Some Analytic generalizations of the Briancon-Skoda Theorem

Licentiate thesis, 2010

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