Degeneration of the archimedean height pairing of algebraically trivial cycles

TL;DR

This paper investigates the asymptotic behavior of the archimedean height pairing of algebraically trivial cycles in degenerating families, proposing a conjecture linking it to non-archimedean pairings and providing new geometric insights.

Contribution

It conjectures that the limit of the archimedean height pairing is governed by the non-archimedean height pairing, verified for algebraically trivial cycles under Griffiths' incidence conjecture.

Findings

Confirmed the conjecture for algebraically trivial cycles

Connected archimedean and non-archimedean height pairings

Provided geometric interpretation of height pairing asymptotics

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.