We introduce a pairing on local intersection cohomology groups of variations of pure Hodge structure, which we call the asymptotic height pairing. Our original application of this pairing was to answer a question on the Ceresa cycle posed by R. Hain and D. Reed. (This question has since been answered independently by Hain.) Here we show that a certain analytic line bundle, called the biextension line bundle, and defined in terms of normal functions, always extends to any smooth partial compactification of the base. We then show that the asymptotic height pairing on intersection cohomology governs the extension of the natural metric on this line bundle studied by Hain and Reed (as well as, more recently, by several other authors). We also prove a positivity property of the asymptotic height pairing, which generalizes the results of a recent preprint of J. Burgos Gil, D. Holmes and R. de Jong, along with a continuity property of the pairing in the normal function case. Moreover, we show that the asymptotic height pairing arises in a natural way from certain Mumford–Grothendieck biextensions associated to normal functions.
Patrick Brosnan. Gregory Pearlstein. "Jumps in the Archimedean height." Duke Math. J. 168 (10) 1737 - 1842, 15 July 2019. https://doi.org/10.1215/00127094-2018-0056