Description
The volume is a fundamental birational invariant of a smooth projective variety X. It measures the asymptotic growth of pluricanonical forms and, when X is minimal, it coincides with the top self-intersection number of K_X.
A striking theorem of Hacon–McKernan–Xu shows that the volume of a log canonical variety of general type is uniformly bounded away from zero in fixed dimension. In this talk I will discuss analogous questions for fibrations and foliations. I will present recent results, examples, and open problems arising from joint works with Codogni–Patakfalvi and Spicer.