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Differentiability of Adelic Volumes and Equidistribution on Quasi-Projective Varieties

Debam Biswas

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.

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.
Paper Structure (7 sections, 29 theorems, 109 equations)

This paper contains 7 sections, 29 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Review of Adelic Divisors
  3. Positive Intersection products
  4. Relation with restricted volumes in the geometric case
  5. Fundamental inequality in the Function Field case
  6. Application to Equidistribution
  7. Relation to other Equidistribution Results

Key Result

Theorem 1

Suppose $\overline{D}$ is a big adelic divisor and $\overline{E}$ is an integrable adelic divisor on a normal essentially quasi-projective variety $U$ over $k$ of dimension $d$. Then the function $t\mapsto \widehat{\emph{vol}}(\overline{D}+t\overline{E})$ is differentiable at $t=0$ with derivative g

Theorems & Definitions (75)

  • Theorem : A
  • Theorem : B
  • Theorem : C
  • Definition 3.1
  • Remark
  • Definition 3.2
  • Remark
  • Lemma 3.3
  • proof
  • Definition 3.4
  • ...and 65 more