Some Analytic Generalizations of the Briançon-skoda Theorem

The Briançon-Skoda theorem appears in many variations in recent literature. The common denominator is that the theorem gives a sufficient condition that implies a membership φ ∈ a, where a is an ideal of some ring R. In the analytic interpretation R is the local ring of an analytic space Z, and the condition is that |φ| ≤ C|a| holds on the space Z. The theorem thus relates the rate of vanishing of φ along the locus of a to actual membership of (powers of) the ideal. The smallest integer N that works for all a ⊂ R and all l ≥ 1 simultaneously will be called the Briançon-Skoda number of the ring R. The thesis contains three papers. The first one gives an elementary proof of the original Briançon-Skoda theorem. This case is simply Z = C. The second paper contains an analytic proof of a generalization by Huneke. The result is also sharper when a has few generators if the geometry is not to complicated in a certain sense. Moreover, the method can give upper bounds for the Briançon-Skoda number for some varieties such as for example the cusp z = w. In the third paper non-reduced analytic spaces are considered. In this setting Huneke’s generalization must be modified to remain valid. More precisely, φ belongs to a if one requires that |Lφ| ≤ C|a| holds on Z for a given family of holomorphic differential operators on Z. We impose the assumption that the local ring OZ is Cohen-Macaulay for technical reasons.