Valuations And Birational Geometry

The main results of this thesis are briefly summarized as follows:1. The notion of quasiplurisubharmonic functions is defined purely algebraically, and this is generalized to a quite general case. The method we use in this paper heavily relies on the techniques of recent valuation theory, which still remains hidden.2. The multiplier ideals of quasiplurisubharmonic functions possess an amount of properties as expected, such as subadditivity.3. The regularization of a quasiplurisubharmonic function is very crucial in complex analysis and geometry. We present an algebraic analogue in this paper:a quasiplurisubharmonic func-tions can be approximated by a decreasing sequence of ideal functions, which correspond to those quasiplurisubharmonic function of the form logΣ|fi|2c in complex analytic settings.4. The Lelong theory of quasiplurisubharmonic function is developed, as well as the computing theory of log canonical thresholds. With the aid of these results, we could explore more structure of the space of quasiplurisubharmonic functions.5. We then study the basic properties of tame functions, in particular those derived from tame valuations. This is expected to describe those valuations which could compute the log canonical threshold of some quasiplurisubharmonic function. The results established here is closely related to the Openness Conjecture.6. We try to define the restriction of a quasiplurisubharmonic function to a regular closed subscheme. We also prove the restriction formulae and the inversion of adjunction.The whole work is highly motivated by the pluricanonical extension problems in complex alge-braic geometry, which is an extremely subtle part of birational classification of higher dimensional algebraic varieties.