Shin-Yao Jow

Introduction to volume functions on projective varieties

Let X be a projective variety of dimension n and D a Cartier divisor. It is an interesting and classical problem to study the behavior of h^0(mD) as m varies. In particular, if one is interested in its asymptotic behavior, then it is natural to consider the "volume" of D, which is roughly speaking the limit of h^0(mD)/m^n as m tends to infinity. As the properties of the volume were more fully explored, it was later realized that instead of looking at one divisor at a time, one should view volume as a function defined on "the space of all divisors on X" (or more precisely the Neron-Severi space of X). This talk will try to give more details on the above ideas, followed by a survey of some knowns and unknowns about the volume function and its higher cohomological analogues.