Birational Geometry and Logarithmic Forms

Abstract

This chapter discusses the birational geometry on non-complete algebraic varieties. However, the reason that any given algebraic variety has a projective model, that is, there exists a projective variety birationally equivalent to the given variety, it is enough to consider complete varieties for the study of birational invariant properties. Such a study is classical birational geometry, which has powerful tools such as regular forms, theory of genera, especially Kodaira dimension. By these, structure of complete algebraic varieties has been studied, in detail. Fortunately, for any non-complete algebraic variety, the spaces of logarithmic forms and the logarithmic genera are defined, which are counterparts (or generalizations) of the spaces of regular forms and the genera for a complete algebraic variety. By using these, a certain kind of birational geometry can be developed, which is proper birational geometry. In proper birational geometry, one uses strictly rational maps, proper birational maps, and logarithmic Kodaira dimension, respectively, in place of rational maps, birational maps, and Kodaira dimension.