Volume function

The volume of a line bundle on a projective variety, which is the asymptotic
growth of the number of global sections, is an interesting invariant that has been studied
a lot recently. As it turns out it can be extended as a continuous log-concave function
over the whole Neron-Severi space of the variety. In a joint work with A. Kuronya and C. Maclean we try to study this function and answer the basic question of how many volume
functions there are in nature for all smooth provective varieties. At the same time we study
the multigraded case, where the picture is more complicated. If time permits, I will try to
show an example of the volume function which is given by a transcendental formula on a
small open set.