Algebraic volumes of divisors

TL;DR

This paper explores the algebraic properties of volumes of Cartier divisors on projective varieties, extending constructions to produce many algebraic volumes and demonstrating that π can also be realized as such a volume.

Contribution

It introduces new methods using real multiplication on abelian varieties to construct a broad class of algebraic volumes, expanding understanding of which numbers can occur as volumes.

Findings

Constructed many algebraic volumes using real multiplication techniques.

Extended previous methods to produce a larger class of algebraic volumes.

Proved that π can be realized as a volume of a Cartier divisor.

Abstract

The volume of a Cartier divisor on a projective variety is a nonnegative real number that measures the asymptotic growth of sections of multiples of the divisor. It is known that the set of these numbers is countable and has the structure of a multiplicative semigroup. At the same time it still remains unknown which nonnegative real algebraic numbers arise as volumes of Cartier divisors on some variety. Here we extend a construction first used by Cutkosky, and use the theory of real multiplication on abelian varieties to obtain a large class of examples of algebraic volumes. We also show that $\pi$ arises as a volume.