Differentiability of Adelic Volumes and Equidistribution on Quasi-Projective Varieties

Abstract: Differentiability of geometric and arithmetic volumes of Hermitian line-bundles leads to the proof of equidistribution results on projective varieties using the variational principle. In this article, we work in the setting of adelic divisors on quasi-projective varieties recently introduced by Xinyi Yuan and Shou-Wu Zhang to show that their geometric and arithmetic adelic volume functions are differentiable on the big cone. We introduce the notions of positive intersection products and show that the differentials are realised as positive intersection products against integrable divisors at big points. We show an analogue of the Fujita Approximation for restricted volumes of adelic divisors in the geometric setting using a construction similar to those of positive intersections and as an application of our differentiability result we derive a slightly weaker quasi-projective analogue of the equidistribution theorem of Berman and Boucksom which generalises the equidistribution obtained by Yuan and Zhang for arithmetically nef adelic divisors on quasi-projective varieties.