Degeneration of the archimedean height pairing of algebraically trivial cycles

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.