Newton-Okounkov bodies and complexity functions

We show that quite universally the holonomicity of the complexity function of a big divisor on a projective variety does not predict the polyhedrality of the Newton-Okounkov body associated to every flag

Local Volumes.

In the present thesis we study a notion of local volume for Cartier divisors on arbitrary blow–ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study a volume for normal isolated singularities, generalizing work of Wahl on surfaces. We also compare this volume of isolated singularities to a different generalization due to Boucksom, de Fernex, and Favre.Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91517/1/mfulger_1.pd