Analytic multiplier ideals and L^2 extension theorems

Abstract

In this thesis, multiplier ideal sheaves and L2 extension theorems are main themes. Multiplier ideals and its jumping numbers play an important role in algebraic geometry and complex geometry because of its applications. Jumping numbers are deeply studied by Ein, Lazardfeld, Smith and Varolin [ELSV] in the algebraic setting. We extend the study of jumping numbers of multiplier ideals due to [ELSV] from the algebraic case to the case of general plurisubharmonic functions. While many properties from [ELSV] are shown to generalize to the plurisubharmonic case, important properties such as periodicity and discreteness do not hold any more. Previously only two particular examples with a cluster point (i.e. failure of discreteness) of jumping numbers were known, due to Guan-Li and to [ELSV] respectively. We generalize them to all toric plurisubharmonic functions in dimension 2 by characterizing precisely when cluster points of jumping numbers exist and by computing all those cluster points. This characterization suggests that clustering of jumping numbers is a rather frequent phenomenon. In particular, we obtain uncountably many new such examples.