L2-serre duality on singular complex spaces and applications

Abstract

In this survey, we explain a version of topological L2-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various L2-vanishing theorems for the (Formula presented)-equation on singular spaces. As one application, we prove Hartogs’ extension theorem for (n − 1)-complete spaces. Another application is the characterization of rational singularities. It is shown that complex spaces with rational singularities behave quite tame with respect to some (Formula presented)-equation in the L2-sense. More precisely: a singular point is rational if and only if the appropriate L2-(Formula presented)-complex is exact in this point. So, we obtain an L2-(Formula presented)-resolution of the structure sheaf in rational singular points.