Algebraic volumes of divisors

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.

In this talk I want to discuss how to extend a construction first used by Cutkosky and the theory of real multiplication on abelian varieties to obtain a large class of examples of algebraic volumes. If time permits I will also show that $\pi$ arises as a volume. This is a joint work with Carsten Bornträger.