On the height of the universal abelian variety
José Ignacio Burgos Gil, Jürg Kramer
TL;DR
The work extends arithmetic intersection theory to adelic divisors that are not nef nor integrable by leveraging finite relative energy and non-pluripolar products. It proves that the adelic line bundle associated to Siegel–Jacobi forms on the universal abelian variety has finite energy and computes its arithmetic self-intersection, establishing the relation $\widehat{\deg}(\overline{\mathscr{J}})=2^{g}\cdot\frac{(d+1)!}{(d'+1)!}\cdot\widehat{\deg}(\overline{\omega})$ and showing the non-integrability of the Hodge bundle in general. The framework unifies analytic and arithmetic tools, extending Yuan–Zhang with relative energy as a robust method for semi-positive yet singular metrics. The results open pathways to applications in mixed Shimura varieties and moduli spaces of stable marked curves, providing new invariants and computational techniques for heights and intersections in more singular settings.
Abstract
In this paper we extend the arithmetic intersection theory of adelic divisors on quasiprojective varieties developed by X. Yuan and S. W. Zhang to cover certain adelic arithmetic divisors that are not nef nor integrable. The key concept used in this extension is the relative finite energy introduced by T. Darvas, E. Di Nezza, and C. H. Lu. As an application, we prove that the line bundle of Siegel--Jacobi forms on the universal abelian variety endowed with its invariant hermitian metric is not integrable but we compute its arithmetic self-intersection number using the new extension. The techniques developed in this paper can be applied in many other situations like mixed Shimura varieties or the moduli space of stable marked curves.