Existence and volume-form independence of the arithmetic equilibrium measure

Establish that, for every smooth projective arithmetic variety over Spec(Z) endowed with a smooth volume form and every smooth Hermitian line bundle on it, the limsup defining the upper arithmetic equilibrium invariant is an actual limit and that this limit is independent of the chosen auxiliary volume form.

Background

The paper introduces the upper arithmetic equilibrium invariant by normalizing the arithmetic distortion function associated with the full lattices of global sections of tensor powers of a Hermitian line bundle and taking a limsup of measures as the tensor power tends to infinity. The authors conjecture that this asymptotic measure has a genuine limit rather than merely a limsup.

Independence from the auxiliary smooth volume form would establish that the resulting arithmetic equilibrium measure is intrinsic to the arithmetic variety and Hermitian line bundle, rather than dependent on the volume form used to define the L2 structure. The paper proves the conjecture under the additional assumptions that the Hermitian line bundle is smooth and weakly nef and that its underlying line bundle is ample, while leaving the stated general case unresolved.

References

The robustness of the preceding results motivates the following conjecture. Let $(\mathcal{X},\mu)$ be a smooth projective arithmetic variety over $\mathbb{Z}$ endowed with a smooth volume form $\mu$, and let $\overline{\mathcal{L}\phi}$ be a smooth Hermitian line bundle on $\mathcal{X}$. Then the $\limsup$ in the definition of $\widehat{\mu}{\sup}{\mathrm{eq}}((\mathcal{X},\mu),\overline{\mathcal{L}_\phi})$ is in fact a limit. Moreover, this limit is independent of the choice of the auxiliary volume form $\mu$.

— Arithmetic theta invariants and arithmetic equilibrium measures  (2608.25450 - Hajli, 26 Aug 2026) in Introduction, Conjecture 1 (labelled \ref{conj1})