in \overline{S}\smallsetminus S$ such that $s_0$ is mapped to $0\in \Delta$; let $t$ be a holomorphic coordinate of $\Delta$. The on $\Delta^*$ extends to a continuous function on $\Delta$.

Theorems & Definitions (118)

  • Theorem 1.3: Brosnan--Pearlstein
  • Remark 1.4
  • Conjecture 1.5
  • Remark 1.7: Silverman's limit theorem
  • Remark 1.8
  • Definition 1.11
  • Theorem 1.12: Brosnan--Pearlstein
  • Conjecture 1.15
  • Theorem A: Theorem \ref{['Main Result, alg. trivial cycles']}
  • Remark 1.16
  • ...and 108 more