THE GEOGRAPHY OF LOG MODELS AND ITS APPLICATIONS.

Abstract

We use the Log Minimal Model Program (LMMP) to investigate the stratification of the set of R-divisors on an algebraic variety X. We first refine the classical definition of Iitaka dimension of an R-divisor D on a variety. Using the equivalence relation on R-boundary divisors defined by LMMP, we prove that the set of R-boundary divisors on some fixed support is decomposed into finitely many equivalence classes which are open and rational polyhedral. Each class is called a country and such decomposition is called the geography log models. As the first application of geography, we prove that the cone of effective divisors on an FT variety up to R-linear equivalence relation ∼R is rational polyhedral. Secondly, we prove the finiteness theorem for projective wlc models of a given log pair (X/Z,B) with klt singularities such that KX +B is big over Z. Advisor: Professor Vyacheslav Shokurov ii ACKNOWLEDGEMENTS