Abstract：

The volume of a divisor on an algebraic variety measures the growth rate of the dimension of the space of sections of tensor powers of the associated line bundle. In the case of certain Calabi-Yau N-folds possessing a large group of pseudo-automorphisms, we show that the behavior of the volume can be highly oscillatory as the divisor class approaches the boundary of the pseudo-effective cone. This is explained by relating the volume function to the dynamical behavior of geodesics on certain hyperbolic manifolds.

After providing an introduction and some context for the above notions, I will discuss some of the ideas that go into the proof. (joint work with John Lesieutre and Valentino Tosatti)